jgraph_4.miz



    begin

    reserve a for Real;

    reserve p,q for Point of ( TOP-REAL 2);

    

     Lm1: p <> ( 0. ( TOP-REAL 2)) implies |.p.| > 0

    proof

      assume p <> ( 0. ( TOP-REAL 2));

      then |.p.| <> 0 by TOPRNS_1: 24;

      hence thesis;

    end;

    theorem :: JGRAPH_4:1

    

     Th1: for X be non empty TopStruct, g be Function of X, R^1 , B be Subset of X, a be Real st g is continuous & B = { p where p be Point of X : (g /. p) > a } holds B is open

    proof

      let X be non empty TopStruct, g be Function of X, R^1 , B be Subset of X, a be Real;

      assume that

       A1: g is continuous and

       A2: B = { p where p be Point of X : (g /. p) > a };

      { r where r be Real : r > a } c= the carrier of R^1

      proof

        let x be object;

        assume x in { r where r be Real : r > a };

        then

        consider r be Real such that

         A3: r = x & r > a;

        r in REAL by XREAL_0:def 1;

        hence thesis by A3, TOPMETR: 17;

      end;

      then

      reconsider D = { r where r be Real : r > a } as Subset of R^1 ;

      

       A4: (g " D) c= B

      proof

        let x be object;

        assume

         A5: x in (g " D);

        then

        reconsider p = x as Point of X;

        (g . x) in D by A5, FUNCT_1:def 7;

        then

         A6: ex r be Real st r = (g . x) & r > a;

        (g /. p) = (g . p);

        hence thesis by A2, A6;

      end;

      

       A7: ( [#] R^1 ) <> {} & D is open by JORDAN2B: 25;

      B c= (g " D)

      proof

        let x be object;

        assume x in B;

        then

        consider p be Point of X such that

         A8: p = x and

         A9: (g /. p) > a by A2;

        ( dom g) = the carrier of X & (g . x) in D by A8, A9, FUNCT_2:def 1;

        hence thesis by A8, FUNCT_1:def 7;

      end;

      then B = (g " D) by A4, XBOOLE_0:def 10;

      hence thesis by A1, A7, TOPS_2: 43;

    end;

    theorem :: JGRAPH_4:2

    

     Th2: for X be non empty TopStruct, g be Function of X, R^1 , B be Subset of X, a be Real st g is continuous & B = { p where p be Point of X : (g /. p) < a } holds B is open

    proof

      let X be non empty TopStruct, g be Function of X, R^1 , B be Subset of X, a be Real;

      assume that

       A1: g is continuous and

       A2: B = { p where p be Point of X : (g /. p) < a };

      { r where r be Real : r < a } c= the carrier of R^1

      proof

        let x be object;

        assume x in { r where r be Real : r < a };

        then

        consider r be Real such that

         A3: r = x & r < a;

        r in REAL by XREAL_0:def 1;

        hence thesis by A3, TOPMETR: 17;

      end;

      then

      reconsider D = { r where r be Real : r < a } as Subset of R^1 ;

      

       A4: (g " D) c= B

      proof

        let x be object;

        assume

         A5: x in (g " D);

        then

        reconsider p = x as Point of X;

        (g . x) in D by A5, FUNCT_1:def 7;

        then

         A6: ex r be Real st r = (g . x) & r < a;

        (g /. p) = (g . p);

        hence thesis by A2, A6;

      end;

      

       A7: ( [#] R^1 ) <> {} & D is open by JORDAN2B: 24;

      B c= (g " D)

      proof

        let x be object;

        assume x in B;

        then

        consider p be Point of X such that

         A8: p = x and

         A9: (g /. p) < a by A2;

        ( dom g) = the carrier of X & (g . x) in D by A8, A9, FUNCT_2:def 1;

        hence thesis by A8, FUNCT_1:def 7;

      end;

      then B = (g " D) by A4, XBOOLE_0:def 10;

      hence thesis by A1, A7, TOPS_2: 43;

    end;

    theorem :: JGRAPH_4:3

    

     Th3: for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = ( [#] ( TOP-REAL 2)) & (for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & (ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2)) holds f is being_homeomorphism

    proof

      let f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume that

       A1: f is continuous one-to-one and

       A2: ( rng f) = ( [#] ( TOP-REAL 2)) and

       A3: for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2;

      reconsider g = (f qua Function " ) as Function of ( TOP-REAL 2), ( TOP-REAL 2) by A1, A2, FUNCT_2: 25;

      

       A4: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      for p be Point of ( TOP-REAL 2), V be Subset of ( TOP-REAL 2) st (g . p) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st p in W & W is open & (g .: W) c= V

      proof

        let p be Point of ( TOP-REAL 2), V be Subset of ( TOP-REAL 2);

        assume that

         A5: (g . p) in V and

         A6: V is open;

        consider K be non empty compact Subset of ( TOP-REAL 2) such that

         A7: K = (f .: K) and

         A8: ex V2 be Subset of ( TOP-REAL 2) st (g . p) in V2 & V2 is open & V2 c= K & (f . (g . p)) in V2 by A3;

        consider V2 be Subset of ( TOP-REAL 2) such that

         A9: (g . p) in V2 and

         A10: V2 is open and

         A11: V2 c= K and

         A12: (f . (g . p)) in V2 by A8;

        

         A13: ( dom (f | K)) = (( dom f) /\ K) by RELAT_1: 61

        .= K by A4, XBOOLE_1: 28;

        

         A14: (g . p) in (V /\ V2) by A5, A9, XBOOLE_0:def 4;

        the carrier of (( TOP-REAL 2) | K) = K by PRE_TOPC: 8;

        then

        reconsider R = ((V /\ V2) /\ K) as Subset of (( TOP-REAL 2) | K) by XBOOLE_1: 17;

        

         A15: R = ((V /\ V2) /\ ( [#] (( TOP-REAL 2) | K))) by PRE_TOPC:def 5;

        (V /\ V2) is open by A6, A10, TOPS_1: 11;

        then

         A16: R is open by A15, TOPS_2: 24;

        

         A17: p in V2 by A1, A2, A12, FUNCT_1: 35;

        then

        reconsider q = p as Point of (( TOP-REAL 2) | K) by A11, PRE_TOPC: 8;

        

         A18: ( rng (f | K)) c= the carrier of ( TOP-REAL 2);

        ( dom (f | K)) = (( dom f) /\ K) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K) by FUNCT_2:def 1

        .= K by XBOOLE_1: 28

        .= the carrier of (( TOP-REAL 2) | K) by PRE_TOPC: 8;

        then

        reconsider h = (f | K) as Function of (( TOP-REAL 2) | K), ( TOP-REAL 2) by A18, FUNCT_2: 2;

        

         A19: h is one-to-one by A1, FUNCT_1: 52;

        

         A20: K = ((f | K) .: K) by A7, RELAT_1: 129

        .= ( rng (f | K)) by A13, RELAT_1: 113;

        then

        consider f1 be Function of (( TOP-REAL 2) | K), (( TOP-REAL 2) | K) such that

         A21: h = f1 and

         A22: f1 is being_homeomorphism by A1, A19, JGRAPH_1: 46, TOPMETR: 7;

        

         A23: ( rng f1) = ( [#] (( TOP-REAL 2) | K)) by A22, TOPS_2:def 5;

        

         A24: f1 is onto by A23, FUNCT_2:def 3;

        ( dom (f1 qua Function " )) = ( rng f1) & ( rng (f1 qua Function " )) = ( dom f1) by A19, A21, FUNCT_1: 33;

        then

        reconsider g1 = (f1 qua Function " ) as Function of (( TOP-REAL 2) | K), (( TOP-REAL 2) | K) by A23, FUNCT_2: 2;

        g1 = (f1 " ) by A19, A21, A24, TOPS_2:def 4;

        then

         A25: g1 is continuous by A22, TOPS_2:def 5;

        

         A26: (f1 . (g . p)) = (f . (g . p)) by A9, A11, A21, FUNCT_1: 49

        .= p by A1, A2, FUNCT_1: 35;

        

         A27: ( dom f1) = (( dom f) /\ K) by A21, RELAT_1: 61

        .= K by A4, XBOOLE_1: 28;

        ( rng f1) = ( dom (f1 qua Function " )) by A19, A21, FUNCT_1: 33;

        then

         A28: ((f1 qua Function " ) . p) in ( rng (f1 qua Function " )) by A11, A17, A20, A21, FUNCT_1: 3;

        

         A29: ( rng (f1 qua Function " )) = ( dom f1) by A19, A21, FUNCT_1: 33;

        (f1 . ((f1 qua Function " ) . p)) = p by A11, A17, A19, A20, A21, FUNCT_1: 35;

        then ((f1 qua Function " ) . p) = (g . p) by A8, A19, A21, A26, A27, A29, A28, FUNCT_1:def 4;

        then ((f1 qua Function " ) . p) in R by A9, A11, A14, XBOOLE_0:def 4;

        then

        consider W3 be Subset of (( TOP-REAL 2) | K) such that

         A30: q in W3 and

         A31: W3 is open and

         A32: ((f1 qua Function " ) .: W3) c= R by A16, A25, JGRAPH_2: 10;

        R = (V /\ (V2 /\ K)) by XBOOLE_1: 16;

        then

         A33: R c= V by XBOOLE_1: 17;

        consider W5 be Subset of ( TOP-REAL 2) such that

         A34: W5 is open and

         A35: W3 = (W5 /\ ( [#] (( TOP-REAL 2) | K))) by A31, TOPS_2: 24;

        reconsider W4 = (W5 /\ V2) as Subset of ( TOP-REAL 2);

        p in W5 by A30, A35, XBOOLE_0:def 4;

        then

         A36: p in W4 by A17, XBOOLE_0:def 4;

        

         A37: ( dom f1) = the carrier of (( TOP-REAL 2) | K) by FUNCT_2:def 1;

        

         A38: ((f qua Function " ) .: W3) c= R

        proof

          let y be object;

          assume y in ((f qua Function " ) .: W3);

          then

          consider x be object such that

           A39: x in ( dom (f qua Function " )) and

           A40: x in W3 and

           A41: y = ((f qua Function " ) . x) by FUNCT_1:def 6;

          

           A42: x in ( rng f) by A1, A39, FUNCT_1: 33;

          then

           A43: y in ( dom f) by A1, A41, FUNCT_1: 32;

          

           A44: (f . y) = x by A1, A41, A42, FUNCT_1: 32;

          the carrier of (( TOP-REAL 2) | K) = K by PRE_TOPC: 8;

          then ex z2 be object st z2 in ( dom f) & z2 in K & (f . y) = (f . z2) by A7, A40, A44, FUNCT_1:def 6;

          then

           A45: y in K by A1, A43, FUNCT_1:def 4;

          then

           A46: y in the carrier of (( TOP-REAL 2) | K) by PRE_TOPC: 8;

          

           A47: ( dom (f1 qua Function " )) = the carrier of (( TOP-REAL 2) | K) by A19, A21, A23, FUNCT_1: 33;

          (f1 . y) = x by A21, A44, A45, FUNCT_1: 49;

          then y = ((f1 qua Function " ) . x) by A19, A21, A37, A46, FUNCT_1: 32;

          then y in ((f1 qua Function " ) .: W3) by A40, A47, FUNCT_1:def 6;

          hence thesis by A32;

        end;

        W4 = (W5 /\ (V2 /\ K)) by A11, XBOOLE_1: 28

        .= ((W5 /\ K) /\ V2) by XBOOLE_1: 16

        .= (W3 /\ V2) by A35, PRE_TOPC:def 5;

        then

         A48: (g .: W4) c= ((g .: W3) /\ (g .: V2)) by RELAT_1: 121;

        ((g .: W3) /\ (g .: V2)) c= (g .: W3) by XBOOLE_1: 17;

        then (g .: W4) c= (g .: W3) by A48;

        then

         A49: (g .: W4) c= R by A38;

        W4 is open by A10, A34, TOPS_1: 11;

        hence thesis by A36, A49, A33, XBOOLE_1: 1;

      end;

      then

       A50: g is continuous by JGRAPH_2: 10;

      f is onto by A2, FUNCT_2:def 3;

      then g = (f " ) by A1, TOPS_2:def 4;

      hence thesis by A1, A2, A4, A50, TOPS_2:def 5;

    end;

    theorem :: JGRAPH_4:4

    

     Th4: for X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real st f1 is continuous & f2 is continuous & b <> 0 & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (((r1 / r2) - a) / b)) & g is continuous

    proof

      let X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real;

      assume that

       A1: f1 is continuous & f2 is continuous and

       A2: b <> 0 and

       A3: for q be Point of X holds (f2 . q) <> 0 ;

      consider g3 be Function of X, R^1 such that

       A4: for p be Point of X, r1,r0 be Real st (f1 . p) = r1 & (f2 . p) = r0 holds (g3 . p) = (r1 / r0) and

       A5: g3 is continuous by A1, A3, JGRAPH_2: 27;

      consider g1 be Function of X, R^1 such that

       A6: for p be Point of X holds (g1 . p) = b & g1 is continuous by JGRAPH_2: 20;

      consider g2 be Function of X, R^1 such that

       A7: for p be Point of X holds (g2 . p) = a & g2 is continuous by JGRAPH_2: 20;

      consider g4 be Function of X, R^1 such that

       A8: for p be Point of X, r1,r0 be Real st (g3 . p) = r1 & (g2 . p) = r0 holds (g4 . p) = (r1 - r0) and

       A9: g4 is continuous by A7, A5, JGRAPH_2: 21;

      for q be Point of X holds (g1 . q) <> 0 by A2, A6;

      then

      consider g5 be Function of X, R^1 such that

       A10: for p be Point of X, r1,r0 be Real st (g4 . p) = r1 & (g1 . p) = r0 holds (g5 . p) = (r1 / r0) and

       A11: g5 is continuous by A6, A9, JGRAPH_2: 27;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g5 . p) = (((r1 / r2) - a) / b)

      proof

        let p be Point of X, r1,r2 be Real;

        set r8 = (r1 / r2);

        

         A12: (g1 . p) = b by A6;

        assume (f1 . p) = r1 & (f2 . p) = r2;

        then

         A13: (g3 . p) = r8 by A4;

        (g2 . p) = a by A7;

        then (g4 . p) = (r8 - a) by A8, A13;

        hence thesis by A10, A12;

      end;

      hence thesis by A11;

    end;

    theorem :: JGRAPH_4:5

    

     Th5: for X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real st f1 is continuous & f2 is continuous & b <> 0 & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r2 * (((r1 / r2) - a) / b))) & g is continuous

    proof

      let X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real;

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous and

       A3: b <> 0 and

       A4: for q be Point of X holds (f2 . q) <> 0 ;

      consider g3 be Function of X, R^1 such that

       A5: for p be Point of X, r1,r0 be Real st (f1 . p) = r1 & (f2 . p) = r0 holds (g3 . p) = (r1 / r0) and

       A6: g3 is continuous by A1, A2, A4, JGRAPH_2: 27;

      consider g1 be Function of X, R^1 such that

       A7: for p be Point of X holds (g1 . p) = b & g1 is continuous by JGRAPH_2: 20;

      consider g2 be Function of X, R^1 such that

       A8: for p be Point of X holds (g2 . p) = a & g2 is continuous by JGRAPH_2: 20;

      consider g4 be Function of X, R^1 such that

       A9: for p be Point of X, r1,r0 be Real st (g3 . p) = r1 & (g2 . p) = r0 holds (g4 . p) = (r1 - r0) and

       A10: g4 is continuous by A8, A6, JGRAPH_2: 21;

      for q be Point of X holds (g1 . q) <> 0 by A3, A7;

      then

      consider g5 be Function of X, R^1 such that

       A11: for p be Point of X, r1,r0 be Real st (g4 . p) = r1 & (g1 . p) = r0 holds (g5 . p) = (r1 / r0) and

       A12: g5 is continuous by A7, A10, JGRAPH_2: 27;

      consider g6 be Function of X, R^1 such that

       A13: for p be Point of X, r1,r0 be Real st (f2 . p) = r1 & (g5 . p) = r0 holds (g6 . p) = (r1 * r0) and

       A14: g6 is continuous by A2, A12, JGRAPH_2: 25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g6 . p) = (r2 * (((r1 / r2) - a) / b))

      proof

        let p be Point of X, r1,r2 be Real;

        assume that

         A15: (f1 . p) = r1 and

         A16: (f2 . p) = r2;

        

         A17: (g2 . p) = a by A8;

        set r8 = (r1 / r2);

        

         A18: (g1 . p) = b by A7;

        (g3 . p) = r8 by A5, A15, A16;

        then (g4 . p) = (r8 - a) by A9, A17;

        then (g5 . p) = (((r1 / r2) - a) / b) by A11, A18;

        hence thesis by A13, A16;

      end;

      hence thesis by A14;

    end;

    theorem :: JGRAPH_4:6

    

     Th6: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (r1 ^2 )) & g is continuous

    proof

      let X be non empty TopSpace, f1 be Function of X, R^1 ;

      assume f1 is continuous;

      then

      consider g1 be Function of X, R^1 such that

       A1: for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 * r1) and

       A2: g1 is continuous by JGRAPH_2: 22;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 ^2 ) by A1;

      hence thesis by A2;

    end;

    theorem :: JGRAPH_4:7

    

     Th7: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = |.r1.|) & g is continuous

    proof

      let X be non empty TopSpace, f1 be Function of X, R^1 ;

      assume f1 is continuous;

      then

      consider g1 be Function of X, R^1 such that

       A1: for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 ^2 ) and

       A2: g1 is continuous by Th6;

      for q be Point of X holds ex r be Real st (g1 . q) = r & r >= 0

      proof

        let q be Point of X;

        reconsider r11 = (f1 . q) as Real;

        (g1 . q) = (r11 ^2 ) by A1;

        hence thesis by XREAL_1: 63;

      end;

      then

      consider g2 be Function of X, R^1 such that

       A3: for p be Point of X, r1 be Real st (g1 . p) = r1 holds (g2 . p) = ( sqrt r1) and

       A4: g2 is continuous by A2, JGRAPH_3: 5;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g2 . p) = |.r1.|

      proof

        let p be Point of X, r1 be Real;

        assume (f1 . p) = r1;

        then (g1 . p) = (r1 ^2 ) by A1;

        

        then (g2 . p) = ( sqrt (r1 ^2 )) by A3

        .= |.r1.| by COMPLEX1: 72;

        hence thesis;

      end;

      hence thesis by A4;

    end;

    theorem :: JGRAPH_4:8

    

     Th8: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = ( - r1)) & g is continuous

    proof

      let X be non empty TopSpace, f1 be Function of X, R^1 ;

      consider g1 be Function of X, R^1 such that

       A1: for p be Point of X holds (g1 . p) = 0 and

       A2: g1 is continuous by JGRAPH_2: 20;

      assume f1 is continuous;

      then

      consider g2 be Function of X, R^1 such that

       A3: for p be Point of X, r1,r2 be Real st (g1 . p) = r1 & (f1 . p) = r2 holds (g2 . p) = (r1 - r2) and

       A4: g2 is continuous by A2, JGRAPH_2: 21;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g2 . p) = ( - r1)

      proof

        let p be Point of X, r1 be Real;

        assume

         A5: (f1 . p) = r1;

        (g1 . p) = 0 by A1;

        then (g2 . p) = ( 0 - r1) by A3, A5;

        hence thesis;

      end;

      hence thesis by A4;

    end;

    theorem :: JGRAPH_4:9

    

     Th9: for X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real st f1 is continuous & f2 is continuous & b <> 0 & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)))) & g is continuous

    proof

      let X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real;

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous and

       A3: b <> 0 & for q be Point of X holds (f2 . q) <> 0 ;

      consider g1 be Function of X, R^1 such that

       A4: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g1 . p) = (((r1 / r2) - a) / b) and

       A5: g1 is continuous by A1, A2, A3, Th4;

      consider g2 be Function of X, R^1 such that

       A6: for p be Point of X, s be Real st (g1 . p) = s holds (g2 . p) = (s ^2 ) and

       A7: g2 is continuous by A5, Th6;

      consider g0 be Function of X, R^1 such that

       A8: for p be Point of X holds (g0 . p) = 1 and

       A9: g0 is continuous by JGRAPH_2: 20;

      consider g3 be Function of X, R^1 such that

       A10: for p be Point of X, s,t be Real st (g0 . p) = s & (g2 . p) = t holds (g3 . p) = (s - t) and

       A11: g3 is continuous by A7, A9, JGRAPH_2: 21;

      consider g4 be Function of X, R^1 such that

       A12: for p be Point of X, s be Real st (g3 . p) = s holds (g4 . p) = |.s.| and

       A13: g4 is continuous by A11, Th7;

      for q be Point of X holds ex r be Real st (g4 . q) = r & r >= 0

      proof

        let q be Point of X;

        reconsider s = (g3 . q) as Real;

        (g4 . q) = |.s.| by A12;

        hence thesis by COMPLEX1: 46;

      end;

      then

      consider g5 be Function of X, R^1 such that

       A14: for p be Point of X, s be Real st (g4 . p) = s holds (g5 . p) = ( sqrt s) and

       A15: g5 is continuous by A13, JGRAPH_3: 5;

      consider g6 be Function of X, R^1 such that

       A16: for p be Point of X, s be Real st (g5 . p) = s holds (g6 . p) = ( - s) and

       A17: g6 is continuous by A15, Th8;

      consider g7 be Function of X, R^1 such that

       A18: for p be Point of X, r1,r0 be Real st (f2 . p) = r1 & (g6 . p) = r0 holds (g7 . p) = (r1 * r0) and

       A19: g7 is continuous by A2, A17, JGRAPH_2: 25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g7 . p) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)))

      proof

        let p be Point of X, r1,r2 be Real;

        assume that

         A20: (f1 . p) = r1 and

         A21: (f2 . p) = r2;

        

         A22: (g0 . p) = 1 by A8;

        (g1 . p) = (((r1 / r2) - a) / b) by A4, A20, A21;

        then (g2 . p) = ((((r1 / r2) - a) / b) ^2 ) by A6;

        then (g3 . p) = (1 - ((((r1 / r2) - a) / b) ^2 )) by A10, A22;

        then (g4 . p) = |.(1 - ((((r1 / r2) - a) / b) ^2 )).| by A12;

        then (g5 . p) = ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|) by A14;

        then (g6 . p) = ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)) by A16;

        hence thesis by A18, A21;

      end;

      hence thesis by A19;

    end;

    theorem :: JGRAPH_4:10

    

     Th10: for X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real st f1 is continuous & f2 is continuous & b <> 0 & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))) & g is continuous

    proof

      let X be non empty TopSpace, f1,f2 be Function of X, R^1 , a,b be Real;

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous and

       A3: b <> 0 & for q be Point of X holds (f2 . q) <> 0 ;

      consider g1 be Function of X, R^1 such that

       A4: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g1 . p) = (((r1 / r2) - a) / b) and

       A5: g1 is continuous by A1, A2, A3, Th4;

      consider g2 be Function of X, R^1 such that

       A6: for p be Point of X, s be Real st (g1 . p) = s holds (g2 . p) = (s ^2 ) and

       A7: g2 is continuous by A5, Th6;

      consider g0 be Function of X, R^1 such that

       A8: for p be Point of X holds (g0 . p) = 1 and

       A9: g0 is continuous by JGRAPH_2: 20;

      consider g3 be Function of X, R^1 such that

       A10: for p be Point of X, s,t be Real st (g0 . p) = s & (g2 . p) = t holds (g3 . p) = (s - t) and

       A11: g3 is continuous by A7, A9, JGRAPH_2: 21;

      consider g4 be Function of X, R^1 such that

       A12: for p be Point of X, s be Real st (g3 . p) = s holds (g4 . p) = |.s.| and

       A13: g4 is continuous by A11, Th7;

      for q be Point of X holds ex r be Real st (g4 . q) = r & r >= 0

      proof

        let q be Point of X;

        reconsider s = (g3 . q) as Real;

        (g4 . q) = |.s.| by A12;

        hence thesis by COMPLEX1: 46;

      end;

      then

      consider g5 be Function of X, R^1 such that

       A14: for p be Point of X, s be Real st (g4 . p) = s holds (g5 . p) = ( sqrt s) and

       A15: g5 is continuous by A13, JGRAPH_3: 5;

      consider g7 be Function of X, R^1 such that

       A16: for p be Point of X, r1,r0 be Real st (f2 . p) = r1 & (g5 . p) = r0 holds (g7 . p) = (r1 * r0) and

       A17: g7 is continuous by A2, A15, JGRAPH_2: 25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g7 . p) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))

      proof

        let p be Point of X, r1,r2 be Real;

        assume that

         A18: (f1 . p) = r1 and

         A19: (f2 . p) = r2;

        

         A20: (g0 . p) = 1 by A8;

        (g1 . p) = (((r1 / r2) - a) / b) by A4, A18, A19;

        then (g2 . p) = ((((r1 / r2) - a) / b) ^2 ) by A6;

        then (g3 . p) = (1 - ((((r1 / r2) - a) / b) ^2 )) by A10, A20;

        then (g4 . p) = |.(1 - ((((r1 / r2) - a) / b) ^2 )).| by A12;

        then (g5 . p) = ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|) by A14;

        hence thesis by A16, A19;

      end;

      hence thesis by A17;

    end;

    definition

      let n be Nat;

      deffunc F( Point of ( TOP-REAL n)) = |.$1.|;

      :: JGRAPH_4:def1

      func n NormF -> Function of ( TOP-REAL n), R^1 means

      : Def1: for q be Point of ( TOP-REAL n) holds (it . q) = |.q.|;

      existence

      proof

        

         A1: for x be Element of ( TOP-REAL n) holds F(x) in the carrier of R^1 by TOPMETR: 17, XREAL_0:def 1;

        thus ex IT be Function of ( TOP-REAL n), R^1 st for q be Point of ( TOP-REAL n) holds (IT . q) = F(q) from FUNCT_2:sch 8( A1);

      end;

      uniqueness

      proof

        thus for f,g be Function of ( TOP-REAL n), R^1 st (for q be Point of ( TOP-REAL n) holds (f . q) = F(q)) & (for q be Point of ( TOP-REAL n) holds (g . q) = F(q)) holds f = g from BINOP_2:sch 1;

      end;

    end

    theorem :: JGRAPH_4:11

    for n be Nat holds ( dom (n NormF )) = the carrier of ( TOP-REAL n) & ( dom (n NormF )) = ( REAL n)

    proof

      let n be Nat;

      thus ( dom (n NormF )) = the carrier of ( TOP-REAL n) by FUNCT_2:def 1;

      hence thesis by EUCLID: 22;

    end;

    theorem :: JGRAPH_4:12

    

     Th12: for n be Nat holds (n NormF ) is continuous

    proof

      let n be Nat;

      for q be Point of ( TOP-REAL n) holds ((n NormF ) . q) = |.q.| by Def1;

      hence thesis by JORDAN2C: 83;

    end;

    registration

      let n be Nat;

      cluster (n NormF ) -> continuous;

      coherence by Th12;

    end

    theorem :: JGRAPH_4:13

    

     Th13: for n be Element of NAT , K0 be Subset of ( TOP-REAL n), f be Function of (( TOP-REAL n) | K0), R^1 st (for p be Point of (( TOP-REAL n) | K0) holds (f . p) = ((n NormF ) . p)) holds f is continuous

    proof

      let n be Element of NAT , K0 be Subset of ( TOP-REAL n), f be Function of (( TOP-REAL n) | K0), R^1 ;

      

       A1: (the carrier of ( TOP-REAL n) /\ K0) = K0 by XBOOLE_1: 28;

      reconsider g = (n NormF ) as Function of ( TOP-REAL n), R^1 ;

      assume for p be Point of (( TOP-REAL n) | K0) holds (f . p) = ((n NormF ) . p);

      then

       A2: for x be object st x in ( dom f) holds (f . x) = ((n NormF ) . x);

      ( dom f) = the carrier of (( TOP-REAL n) | K0) & the carrier of (( TOP-REAL n) | K0) = K0 by FUNCT_2:def 1, PRE_TOPC: 8;

      then ( dom f) = (( dom (n NormF )) /\ K0) by A1, FUNCT_2:def 1;

      then f = (g | K0) by A2, FUNCT_1: 46;

      hence thesis by TOPMETR: 7;

    end;

    theorem :: JGRAPH_4:14

    

     Th14: for n be Element of NAT , p be Point of ( Euclid n), r be Real, B be Subset of ( TOP-REAL n) st B = ( cl_Ball (p,r)) holds B is bounded closed

    proof

      let n be Element of NAT , p be Point of ( Euclid n), r be Real, B be Subset of ( TOP-REAL n);

      assume

       A1: B = ( cl_Ball (p,r));

      ( cl_Ball (p,r)) c= ( Ball (p,(r + 1)))

      proof

        let x be object;

        

         A2: r < (r + 1) by XREAL_1: 29;

        assume

         A3: x in ( cl_Ball (p,r));

        then

        reconsider q = x as Point of ( Euclid n);

        ( dist (p,q)) <= r by A3, METRIC_1: 12;

        then ( dist (p,q)) < (r + 1) by A2, XXREAL_0: 2;

        hence thesis by METRIC_1: 11;

      end;

      then ( cl_Ball (p,r)) is bounded by TBSP_1: 14;

      hence B is bounded by A1, JORDAN2C: 11;

      

       A4: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider BB = B as Subset of ( TopSpaceMetr ( Euclid n));

      BB is closed by A1, TOPREAL6: 57;

      hence thesis by A4, PRE_TOPC: 31;

    end;

    theorem :: JGRAPH_4:15

    

     Th15: for p be Point of ( Euclid 2), r be Real, B be Subset of ( TOP-REAL 2) st B = ( cl_Ball (p,r)) holds B is compact

    proof

      let p be Point of ( Euclid 2), r be Real, B be Subset of ( TOP-REAL 2);

      assume B = ( cl_Ball (p,r));

      then B is bounded closed by Th14;

      hence thesis by TOPREAL6: 79;

    end;

    begin

    definition

      let s be Real, q be Point of ( TOP-REAL 2);

      :: JGRAPH_4:def2

      func FanW (s,q) -> Point of ( TOP-REAL 2) equals

      : Def2: ( |.q.| * |[( - ( sqrt (1 - (((((q `2 ) / |.q.|) - s) / (1 - s)) ^2 )))), ((((q `2 ) / |.q.|) - s) / (1 - s))]|) if ((q `2 ) / |.q.|) >= s & (q `1 ) < 0 ,

( |.q.| * |[( - ( sqrt (1 - (((((q `2 ) / |.q.|) - s) / (1 + s)) ^2 )))), ((((q `2 ) / |.q.|) - s) / (1 + s))]|) if ((q `2 ) / |.q.|) < s & (q `1 ) < 0

      otherwise q;

      correctness ;

    end

    definition

      let s be Real;

      :: JGRAPH_4:def3

      func s -FanMorphW -> Function of ( TOP-REAL 2), ( TOP-REAL 2) means

      : Def3: for q be Point of ( TOP-REAL 2) holds (it . q) = ( FanW (s,q));

      existence

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanW (s,$1));

        thus ex IT be Function of ( TOP-REAL 2), ( TOP-REAL 2) st for q be Point of ( TOP-REAL 2) holds (IT . q) = F(q) from FUNCT_2:sch 4;

      end;

      uniqueness

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanW (s,$1));

        thus for f,g be Function of ( TOP-REAL 2), ( TOP-REAL 2) st (for q be Point of ( TOP-REAL 2) holds (f . q) = F(q)) & (for q be Point of ( TOP-REAL 2) holds (g . q) = F(q)) holds f = g from BINOP_2:sch 1;

      end;

    end

    theorem :: JGRAPH_4:16

    

     Th16: for sn be Real holds (((q `2 ) / |.q.|) >= sn & (q `1 ) < 0 implies ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|) & ((q `1 ) >= 0 implies ((sn -FanMorphW ) . q) = q)

    proof

      let sn be Real;

      hereby

        assume ((q `2 ) / |.q.|) >= sn & (q `1 ) < 0 ;

        

        then ( FanW (sn,q)) = ( |.q.| * |[( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ((((q `2 ) / |.q.|) - sn) / (1 - sn))]|) by Def2

        .= |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by EUCLID: 58;

        hence ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by Def3;

      end;

      assume

       A1: (q `1 ) >= 0 ;

      ((sn -FanMorphW ) . q) = ( FanW (sn,q)) by Def3;

      hence thesis by A1, Def2;

    end;

    theorem :: JGRAPH_4:17

    

     Th17: for sn be Real st ((q `2 ) / |.q.|) <= sn & (q `1 ) < 0 holds ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|

    proof

      let sn be Real;

      assume that

       A1: ((q `2 ) / |.q.|) <= sn and

       A2: (q `1 ) < 0 ;

      per cases by A1, XXREAL_0: 1;

        suppose ((q `2 ) / |.q.|) < sn;

        

        then ( FanW (sn,q)) = ( |.q.| * |[( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ((((q `2 ) / |.q.|) - sn) / (1 + sn))]|) by A2, Def2

        .= |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by EUCLID: 58;

        hence thesis by Def3;

      end;

        suppose

         A3: ((q `2 ) / |.q.|) = sn;

        then ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = 0 ;

        hence thesis by A2, A3, Th16;

      end;

    end;

    theorem :: JGRAPH_4:18

    

     Th18: for sn be Real st ( - 1) < sn & sn < 1 holds (((q `2 ) / |.q.|) >= sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|) & (((q `2 ) / |.q.|) <= sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|)

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      per cases ;

        suppose

         A3: ((q `2 ) / |.q.|) >= sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose

           A4: (q `1 ) < 0 ;

          

          then ( FanW (sn,q)) = ( |.q.| * |[( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ((((q `2 ) / |.q.|) - sn) / (1 - sn))]|) by A3, Def2

          .= |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by EUCLID: 58;

          hence thesis by A4, Def3, Th17;

        end;

          suppose

           A5: (q `1 ) >= 0 ;

          then

           A6: ((sn -FanMorphW ) . q) = q by Th16;

          

           A7: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

          

           A8: (1 - sn) > 0 by A2, XREAL_1: 149;

          

           A9: (q `1 ) = 0 by A3, A5;

           |.q.| <> 0 by A3, TOPRNS_1: 24;

          then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          then (((q `2 ) ^2 ) / ( |.q.| ^2 )) = (1 ^2 ) by A7, A9, XCMPLX_1: 60;

          then (((q `2 ) / |.q.|) ^2 ) = (1 ^2 ) by XCMPLX_1: 76;

          then

           A10: ( sqrt (((q `2 ) / |.q.|) ^2 )) = 1 by SQUARE_1: 22;

           A11:

          now

            assume (q `2 ) < 0 ;

            then ( - ((q `2 ) / |.q.|)) = 1 by A10, SQUARE_1: 23;

            hence contradiction by A1, A3;

          end;

          ( sqrt ( |.q.| ^2 )) = |.q.| by SQUARE_1: 22;

          then

           A12: |.q.| = (q `2 ) by A7, A9, A11, SQUARE_1: 22;

          then 1 = ((q `2 ) / |.q.|) by A3, TOPRNS_1: 24, XCMPLX_1: 60;

          then ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = 1 by A8, XCMPLX_1: 60;

          hence thesis by A2, A6, A9, A12, EUCLID: 53, SQUARE_1: 17, TOPRNS_1: 24, XCMPLX_1: 60;

        end;

      end;

        suppose

         A13: ((q `2 ) / |.q.|) <= sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose (q `1 ) < 0 ;

          hence thesis by Th16, Th17;

        end;

          suppose

           A14: (q `1 ) >= 0 ;

          

           A15: (1 + sn) > 0 by A1, XREAL_1: 148;

          

           A16: |.q.| <> 0 by A13, TOPRNS_1: 24;

          

           A17: (q `1 ) = 0 by A13, A14;

           |.q.| > 0 & 1 > ((q `2 ) / |.q.|) by A2, A13, Lm1, XXREAL_0: 2;

          then (1 * |.q.|) > (((q `2 ) / |.q.|) * |.q.|) by XREAL_1: 68;

          then

           A18: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & |.q.| > (q `2 ) by A13, JGRAPH_3: 1, TOPRNS_1: 24, XCMPLX_1: 87;

          then

           A19: (q `2 ) = ( - |.q.|) by A17, SQUARE_1: 40;

          then ( - 1) = ((q `2 ) / |.q.|) by A13, TOPRNS_1: 24, XCMPLX_1: 197;

          

          then

           A20: ((((q `2 ) / |.q.|) - sn) / (1 + sn)) = (( - (1 + sn)) / (1 + sn))

          .= ( - 1) by A15, XCMPLX_1: 197;

           |.q.| = ( - (q `2 )) by A17, A18, SQUARE_1: 40;

          then |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| = q by A17, A20, EUCLID: 53, SQUARE_1: 17;

          hence thesis by A1, A14, A16, A19, Th16, XCMPLX_1: 197;

        end;

      end;

        suppose (q `1 ) > 0 or q = ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

    end;

    

     Lm2: for K be non empty Subset of ( TOP-REAL 2) holds ( proj1 | K) is continuous Function of (( TOP-REAL 2) | K), R^1 & for q be Point of (( TOP-REAL 2) | K) holds (( proj1 | K) . q) = ( proj1 . q)

    proof

      let K be non empty Subset of ( TOP-REAL 2);

      reconsider g2 = ( proj1 | K) as Function of (( TOP-REAL 2) | K), R^1 by TOPMETR: 17;

      

       A1: the carrier of (( TOP-REAL 2) | K) = K by PRE_TOPC: 8;

      for q be Point of (( TOP-REAL 2) | K) holds (g2 . q) = ( proj1 . q)

      proof

        let q be Point of (( TOP-REAL 2) | K);

        q in the carrier of (( TOP-REAL 2) | K) & ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj1 ) /\ K) by A1, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      hence thesis by JGRAPH_2: 29;

    end;

    

     Lm3: for K be non empty Subset of ( TOP-REAL 2) holds ( proj2 | K) is continuous Function of (( TOP-REAL 2) | K), R^1 & for q be Point of (( TOP-REAL 2) | K) holds (( proj2 | K) . q) = ( proj2 . q)

    proof

      let K be non empty Subset of ( TOP-REAL 2);

      reconsider g2 = ( proj2 | K) as Function of (( TOP-REAL 2) | K), R^1 by TOPMETR: 17;

      

       A1: the carrier of (( TOP-REAL 2) | K) = K by PRE_TOPC: 8;

      for q be Point of (( TOP-REAL 2) | K) holds (g2 . q) = ( proj2 . q)

      proof

        let q be Point of (( TOP-REAL 2) | K);

        q in the carrier of (( TOP-REAL 2) | K) & ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj2 ) /\ K) by A1, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      hence thesis by JGRAPH_2: 30;

    end;

    

     Lm4: ( dom (2 NormF )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

    

     Lm5: for K be non empty Subset of ( TOP-REAL 2) holds ((2 NormF ) | K) is continuous Function of (( TOP-REAL 2) | K), R^1 & for q be Point of (( TOP-REAL 2) | K) holds (((2 NormF ) | K) . q) = ((2 NormF ) . q)

    proof

      let K1 be non empty Subset of ( TOP-REAL 2);

      

       A1: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then

      reconsider g1 = ((2 NormF ) | K1) as Function of (( TOP-REAL 2) | K1), R^1 by FUNCT_2: 32;

      for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) = ((2 NormF ) . q)

      proof

        let q be Point of (( TOP-REAL 2) | K1);

        q in the carrier of (( TOP-REAL 2) | K1);

        then q in (( dom (2 NormF )) /\ K1) by A1, Lm4, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      hence thesis by Th13;

    end;

    

     Lm6: for K1 be non empty Subset of ( TOP-REAL 2), g1 be Function of (( TOP-REAL 2) | K1), R^1 st g1 = ((2 NormF ) | K1) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2))) holds for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0

    proof

      let K1 be non empty Subset of ( TOP-REAL 2), g1 be Function of (( TOP-REAL 2) | K1), R^1 ;

      assume that

       A1: g1 = ((2 NormF ) | K1) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2));

      let q be Point of (( TOP-REAL 2) | K1);

      the carrier of (( TOP-REAL 2) | K1) = K1 & q in the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider q2 = q as Point of ( TOP-REAL 2);

      (g1 . q) = ((2 NormF ) . q) by A1, Lm5

      .= |.q2.| by Def1;

      hence thesis by A2, TOPRNS_1: 24;

    end;

    theorem :: JGRAPH_4:19

    

     Th19: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st sn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 - sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: sn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in K1 by A7, A8, A9, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        

         A11: (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        (f . r) = ( |.r.| * ((((r `2 ) / |.r.|) - sn) / (1 - sn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:20

    

     Th20: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < sn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 + sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: ( - 1) < sn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      (1 + sn) > 0 by A1, XREAL_1: 148;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A8: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in ( dom g3) by A7, A9;

        then x in K1 by A7, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        

         A11: (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        (f . r) = ( |.r.| * ((((r `2 ) / |.r.|) - sn) / (1 + sn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      ( dom f) = ( dom g3) by A7, FUNCT_2:def 1;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:21

    

     Th21: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st sn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 - sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: sn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))) and

       A6: g3 is continuous by A4, Th9;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        

         A9: (1 - sn) > 0 by A1, XREAL_1: 149;

        assume

         A10: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A11: |.r.| <> 0 by A3, A10, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `2 ) - |.r.|) * ((r `2 ) + |.r.|)) = ( - ((r `1 ) ^2 ));

        ((r `1 ) ^2 ) >= 0 by XREAL_1: 63;

        then (r `2 ) <= |.r.| by A12, XREAL_1: 93;

        then ((r `2 ) / |.r.|) <= ( |.r.| / |.r.|) by XREAL_1: 72;

        then ((r `2 ) / |.r.|) <= 1 by A11, XCMPLX_1: 60;

        then

         A13: (((r `2 ) / |.r.|) - sn) <= (1 - sn) by XREAL_1: 9;

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A10;

         A14:

        now

          assume ((1 - sn) ^2 ) = 0 ;

          then ((1 - sn) + sn) = ( 0 + sn) by XCMPLX_1: 6;

          hence contradiction by A1;

        end;

        (sn - ((r `2 ) / |.r.|)) <= 0 by A3, A10, XREAL_1: 47;

        then ( - (sn - ((r `2 ) / |.r.|))) >= ( - (1 - sn)) by A9, XREAL_1: 24;

        then ((1 - sn) ^2 ) >= 0 & ((((r `2 ) / |.r.|) - sn) ^2 ) <= ((1 - sn) ^2 ) by A13, SQUARE_1: 49, XREAL_1: 63;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 - sn) ^2 )) <= (((1 - sn) ^2 ) / ((1 - sn) ^2 )) by XREAL_1: 72;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 - sn) ^2 )) <= 1 by A14, XCMPLX_1: 60;

        then (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )).| = (1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )) by ABSVALUE:def 1;

        then

         A15: (f . r) = ( |.r.| * ( - ( sqrt |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )).|))) by A2, A10;

        

         A16: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        hence thesis by A5, A15, A16;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:22

    

     Th22: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < sn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 + sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: ( - 1) < sn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2));

      

       A4: (1 + sn) > 0 by A1, XREAL_1: 148;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))) and

       A6: g3 is continuous by A4, Th9;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A9;

        

         A10: ((1 + sn) ^2 ) > 0 by A4, SQUARE_1: 12;

        

         A11: |.r.| <> 0 by A3, A9, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `2 ) - |.r.|) * ((r `2 ) + |.r.|)) = ( - ((r `1 ) ^2 ));

        ((r `1 ) ^2 ) >= 0 by XREAL_1: 63;

        then ( - |.r.|) <= (r `2 ) by A12, XREAL_1: 93;

        then ((r `2 ) / |.r.|) >= (( - |.r.|) / |.r.|) by XREAL_1: 72;

        then ((r `2 ) / |.r.|) >= ( - 1) by A11, XCMPLX_1: 197;

        then (((r `2 ) / |.r.|) - sn) >= (( - 1) - sn) by XREAL_1: 9;

        then

         A13: (((r `2 ) / |.r.|) - sn) >= ( - (1 + sn));

        (sn - ((r `2 ) / |.r.|)) >= 0 by A3, A9, XREAL_1: 48;

        then ( - (sn - ((r `2 ) / |.r.|))) <= ( - 0 );

        then ((((r `2 ) / |.r.|) - sn) ^2 ) <= ((1 + sn) ^2 ) by A4, A13, SQUARE_1: 49;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 + sn) ^2 )) <= (((1 + sn) ^2 ) / ((1 + sn) ^2 )) by A4, XREAL_1: 72;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 + sn) ^2 )) <= 1 by A10, XCMPLX_1: 60;

        then (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )).| = (1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )) by ABSVALUE:def 1;

        then

         A14: (f . r) = ( |.r.| * ( - ( sqrt |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )).|))) by A2, A9;

        

         A15: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        hence thesis by A5, A14, A15;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:23

    

     Th23: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[( - cn), sn]|;

      

       A1: (p0 `1 ) = ( - cn) by EUCLID: 52;

      (p0 `2 ) = sn by EUCLID: 52;

      

      then

       A2: |.p0.| = ( sqrt ((( - cn) ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1

      .= ( sqrt ((cn ^2 ) + (sn ^2 )));

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - ( - cn)) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - cn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A7: p0 in K0 by A3, A1, A6, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A8: ( rng ( proj1 * ((sn -FanMorphW ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A9: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) / |.p8.|) >= sn & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A10: ( dom ((sn -FanMorphW ) | K1)) c= ( dom ( proj2 * ((sn -FanMorphW ) | K1)))

      proof

        let x be object;

        assume

         A11: x in ( dom ((sn -FanMorphW ) | K1));

        then x in (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphW )) by XBOOLE_0:def 4;

        then

         A12: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphW ) . x) in ( rng (sn -FanMorphW )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphW ) | K1) . x) = ((sn -FanMorphW ) . x) by A11, FUNCT_1: 47;

        hence thesis by A11, A12, FUNCT_1: 11;

      end;

      

       A13: ( rng ( proj2 * ((sn -FanMorphW ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj2 * ((sn -FanMorphW ) | K1))) c= ( dom ((sn -FanMorphW ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((sn -FanMorphW ) | K1))) = ( dom ((sn -FanMorphW ) | K1)) by A10, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj2 * ((sn -FanMorphW ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A13, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A14: ( dom ((sn -FanMorphW ) | K1)) = (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A15: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A16: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A15;

        then

         A17: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A3, Th18;

        (((sn -FanMorphW ) | K1) . p) = ((sn -FanMorphW ) . p) by A16, A15, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|) by A16, A14, A15, A17, FUNCT_1: 13

        .= ( |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A18: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)));

      

       A19: ( dom ((sn -FanMorphW ) | K1)) c= ( dom ( proj1 * ((sn -FanMorphW ) | K1)))

      proof

        let x be object;

        assume

         A20: x in ( dom ((sn -FanMorphW ) | K1));

        then x in (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphW )) by XBOOLE_0:def 4;

        then

         A21: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphW ) . x) in ( rng (sn -FanMorphW )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphW ) | K1) . x) = ((sn -FanMorphW ) . x) by A20, FUNCT_1: 47;

        hence thesis by A20, A21, FUNCT_1: 11;

      end;

      ( dom ( proj1 * ((sn -FanMorphW ) | K1))) c= ( dom ((sn -FanMorphW ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((sn -FanMorphW ) | K1))) = ( dom ((sn -FanMorphW ) | K1)) by A19, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj1 * ((sn -FanMorphW ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A8, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A22: ( dom ((sn -FanMorphW ) | K1)) = (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A23: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A24: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A23;

        then

         A25: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A3, Th18;

        (((sn -FanMorphW ) | K1) . p) = ((sn -FanMorphW ) . p) by A24, A23, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|) by A24, A22, A23, A25, FUNCT_1: 13

        .= ( |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A26: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A27: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A27;

        hence thesis;

      end;

      then

       A28: f1 is continuous by A3, A26, Th21;

      

       A29: for x,y,r,s be Real st |[x, y]| in K1 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that

         A30: |[x, y]| in K1 and

         A31: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A32: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A30;

        

         A33: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A34: (f1 . p99) = ( |.p99.| * ( - ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)) ^2 ))))) by A26, A30;

        (((sn -FanMorphW ) | K0) . |[x, y]|) = ((sn -FanMorphW ) . |[x, y]|) by A30, FUNCT_1: 49

        .= |[( |.p99.| * ( - ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)) ^2 ))))), ( |.p99.| * ((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)))]| by A3, A32, Th18

        .= |[r, s]| by A18, A30, A31, A33, A34;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A35: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A35;

        hence thesis;

      end;

      then f2 is continuous by A3, A18, Th19;

      hence thesis by A7, A9, A28, A29, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:24

    

     Th24: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[( - cn), sn]|;

      

       A1: (p0 `1 ) = ( - cn) by EUCLID: 52;

      (p0 `2 ) = sn by EUCLID: 52;

      

      then

       A2: |.p0.| = ( sqrt ((( - cn) ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1

      .= ( sqrt ((cn ^2 ) + (sn ^2 )));

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - ( - cn)) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - cn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A7: p0 in K0 by A3, A1, A6, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A8: ( rng ( proj1 * ((sn -FanMorphW ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A9: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) / |.p8.|) <= sn & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A10: ( dom ((sn -FanMorphW ) | K1)) c= ( dom ( proj2 * ((sn -FanMorphW ) | K1)))

      proof

        let x be object;

        assume

         A11: x in ( dom ((sn -FanMorphW ) | K1));

        then x in (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphW )) by XBOOLE_0:def 4;

        then

         A12: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphW ) . x) in ( rng (sn -FanMorphW )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphW ) | K1) . x) = ((sn -FanMorphW ) . x) by A11, FUNCT_1: 47;

        hence thesis by A11, A12, FUNCT_1: 11;

      end;

      

       A13: ( rng ( proj2 * ((sn -FanMorphW ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj2 * ((sn -FanMorphW ) | K1))) c= ( dom ((sn -FanMorphW ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((sn -FanMorphW ) | K1))) = ( dom ((sn -FanMorphW ) | K1)) by A10, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj2 * ((sn -FanMorphW ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A13, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A14: ( dom ((sn -FanMorphW ) | K1)) = (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A15: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A16: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A15;

        then

         A17: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A3, Th18;

        (((sn -FanMorphW ) | K1) . p) = ((sn -FanMorphW ) . p) by A16, A15, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|) by A16, A14, A15, A17, FUNCT_1: 13

        .= ( |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A18: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)));

      

       A19: ( dom ((sn -FanMorphW ) | K1)) c= ( dom ( proj1 * ((sn -FanMorphW ) | K1)))

      proof

        let x be object;

        assume

         A20: x in ( dom ((sn -FanMorphW ) | K1));

        then x in (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphW )) by XBOOLE_0:def 4;

        then

         A21: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphW ) . x) in ( rng (sn -FanMorphW )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphW ) | K1) . x) = ((sn -FanMorphW ) . x) by A20, FUNCT_1: 47;

        hence thesis by A20, A21, FUNCT_1: 11;

      end;

      ( dom ( proj1 * ((sn -FanMorphW ) | K1))) c= ( dom ((sn -FanMorphW ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((sn -FanMorphW ) | K1))) = ( dom ((sn -FanMorphW ) | K1)) by A19, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj1 * ((sn -FanMorphW ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A8, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A22: ( dom ((sn -FanMorphW ) | K1)) = (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A23: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A24: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A23;

        then

         A25: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A3, Th18;

        (((sn -FanMorphW ) | K1) . p) = ((sn -FanMorphW ) . p) by A24, A23, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|) by A24, A22, A23, A25, FUNCT_1: 13

        .= ( |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A26: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A27: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A27;

        hence thesis;

      end;

      then

       A28: f1 is continuous by A3, A26, Th22;

      

       A29: for x,y,r,s be Real st |[x, y]| in K1 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that

         A30: |[x, y]| in K1 and

         A31: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A32: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A30;

        

         A33: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A34: (f1 . p99) = ( |.p99.| * ( - ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)) ^2 ))))) by A26, A30;

        (((sn -FanMorphW ) | K0) . |[x, y]|) = ((sn -FanMorphW ) . |[x, y]|) by A30, FUNCT_1: 49

        .= |[( |.p99.| * ( - ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)) ^2 ))))), ( |.p99.| * ((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)))]| by A3, A32, Th18

        .= |[r, s]| by A18, A30, A31, A33, A34;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A35: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A35;

        hence thesis;

      end;

      then f2 is continuous by A3, A18, Th20;

      hence thesis by A7, A9, A28, A29, JGRAPH_2: 35;

    end;

    

     Lm7: for sn be Real, K1 be Subset of ( TOP-REAL 2) st K1 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) >= (sn * |.p7.|) } holds K1 is closed

    proof

      set K10 = ( [#] ( TOP-REAL 2));

      reconsider g0 = ((2 NormF ) | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm5;

      reconsider g1 = ( proj2 | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm3;

      let sn be Real, K1 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) >= (sn * |.$1.|));

      consider g2 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A1: for q be Point of (( TOP-REAL 2) | K10), r1 be Real st (g0 . q) = r1 holds (g2 . q) = (sn * r1) and

       A2: g2 is continuous by JGRAPH_2: 23;

      consider g3 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A3: for q be Point of (( TOP-REAL 2) | K10), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 - r2) and

       A4: g3 is continuous by A2, JGRAPH_2: 21;

      

       A5: (( TOP-REAL 2) | K10) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then

      reconsider g = g3 as Function of ( TOP-REAL 2), R^1 ;

      reconsider K2 = K1 as Subset of the TopStruct of ( TOP-REAL 2);

      assume K1 = { p : (p `2 ) >= (sn * |.p.|) };

      then

       A6: K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] };

      

       A7: (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] } from JGRAPH_2:sch 2( A6);

      

       A8: for p be Point of ( TOP-REAL 2) holds (g3 . p) = ((sn * |.p.|) - (p `2 ))

      proof

        let p be Point of ( TOP-REAL 2);

        (g0 . p) = ((2 NormF ) . p) by A5, Lm5

        .= |.p.| by Def1;

        then

         A9: (g2 . p) = (sn * |.p.|) by A1, A5;

        (g1 . p) = ( proj2 . p) by A5, Lm3

        .= (p `2 ) by PSCOMP_1:def 6;

        hence thesis by A3, A5, A9;

      end;

      

       A10: (K1 ` ) c= { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 }

      proof

        let x be object;

        assume x in (K1 ` );

        then

        consider p9 be Point of ( TOP-REAL 2) such that

         A11: x = p9 and

         A12: (p9 `2 ) < (sn * |.p9.|) by A7;

        

         A13: (g /. p9) = ((sn * |.p9.|) - (p9 `2 )) by A8;

        ((sn * |.p9.|) - (p9 `2 )) > 0 by A12, XREAL_1: 50;

        hence thesis by A11, A13;

      end;

      { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 } c= (K1 ` )

      proof

        let x be object;

        assume x in { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 };

        then

        consider p7 be Point of ( TOP-REAL 2) such that

         A14: p7 = x and

         A15: (g /. p7) > 0 ;

        (g /. p7) = ((sn * |.p7.|) - (p7 `2 )) by A8;

        then (((sn * |.p7.|) - (p7 `2 )) + (p7 `2 )) > ( 0 + (p7 `2 )) by A15, XREAL_1: 8;

        hence thesis by A7, A14;

      end;

      then (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 } by A10, XBOOLE_0:def 10;

      then (K2 ` ) is open by A4, A5, Th1;

      then (K1 ` ) is open by PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    

     Lm8: for sn be Real, K1 be Subset of ( TOP-REAL 2) st K1 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) >= (sn * |.p7.|) } holds K1 is closed

    proof

      set K10 = ( [#] ( TOP-REAL 2));

      reconsider g0 = ((2 NormF ) | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm5;

      reconsider g1 = ( proj1 | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm2;

      let sn be Real, K1 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) >= (sn * |.$1.|));

      consider g2 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A1: for q be Point of (( TOP-REAL 2) | K10), r1 be Real st (g0 . q) = r1 holds (g2 . q) = (sn * r1) and

       A2: g2 is continuous by JGRAPH_2: 23;

      consider g3 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A3: for q be Point of (( TOP-REAL 2) | K10), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 - r2) and

       A4: g3 is continuous by A2, JGRAPH_2: 21;

      

       A5: (( TOP-REAL 2) | K10) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then

      reconsider g = g3 as Function of ( TOP-REAL 2), R^1 ;

      reconsider K2 = K1 as Subset of the TopStruct of ( TOP-REAL 2);

      assume K1 = { p : (p `1 ) >= (sn * |.p.|) };

      then

       A6: K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] };

      

       A7: (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] } from JGRAPH_2:sch 2( A6);

      

       A8: for p be Point of ( TOP-REAL 2) holds (g3 . p) = ((sn * |.p.|) - (p `1 ))

      proof

        let p be Point of ( TOP-REAL 2);

        (g0 . p) = ((2 NormF ) . p) by A5, Lm5

        .= |.p.| by Def1;

        then

         A9: (g2 . p) = (sn * |.p.|) by A1, A5;

        (g1 . p) = ( proj1 . p) by A5, Lm2

        .= (p `1 ) by PSCOMP_1:def 5;

        hence thesis by A3, A5, A9;

      end;

      

       A10: (K1 ` ) c= { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 }

      proof

        let x be object;

        assume x in (K1 ` );

        then

        consider p9 be Point of ( TOP-REAL 2) such that

         A11: x = p9 and

         A12: (p9 `1 ) < (sn * |.p9.|) by A7;

        

         A13: (g /. p9) = ((sn * |.p9.|) - (p9 `1 )) by A8;

        ((sn * |.p9.|) - (p9 `1 )) > 0 by A12, XREAL_1: 50;

        hence thesis by A11, A13;

      end;

      { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 } c= (K1 ` )

      proof

        let x be object;

        assume x in { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 };

        then

        consider p7 be Point of ( TOP-REAL 2) such that

         A14: p7 = x and

         A15: (g /. p7) > 0 ;

        (g /. p7) = ((sn * |.p7.|) - (p7 `1 )) by A8;

        then (((sn * |.p7.|) - (p7 `1 )) + (p7 `1 )) > ( 0 + (p7 `1 )) by A15, XREAL_1: 8;

        hence thesis by A7, A14;

      end;

      then (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) > 0 } by A10, XBOOLE_0:def 10;

      then (K2 ` ) is open by A4, A5, Th1;

      then (K1 ` ) is open by PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    theorem :: JGRAPH_4:25

    

     Th25: for sn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) <= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) >= (sn * |.$1.|));

      assume

       A1: K003 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) <= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm7, JORDAN6: 5;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    

     Lm9: for sn be Real, K1 be Subset of ( TOP-REAL 2) st K1 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (sn * |.p7.|) } holds K1 is closed

    proof

      set K10 = ( [#] ( TOP-REAL 2));

      reconsider g0 = ((2 NormF ) | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm5;

      reconsider g1 = ( proj2 | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm3;

      let sn be Real, K1 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) <= (sn * |.$1.|));

      consider g2 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A1: for q be Point of (( TOP-REAL 2) | K10), r1 be Real st (g0 . q) = r1 holds (g2 . q) = (sn * r1) and

       A2: g2 is continuous by JGRAPH_2: 23;

      consider g3 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A3: for q be Point of (( TOP-REAL 2) | K10), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 - r2) and

       A4: g3 is continuous by A2, JGRAPH_2: 21;

      

       A5: (( TOP-REAL 2) | K10) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then

      reconsider g = g3 as Function of ( TOP-REAL 2), R^1 ;

      reconsider K2 = K1 as Subset of the TopStruct of ( TOP-REAL 2);

      assume K1 = { p : (p `2 ) <= (sn * |.p.|) };

      then

       A6: K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] };

      

       A7: (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] } from JGRAPH_2:sch 2( A6);

      

       A8: for p be Point of ( TOP-REAL 2) holds (g3 . p) = ((sn * |.p.|) - (p `2 ))

      proof

        let p be Point of ( TOP-REAL 2);

        (g0 . p) = ((2 NormF ) . p) by A5, Lm5

        .= |.p.| by Def1;

        then

         A9: (g2 . p) = (sn * |.p.|) by A1, A5;

        (g1 . p) = ( proj2 . p) by A5, Lm3

        .= (p `2 ) by PSCOMP_1:def 6;

        hence thesis by A3, A5, A9;

      end;

      

       A10: (K1 ` ) c= { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 }

      proof

        let x be object;

        assume x in (K1 ` );

        then

        consider p9 be Point of ( TOP-REAL 2) such that

         A11: x = p9 and

         A12: (p9 `2 ) > (sn * |.p9.|) by A7;

        

         A13: (g /. p9) = ((sn * |.p9.|) - (p9 `2 )) by A8;

        ((sn * |.p9.|) - (p9 `2 )) < 0 by A12, XREAL_1: 49;

        hence thesis by A11, A13;

      end;

      { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 } c= (K1 ` )

      proof

        let x be object;

        assume x in { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 };

        then

        consider p7 be Point of ( TOP-REAL 2) such that

         A14: p7 = x and

         A15: (g /. p7) < 0 ;

        (g /. p7) = ((sn * |.p7.|) - (p7 `2 )) by A8;

        then (((sn * |.p7.|) - (p7 `2 )) + (p7 `2 )) < ( 0 + (p7 `2 )) by A15, XREAL_1: 8;

        hence thesis by A7, A14;

      end;

      then (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 } by A10, XBOOLE_0:def 10;

      then (K2 ` ) is open by A4, A5, Th2;

      then (K1 ` ) is open by PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    

     Lm10: for sn be Real, K1 be Subset of ( TOP-REAL 2) st K1 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (sn * |.p7.|) } holds K1 is closed

    proof

      set K10 = ( [#] ( TOP-REAL 2));

      reconsider g0 = ((2 NormF ) | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm5;

      reconsider g1 = ( proj1 | K10) as continuous Function of (( TOP-REAL 2) | K10), R^1 by Lm2;

      let sn be Real, K1 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) <= (sn * |.$1.|));

      consider g2 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A1: for q be Point of (( TOP-REAL 2) | K10), r1 be Real st (g0 . q) = r1 holds (g2 . q) = (sn * r1) and

       A2: g2 is continuous by JGRAPH_2: 23;

      consider g3 be Function of (( TOP-REAL 2) | K10), R^1 such that

       A3: for q be Point of (( TOP-REAL 2) | K10), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 - r2) and

       A4: g3 is continuous by A2, JGRAPH_2: 21;

      

       A5: (( TOP-REAL 2) | K10) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then

      reconsider g = g3 as Function of ( TOP-REAL 2), R^1 ;

      reconsider K2 = K1 as Subset of the TopStruct of ( TOP-REAL 2);

      assume K1 = { p : (p `1 ) <= (sn * |.p.|) };

      then

       A6: K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] };

      

       A7: (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] } from JGRAPH_2:sch 2( A6);

      

       A8: for p be Point of ( TOP-REAL 2) holds (g3 . p) = ((sn * |.p.|) - (p `1 ))

      proof

        let p be Point of ( TOP-REAL 2);

        (g0 . p) = ((2 NormF ) . p) by A5, Lm5

        .= |.p.| by Def1;

        then

         A9: (g2 . p) = (sn * |.p.|) by A1, A5;

        (g1 . p) = ( proj1 . p) by A5, Lm2

        .= (p `1 ) by PSCOMP_1:def 5;

        hence thesis by A3, A5, A9;

      end;

      

       A10: (K1 ` ) c= { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 }

      proof

        let x be object;

        assume x in (K1 ` );

        then

        consider p9 be Point of ( TOP-REAL 2) such that

         A11: x = p9 and

         A12: (p9 `1 ) > (sn * |.p9.|) by A7;

        

         A13: (g /. p9) = ((sn * |.p9.|) - (p9 `1 )) by A8;

        ((sn * |.p9.|) - (p9 `1 )) < 0 by A12, XREAL_1: 49;

        hence thesis by A11, A13;

      end;

      { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 } c= (K1 ` )

      proof

        let x be object;

        assume x in { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 };

        then

        consider p7 be Point of ( TOP-REAL 2) such that

         A14: p7 = x and

         A15: (g /. p7) < 0 ;

        (g /. p7) = ((sn * |.p7.|) - (p7 `1 )) by A8;

        then (((sn * |.p7.|) - (p7 `1 )) + (p7 `1 )) < ( 0 + (p7 `1 )) by A15, XREAL_1: 8;

        hence thesis by A7, A14;

      end;

      then (K1 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : (g /. p7) < 0 } by A10, XBOOLE_0:def 10;

      then (K2 ` ) is open by A4, A5, Th2;

      then (K1 ` ) is open by PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    theorem :: JGRAPH_4:26

    

     Th26: for sn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `2 ) <= (sn * |.p.|) & (p `1 ) <= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) <= (sn * |.$1.|));

      assume

       A1: K003 = { p : P[p] & Q[p] };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm9, JORDAN6: 5;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:27

    

     Th27: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[( - cn), sn]|;

      

       A1: (p0 `1 ) = ( - cn) by EUCLID: 52;

      (p0 `2 ) = sn by EUCLID: 52;

      

      then

       A2: |.p0.| = ( sqrt ((( - cn) ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1

      .= ( sqrt ((cn ^2 ) + (sn ^2 )));

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - ( - cn)) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - cn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      then p0 in K0 by A3, A1, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then

       A7: ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A8: p0 in { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5;

       not p0 in {( 0. ( TOP-REAL 2))} by A6, TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A3, XBOOLE_0:def 5;

      K1 c= D

      proof

        let x be object;

        assume

         A9: x in K1;

        then ex p6 be Point of ( TOP-REAL 2) st p6 = x & (p6 `1 ) <= 0 & p6 <> ( 0. ( TOP-REAL 2)) by A3;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A3, A9, XBOOLE_0:def 5;

      end;

      then D = (K1 \/ D) by XBOOLE_1: 12;

      then

       A10: (( TOP-REAL 2) | K1) is SubSpace of (( TOP-REAL 2) | D) by TOPMETR: 4;

      

       A11: { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      

       A12: { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      then

      reconsider K00 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A8, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A13: ( rng (f | K00)) c= D;

      p0 in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5, A7;

      then

      reconsider K11 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A11, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A14: ( rng (f | K11)) c= D;

      the carrier of (( TOP-REAL 2) | B0) = the carrier of (( TOP-REAL 2) | D);

      

      then

       A15: ( dom f) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1

      .= K1 by PRE_TOPC: 8;

      

      then ( dom (f | K00)) = K00 by A12, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K00) by PRE_TOPC: 8;

      then

      reconsider f1 = (f | K00) as Function of ((( TOP-REAL 2) | K1) | K00), (( TOP-REAL 2) | D) by A13, FUNCT_2: 2;

      ( dom (f | K11)) = K11 by A11, A15, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K11) by PRE_TOPC: 8;

      then

      reconsider f2 = (f | K11) as Function of ((( TOP-REAL 2) | K1) | K11), (( TOP-REAL 2) | D) by A14, FUNCT_2: 2;

      

       A16: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) / |.$1.|) >= sn & ($1 `1 ) <= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A17: ( dom f2) = the carrier of ((( TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1

      .= K11 by PRE_TOPC: 8;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K001 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A8;

      

       A18: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= (sn * |.$1.|) & ($1 `1 ) <= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K003 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) <= 0 } as Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) / |.$1.|) <= sn & ($1 `1 ) <= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A19: { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      

       A20: ( rng ((sn -FanMorphW ) | K001)) c= K1

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphW ) | K001));

        then

        consider x be object such that

         A21: x in ( dom ((sn -FanMorphW ) | K001)) and

         A22: y = (((sn -FanMorphW ) | K001) . x) by FUNCT_1:def 3;

        x in ( dom (sn -FanMorphW )) by A21, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A23: y = ((sn -FanMorphW ) . q) by A21, A22, FUNCT_1: 47;

        ( dom ((sn -FanMorphW ) | K001)) = (( dom (sn -FanMorphW )) /\ K001) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K001) by FUNCT_2:def 1

        .= K001 by XBOOLE_1: 28;

        then

         A24: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `2 ) / |.p2.|) >= sn & (p2 `1 ) <= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A21;

        then

         A25: (((q `2 ) / |.q.|) - sn) >= 0 by XREAL_1: 48;

         |.q.| <> 0 by A24, TOPRNS_1: 24;

        then

         A26: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

        

         A27: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

        

         A28: (1 - sn) > 0 by A3, XREAL_1: 149;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then ((q `2 ) ^2 ) <= ( |.q.| ^2 ) by JGRAPH_3: 1;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A26, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

        then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A28, XREAL_1: 72;

        then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A28, XCMPLX_1: 197;

        then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A28, A25, SQUARE_1: 49;

        then

         A29: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

        then

         A30: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

        ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by A29, SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

        then

         A31: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A32: (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

        

        then

         A33: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A30, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A27, A33;

        then

         A34: q4 <> ( 0. ( TOP-REAL 2)) by A26, TOPRNS_1: 23;

        ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A3, A24, Th18;

        hence thesis by A3, A23, A32, A31, A34;

      end;

      

       A35: ( dom (sn -FanMorphW )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((sn -FanMorphW ) | K001)) = K001 by RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K001) by PRE_TOPC: 8;

      then

      reconsider f3 = ((sn -FanMorphW ) | K001) as Function of (( TOP-REAL 2) | K001), (( TOP-REAL 2) | K1) by A18, A20, FUNCT_2: 2;

      

       A36: K003 is closed by Th25;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= (sn * |.$1.|) & ($1 `1 ) <= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K004 = { p : (p `2 ) <= (sn * |.p.|) & (p `1 ) <= 0 } as Subset of ( TOP-REAL 2);

      

       A37: (K004 /\ K1) c= K11

      proof

        let x be object;

        assume

         A38: x in (K004 /\ K1);

        then x in K004 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A39: q1 = x and

         A40: (q1 `2 ) <= (sn * |.q1.|) and (q1 `1 ) <= 0 ;

        x in K1 by A38, XBOOLE_0:def 4;

        then

         A41: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `1 ) <= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `2 ) / |.q1.|) <= ((sn * |.q1.|) / |.q1.|) by A40, XREAL_1: 72;

        then ((q1 `2 ) / |.q1.|) <= sn by A39, A41, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A39, A41;

      end;

      

       A42: K004 is closed by Th26;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K00) = (( TOP-REAL 2) | K001) & f1 = f3 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A43: f1 is continuous by A3, A10, Th23, PRE_TOPC: 26;

      

       A44: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

      p0 in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5, A7;

      then

      reconsider K111 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A19;

      

       A45: ( rng ((sn -FanMorphW ) | K111)) c= K1

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphW ) | K111));

        then

        consider x be object such that

         A46: x in ( dom ((sn -FanMorphW ) | K111)) and

         A47: y = (((sn -FanMorphW ) | K111) . x) by FUNCT_1:def 3;

        x in ( dom (sn -FanMorphW )) by A46, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A48: y = ((sn -FanMorphW ) . q) by A46, A47, FUNCT_1: 47;

        ( dom ((sn -FanMorphW ) | K111)) = (( dom (sn -FanMorphW )) /\ K111) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K111) by FUNCT_2:def 1

        .= K111 by XBOOLE_1: 28;

        then

         A49: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `2 ) / |.p2.|) <= sn & (p2 `1 ) <= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A46;

        then

         A50: (((q `2 ) / |.q.|) - sn) <= 0 by XREAL_1: 47;

         |.q.| <> 0 by A49, TOPRNS_1: 24;

        then

         A51: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

        

         A52: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

        

         A53: (1 + sn) > 0 by A3, XREAL_1: 148;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A51, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        then (( - (1 + sn)) / (1 + sn)) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A53, XREAL_1: 72;

        then ( - 1) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A53, XCMPLX_1: 197;

        then

         A54: (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) <= (1 ^2 ) by A53, A50, SQUARE_1: 49;

        then

         A55: (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

        (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by A54, XREAL_1: 48;

        then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XCMPLX_1: 187;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

        then

         A56: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A57: (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

        

        then

         A58: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A55, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A52, A58;

        then

         A59: q4 <> ( 0. ( TOP-REAL 2)) by A51, TOPRNS_1: 23;

        ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A3, A49, Th18;

        hence thesis by A3, A48, A57, A56, A59;

      end;

      ( dom ((sn -FanMorphW ) | K111)) = K111 by A35, RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K111) by PRE_TOPC: 8;

      then

      reconsider f4 = ((sn -FanMorphW ) | K111) as Function of (( TOP-REAL 2) | K111), (( TOP-REAL 2) | K1) by A16, A45, FUNCT_2: 2;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K11) = (( TOP-REAL 2) | K111) & f2 = f4 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A60: f2 is continuous by A3, A10, Th24, PRE_TOPC: 26;

      set T1 = ((( TOP-REAL 2) | K1) | K00), T2 = ((( TOP-REAL 2) | K1) | K11);

      

       A61: ( [#] ((( TOP-REAL 2) | K1) | K11)) = K11 by PRE_TOPC:def 5;

      K11 c= (K004 /\ K1)

      proof

        let x be object;

        assume x in K11;

        then

        consider p such that

         A62: p = x and

         A63: ((p `2 ) / |.p.|) <= sn and

         A64: (p `1 ) <= 0 and

         A65: p <> ( 0. ( TOP-REAL 2));

        (((p `2 ) / |.p.|) * |.p.|) <= (sn * |.p.|) by A63, XREAL_1: 64;

        then (p `2 ) <= (sn * |.p.|) by A65, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A66: x in K004 by A62, A64;

        x in K1 by A3, A62, A64, A65;

        hence thesis by A66, XBOOLE_0:def 4;

      end;

      then K11 = (K004 /\ ( [#] (( TOP-REAL 2) | K1))) by A44, A37, XBOOLE_0:def 10;

      then

       A67: K11 is closed by A42, PRE_TOPC: 13;

      

       A68: (K003 /\ K1) c= K00

      proof

        let x be object;

        assume

         A69: x in (K003 /\ K1);

        then x in K003 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A70: q1 = x and

         A71: (q1 `2 ) >= (sn * |.q1.|) and (q1 `1 ) <= 0 ;

        x in K1 by A69, XBOOLE_0:def 4;

        then

         A72: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `1 ) <= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `2 ) / |.q1.|) >= ((sn * |.q1.|) / |.q1.|) by A71, XREAL_1: 72;

        then ((q1 `2 ) / |.q1.|) >= sn by A70, A72, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A70, A72;

      end;

      

       A73: the carrier of (( TOP-REAL 2) | K1) = K0 by PRE_TOPC: 8;

      

       A74: D <> {} ;

      

       A75: ( [#] ((( TOP-REAL 2) | K1) | K00)) = K00 by PRE_TOPC:def 5;

      

       A76: for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f1 . p) = (f2 . p)

      proof

        let p be object;

        assume

         A77: p in (( [#] T1) /\ ( [#] T2));

        then p in K00 by A75, XBOOLE_0:def 4;

        

        hence (f1 . p) = (f . p) by FUNCT_1: 49

        .= (f2 . p) by A61, A77, FUNCT_1: 49;

      end;

      K00 c= (K003 /\ K1)

      proof

        let x be object;

        assume x in K00;

        then

        consider p such that

         A78: p = x and

         A79: ((p `2 ) / |.p.|) >= sn and

         A80: (p `1 ) <= 0 and

         A81: p <> ( 0. ( TOP-REAL 2));

        (((p `2 ) / |.p.|) * |.p.|) >= (sn * |.p.|) by A79, XREAL_1: 64;

        then (p `2 ) >= (sn * |.p.|) by A81, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A82: x in K003 by A78, A80;

        x in K1 by A3, A78, A80, A81;

        hence thesis by A82, XBOOLE_0:def 4;

      end;

      then K00 = (K003 /\ ( [#] (( TOP-REAL 2) | K1))) by A44, A68, XBOOLE_0:def 10;

      then

       A83: K00 is closed by A36, PRE_TOPC: 13;

      

       A84: K1 c= (K00 \/ K11)

      proof

        let x be object;

        assume x in K1;

        then

        consider p such that

         A85: p = x & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) by A3;

        per cases ;

          suppose ((p `2 ) / |.p.|) >= sn;

          then x in K00 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

          suppose ((p `2 ) / |.p.|) < sn;

          then x in K11 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

      end;

      then (( [#] ((( TOP-REAL 2) | K1) | K00)) \/ ( [#] ((( TOP-REAL 2) | K1) | K11))) = ( [#] (( TOP-REAL 2) | K1)) by A75, A61, A44, XBOOLE_0:def 10;

      then

      consider h be Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D) such that

       A86: h = (f1 +* f2) and

       A87: h is continuous by A75, A61, A83, A67, A43, A60, A76, JGRAPH_2: 1;

      

       A88: ( dom h) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A89: ( dom f1) = the carrier of ((( TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1

      .= K00 by PRE_TOPC: 8;

      

       A90: for y be object st y in ( dom h) holds (h . y) = (f . y)

      proof

        let y be object;

        assume

         A91: y in ( dom h);

        now

          per cases by A84, A88, A73, A91, XBOOLE_0:def 3;

            case

             A92: y in K00 & not y in K11;

            then y in (( dom f1) \/ ( dom f2)) by A89, XBOOLE_0:def 3;

            

            hence (h . y) = (f1 . y) by A17, A86, A92, FUNCT_4:def 1

            .= (f . y) by A92, FUNCT_1: 49;

          end;

            case

             A93: y in K11;

            then y in (( dom f1) \/ ( dom f2)) by A17, XBOOLE_0:def 3;

            

            hence (h . y) = (f2 . y) by A17, A86, A93, FUNCT_4:def 1

            .= (f . y) by A93, FUNCT_1: 49;

          end;

        end;

        hence thesis;

      end;

      K0 = the carrier of (( TOP-REAL 2) | K0) by PRE_TOPC: 8

      .= ( dom f) by A74, FUNCT_2:def 1;

      hence thesis by A87, A88, A90, FUNCT_1: 2, PRE_TOPC: 8;

    end;

    theorem :: JGRAPH_4:28

    

     Th28: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[cn, ( - sn)]|;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A2: (p0 `1 ) = cn & (1 - (sn ^2 )) > 0 by EUCLID: 52, XREAL_1: 50;

      then p0 <> ( 0. ( TOP-REAL 2)) by JGRAPH_2: 3, SQUARE_1: 25;

      then not p0 in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A1, XBOOLE_0:def 5;

      (p0 `1 ) > 0 by A2, SQUARE_1: 25;

      then p0 in K0 by A1, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A3: K1 c= D

      proof

        let x be object;

        assume x in K1;

        then

        consider p2 be Point of ( TOP-REAL 2) such that

         A4: p2 = x and (p2 `1 ) >= 0 and

         A5: p2 <> ( 0. ( TOP-REAL 2)) by A1;

         not p2 in {( 0. ( TOP-REAL 2))} by A5, TARSKI:def 1;

        hence thesis by A1, A4, XBOOLE_0:def 5;

      end;

      for p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D) st (f . p) in V & V is open holds ex W be Subset of (( TOP-REAL 2) | K1) st p in W & W is open & (f .: W) c= V

      proof

        let p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D);

        assume that

         A6: (f . p) in V and

         A7: V is open;

        consider V2 be Subset of ( TOP-REAL 2) such that

         A8: V2 is open and

         A9: (V2 /\ ( [#] (( TOP-REAL 2) | D))) = V by A7, TOPS_2: 24;

        reconsider W2 = (V2 /\ ( [#] (( TOP-REAL 2) | K1))) as Subset of (( TOP-REAL 2) | K1);

        

         A10: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

        then

         A11: (f . p) = ((sn -FanMorphW ) . p) by A1, FUNCT_1: 49;

        

         A12: (f .: W2) c= V

        proof

          let y be object;

          assume y in (f .: W2);

          then

          consider x be object such that

           A13: x in ( dom f) and

           A14: x in W2 and

           A15: y = (f . x) by FUNCT_1:def 6;

          f is Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D);

          then ( dom f) = K1 by A10, FUNCT_2:def 1;

          then

          consider p4 be Point of ( TOP-REAL 2) such that

           A16: x = p4 and

           A17: (p4 `1 ) >= 0 and p4 <> ( 0. ( TOP-REAL 2)) by A1, A13;

          

           A18: p4 in V2 by A14, A16, XBOOLE_0:def 4;

          p4 in ( [#] (( TOP-REAL 2) | K1)) by A13, A16;

          then p4 in D by A3, A10;

          then

           A19: p4 in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

          (f . p4) = ((sn -FanMorphW ) . p4) by A1, A10, A13, A16, FUNCT_1: 49

          .= p4 by A17, Th16;

          hence thesis by A9, A15, A16, A18, A19, XBOOLE_0:def 4;

        end;

        p in the carrier of (( TOP-REAL 2) | K1);

        then

        consider q be Point of ( TOP-REAL 2) such that

         A20: q = p and

         A21: (q `1 ) >= 0 and q <> ( 0. ( TOP-REAL 2)) by A1, A10;

        ((sn -FanMorphW ) . q) = q by A21, Th16;

        then p in V2 by A6, A9, A11, A20, XBOOLE_0:def 4;

        then

         A22: p in W2 by XBOOLE_0:def 4;

        W2 is open by A8, TOPS_2: 24;

        hence thesis by A22, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JGRAPH_4:29

    

     Th29: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0) st B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds K0 is closed

    proof

      set J0 = ( NonZero ( TOP-REAL 2));

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 ;

      set I1 = { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      let B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0);

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A1: I1 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ J0) from JGRAPH_3:sch 2;

      assume B0 = J0 & K0 = I1;

      then K1 is closed & K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, JORDAN6: 5, PRE_TOPC:def 5;

      hence thesis by PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_4:30

    

     Th30: for sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 & $1 <> ( 0. ( TOP-REAL 2));

      let sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      reconsider K1 = { p : P[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by A1, PRE_TOPC: 7;

      hence thesis by A1, Th27;

    end;

    theorem :: JGRAPH_4:31

    

     Th31: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0) st B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds K0 is closed

    proof

      set J0 = ( NonZero ( TOP-REAL 2));

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= 0 ;

      set I1 = { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      let B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0);

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A1: I1 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ J0) from JGRAPH_3:sch 2;

      assume B0 = J0 & K0 = I1;

      then K1 is closed & K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, JORDAN6: 4, PRE_TOPC:def 5;

      hence thesis by PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_4:32

    

     Th32: for sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphW ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th28;

    end;

    theorem :: JGRAPH_4:33

    

     Th33: for sn be Real, p be Point of ( TOP-REAL 2) holds |.((sn -FanMorphW ) . p).| = |.p.|

    proof

      let sn be Real, p be Point of ( TOP-REAL 2);

      set z = ((sn -FanMorphW ) . p);

      reconsider q = p, qz = z as Point of ( TOP-REAL 2);

      per cases ;

        suppose

         A1: ((q `2 ) / |.q.|) >= sn & (q `1 ) < 0 ;

        then

         A2: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by Th16;

        then

         A3: (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

        

         A4: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by A2, EUCLID: 52;

        

         A5: (((q `2 ) / |.q.|) - sn) >= 0 by A1, XREAL_1: 48;

        

         A6: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         |.q.| <> 0 by A1, JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A6, XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A7, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then

         A8: (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        per cases ;

          suppose

           A9: (1 - sn) = 0 ;

          

           A10: ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = ((((q `2 ) / |.q.|) - sn) * ((1 - sn) " )) by XCMPLX_0:def 9

          .= ((((q `2 ) / |.q.|) - sn) * 0 ) by A9

          .= 0 ;

          then ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) = ( - 1) by SQUARE_1: 18;

          

          then ((sn -FanMorphW ) . q) = |[( |.q.| * ( - 1)), ( |.q.| * 0 )]| by A1, A10, Th16

          .= |[( - |.q.|), 0 ]|;

          then (((sn -FanMorphW ) . q) `1 ) = ( - |.q.|) & (((sn -FanMorphW ) . q) `2 ) = 0 by EUCLID: 52;

          

          then |.((sn -FanMorphW ) . p).| = ( sqrt ((( - |.q.|) ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ( |.q.| ^2 ))

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A11: (1 - sn) <> 0 ;

          per cases by A11;

            suppose

             A12: (1 - sn) > 0 ;

            ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by A8, XREAL_1: 24;

            then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A12, XREAL_1: 72;

            then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A12, XCMPLX_1: 197;

            then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A5, A12, SQUARE_1: 49;

            then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A13: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A14: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 )) by A3

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A13, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A4, A14;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A15: (1 - sn) < 0 ;

            ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A1, SQUARE_1: 12, XREAL_1: 8;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, A6, XREAL_1: 74;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then

             A16: 1 > ((q `2 ) / |.p.|) by SQUARE_1: 52;

            (((q `2 ) / |.q.|) - sn) >= 0 by A1, XREAL_1: 48;

            hence thesis by A15, A16, XREAL_1: 9;

          end;

        end;

      end;

        suppose

         A17: ((q `2 ) / |.q.|) < sn & (q `1 ) < 0 ;

        then |.q.| <> 0 by JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A18: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

        

         A19: (((q `2 ) / |.q.|) - sn) < 0 by A17, XREAL_1: 49;

        

         A20: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A20, XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A18, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then

         A21: (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        

         A22: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A17, Th17;

        then

         A23: (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

        

         A24: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A22, EUCLID: 52;

        per cases ;

          suppose

           A25: (1 + sn) = 0 ;

          ((((q `2 ) / |.q.|) - sn) / (1 + sn)) = ((((q `2 ) / |.q.|) - sn) * ((1 + sn) " )) by XCMPLX_0:def 9

          .= ((((q `2 ) / |.q.|) - sn) * 0 ) by A25

          .= 0 ;

          then (((sn -FanMorphW ) . q) `1 ) = ( - |.q.|) & (((sn -FanMorphW ) . q) `2 ) = 0 by A22, EUCLID: 52, SQUARE_1: 18;

          

          then |.((sn -FanMorphW ) . p).| = ( sqrt ((( - |.q.|) ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ( |.q.| ^2 ))

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A26: (1 + sn) <> 0 ;

          per cases by A26;

            suppose

             A27: (1 + sn) > 0 ;

            then (( - (1 + sn)) / (1 + sn)) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A21, XREAL_1: 72;

            then ( - 1) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A27, XCMPLX_1: 197;

            then (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) <= (1 ^2 ) by A19, A27, SQUARE_1: 49;

            then

             A28: (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

            

             A29: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 )) by A23

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A28, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A24, A29;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A30: (1 + sn) < 0 ;

            ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A17, SQUARE_1: 12, XREAL_1: 8;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A18, A20, XREAL_1: 74;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A18, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then ( - 1) < ((q `2 ) / |.p.|) by SQUARE_1: 52;

            then

             A31: (((q `2 ) / |.q.|) - sn) > (( - 1) - sn) by XREAL_1: 9;

            ( - (1 + sn)) > ( - 0 ) by A30, XREAL_1: 24;

            hence thesis by A17, A31, XREAL_1: 49;

          end;

        end;

      end;

        suppose (q `1 ) >= 0 ;

        hence thesis by Th16;

      end;

    end;

    theorem :: JGRAPH_4:34

    

     Th34: for sn be Real, x,K0 be set st ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((sn -FanMorphW ) . x) in K0

    proof

      let sn be Real, x,K0 be set;

      assume

       A1: ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then

      consider p such that

       A2: p = x and

       A3: (p `1 ) <= 0 and

       A4: p <> ( 0. ( TOP-REAL 2));

       A5:

      now

        assume |.p.| <= 0 ;

        then |.p.| = 0 ;

        hence contradiction by A4, TOPRNS_1: 24;

      end;

      then

       A6: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

      per cases ;

        suppose

         A7: ((p `2 ) / |.p.|) <= sn;

        reconsider p9 = ((sn -FanMorphW ) . p) as Point of ( TOP-REAL 2);

        ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A3, A4, A7, Th18;

        then

         A8: (p9 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

        

         A9: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A10: (1 + sn) > 0 by A1, XREAL_1: 148;

        per cases ;

          suppose (p `1 ) = 0 ;

          hence thesis by A1, A2, Th16;

        end;

          suppose (p `1 ) <> 0 ;

          then ( 0 + ((p `2 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A9, XREAL_1: 74;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `2 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ( - 1) < ((p `2 ) / |.p.|) by SQUARE_1: 52;

          then (( - 1) - sn) < (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

          then ((( - 1) * (1 + sn)) / (1 + sn)) < ((((p `2 ) / |.p.|) - sn) / (1 + sn)) by A10, XREAL_1: 74;

          then

           A11: ( - 1) < ((((p `2 ) / |.p.|) - sn) / (1 + sn)) by A10, XCMPLX_1: 89;

          (((p `2 ) / |.p.|) - sn) <= 0 by A7, XREAL_1: 47;

          then (1 ^2 ) > (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ) by A10, A11, SQUARE_1: 50;

          then (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )) > 0 by XREAL_1: 50;

          then ( - ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) > 0 by SQUARE_1: 25;

          then ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) < 0 ;

          then ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) < 0 by A5, XREAL_1: 132;

          hence thesis by A1, A2, A8, JGRAPH_2: 3;

        end;

      end;

        suppose

         A12: ((p `2 ) / |.p.|) > sn;

        reconsider p9 = ((sn -FanMorphW ) . p) as Point of ( TOP-REAL 2);

        ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A3, A4, A12, Th18;

        then

         A13: (p9 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

        

         A14: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A15: (1 - sn) > 0 by A1, XREAL_1: 149;

        per cases ;

          suppose (p `1 ) = 0 ;

          hence thesis by A1, A2, Th16;

        end;

          suppose (p `1 ) <> 0 ;

          then ( 0 + ((p `2 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A14, XREAL_1: 74;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `2 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ((p `2 ) / |.p.|) < 1 by SQUARE_1: 52;

          then (((p `2 ) / |.p.|) - sn) < (1 - sn) by XREAL_1: 9;

          then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) < ((1 - sn) / (1 - sn)) by A15, XREAL_1: 74;

          then

           A16: ((((p `2 ) / |.p.|) - sn) / (1 - sn)) < 1 by A15, XCMPLX_1: 60;

          ( - (1 - sn)) < ( - 0 ) & (((p `2 ) / |.p.|) - sn) >= (sn - sn) by A12, A15, XREAL_1: 9, XREAL_1: 24;

          then ((( - 1) * (1 - sn)) / (1 - sn)) < ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A15, XREAL_1: 74;

          then ( - 1) < ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A15, XCMPLX_1: 89;

          then (1 ^2 ) > (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ) by A16, SQUARE_1: 50;

          then (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )) > 0 by XREAL_1: 50;

          then ( - ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) > 0 by SQUARE_1: 25;

          then ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) < 0 ;

          then (p9 `1 ) < 0 by A5, A13, XREAL_1: 132;

          hence thesis by A1, A2, JGRAPH_2: 3;

        end;

      end;

    end;

    theorem :: JGRAPH_4:35

    

     Th35: for sn be Real, x,K0 be set st ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((sn -FanMorphW ) . x) in K0

    proof

      let sn be Real, x,K0 be set;

      assume

       A1: ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then ex p st p = x & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

      hence thesis by A1, Th16;

    end;

    scheme :: JGRAPH_4:sch1

    InclSub { D() -> non empty Subset of ( TOP-REAL 2) , P[ set] } :

{ p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D())

      provided

       A1: D() = ( NonZero ( TOP-REAL 2));

      let x be object;

      assume x in { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      then

       A2: ex p st x = p & P[p] & p <> ( 0. ( TOP-REAL 2));

      

       A3: (D() ` ) = {( 0. ( TOP-REAL 2))} by A1, JGRAPH_3: 20;

      now

        assume not x in D();

        then x in (the carrier of ( TOP-REAL 2) \ D()) by A2, XBOOLE_0:def 5;

        then x in (D() ` ) by SUBSET_1:def 4;

        hence contradiction by A3, A2, TARSKI:def 1;

      end;

      hence thesis by PRE_TOPC: 8;

    end;

    theorem :: JGRAPH_4:36

    

     Th36: for sn be Real, D be non empty Subset of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (D ` ) = {( 0. ( TOP-REAL 2))} holds ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((sn -FanMorphW ) | D) & h is continuous

    proof

      ( |[ 0 , 1]| `1 ) = 0 & ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

      then

       A1: |[ 0 , 1]| in { p where p be Point of ( TOP-REAL 2) : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      set Y1 = |[ 0 , 1]|;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 ;

      reconsider B0 = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

      let sn be Real, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A2: ( - 1) < sn & sn < 1 and

       A3: (D ` ) = {( 0. ( TOP-REAL 2))};

      

       A4: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A5: D = (B0 ` ) by A3

      .= ( NonZero ( TOP-REAL 2)) by SUBSET_1:def 4;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A1;

      

       A6: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      

       A7: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A8: ( rng ((sn -FanMorphW ) | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphW ) | K0));

        then

        consider x be object such that

         A9: x in ( dom ((sn -FanMorphW ) | K0)) and

         A10: y = (((sn -FanMorphW ) | K0) . x) by FUNCT_1:def 3;

        x in (( dom (sn -FanMorphW )) /\ K0) by A9, RELAT_1: 61;

        then

         A11: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A11;

        ((sn -FanMorphW ) . p) = y by A10, A11, FUNCT_1: 49;

        then y in K0 by A2, A11, Th34;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A12: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      (Y1 `1 ) = 0 & (Y1 `2 ) = 1 by EUCLID: 52;

      then

       A13: Y1 in { p where p be Point of ( TOP-REAL 2) : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      

       A14: the carrier of (( TOP-REAL 2) | D) = ( NonZero ( TOP-REAL 2)) by A5, PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= 0 ;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K1 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A13;

      

       A15: K0 is closed & K1 is closed by A5, Th29, Th31;

      ( dom ((sn -FanMorphW ) | K0)) = (( dom (sn -FanMorphW )) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by FUNCT_2:def 1

      .= K0 by A12, XBOOLE_1: 28;

      then

      reconsider f = ((sn -FanMorphW ) | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A6, A8, FUNCT_2: 2, XBOOLE_1: 1;

      

       A16: K1 = the carrier of ((( TOP-REAL 2) | D) | K1) by PRE_TOPC: 8;

      

       A17: ( rng ((sn -FanMorphW ) | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphW ) | K1));

        then

        consider x be object such that

         A18: x in ( dom ((sn -FanMorphW ) | K1)) and

         A19: y = (((sn -FanMorphW ) | K1) . x) by FUNCT_1:def 3;

        x in (( dom (sn -FanMorphW )) /\ K1) by A18, RELAT_1: 61;

        then

         A20: x in K1 by XBOOLE_0:def 4;

        K1 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A20;

        ((sn -FanMorphW ) . p) = y by A19, A20, FUNCT_1: 49;

        then y in K1 by A2, A20, Th35;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A21: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ((sn -FanMorphW ) | K1)) = (( dom (sn -FanMorphW )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by A21, XBOOLE_1: 28;

      then

      reconsider g = ((sn -FanMorphW ) | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A16, A17, FUNCT_2: 2, XBOOLE_1: 1;

      

       A22: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A23: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A24: x in D;

        then

        reconsider px = x as Point of ( TOP-REAL 2);

         not x in {( 0. ( TOP-REAL 2))} by A5, A24, XBOOLE_0:def 5;

        then (px `1 ) <= 0 & px <> ( 0. ( TOP-REAL 2)) or (px `1 ) >= 0 & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        then x in K0 or x in K1;

        hence thesis by XBOOLE_0:def 3;

      end;

      

       A25: ( dom f) = K0 by A6, FUNCT_2:def 1;

      

       A26: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A27: for x be object st x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1))) holds (f . x) = (g . x)

      proof

        let x be object;

        assume

         A28: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A26, XBOOLE_0:def 4;

        then (f . x) = ((sn -FanMorphW ) . x) by FUNCT_1: 49;

        hence thesis by A22, A28, FUNCT_1: 49;

      end;

      D = ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

      then

       A29: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A26, A22, A23, XBOOLE_0:def 10;

      

       A30: f is continuous & g is continuous by A2, A5, Th30, Th32;

      then

      consider h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) such that

       A31: h = (f +* g) and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2: 1;

      

       A32: ( dom h) = the carrier of (( TOP-REAL 2) | D) by FUNCT_2:def 1;

      

       A33: ( dom g) = K1 by A16, FUNCT_2:def 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A34: f tolerates g by A27, A25, A33, PARTFUN1:def 4;

      

       A35: for x be object st x in ( dom h) holds (h . x) = (((sn -FanMorphW ) | D) . x)

      proof

        let x be object;

        assume

         A36: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A14, XBOOLE_0:def 5;

        

         A37: x in ((D ` ) ` ) by A32, A36, PRE_TOPC: 8;

         not x in {( 0. ( TOP-REAL 2))} by A14, A36, XBOOLE_0:def 5;

        then

         A38: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

           A39: x in K0;

          

           A40: (((sn -FanMorphW ) | D) . p) = ((sn -FanMorphW ) . p) by A37, FUNCT_1: 49

          .= (f . p) by A39, FUNCT_1: 49;

          (h . p) = ((g +* f) . p) by A31, A34, FUNCT_4: 34

          .= (f . p) by A25, A39, FUNCT_4: 13;

          hence thesis by A40;

        end;

          suppose not x in K0;

          then not (p `1 ) <= 0 by A38;

          then

           A41: x in K1 by A38;

          (((sn -FanMorphW ) | D) . p) = ((sn -FanMorphW ) . p) by A37, FUNCT_1: 49

          .= (g . p) by A41, FUNCT_1: 49;

          hence thesis by A31, A33, A41, FUNCT_4: 13;

        end;

      end;

      ( dom (sn -FanMorphW )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((sn -FanMorphW ) | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A4, XBOOLE_1: 28;

      then (f +* g) = ((sn -FanMorphW ) | D) by A31, A32, A35, FUNCT_1: 2;

      hence thesis by A26, A22, A29, A30, A15, A27, JGRAPH_2: 1;

    end;

    

     Lm11: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

    theorem :: JGRAPH_4:37

    

     Th37: for sn be Real st ( - 1) < sn & sn < 1 holds ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (sn -FanMorphW ) & h is continuous

    proof

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      reconsider f = (sn -FanMorphW ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A3: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Th16, JGRAPH_2: 3;

      

       A4: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

        let p be Point of (( TOP-REAL 2) | D);

        

         A5: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A5, XBOOLE_0:def 5;

        then

         A6: p <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

           A7: ((q `2 ) / |.q.|) >= sn & (q `1 ) <= 0 ;

          set q9 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

          

           A8: (q9 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

          

           A9: (q9 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

          now

            assume

             A10: q9 = ( 0. ( TOP-REAL 2));

            

             A11: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

            

            then ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) = ( - ( sqrt (1 - 0 ))) by A8, A10, JGRAPH_2: 3, XCMPLX_1: 6

            .= ( - 1) by SQUARE_1: 18;

            hence contradiction by A9, A10, A11, JGRAPH_2: 3, XCMPLX_1: 6;

          end;

          hence thesis by A1, A2, A3, A6, A7, Th18;

        end;

          suppose

           A12: ((q `2 ) / |.q.|) < sn & (q `1 ) <= 0 ;

          set q9 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

          

           A13: (q9 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

          

           A14: (q9 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

          now

            assume

             A15: q9 = ( 0. ( TOP-REAL 2));

            

             A16: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

            

            then ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) = ( - ( sqrt (1 - 0 ))) by A13, A15, JGRAPH_2: 3, XCMPLX_1: 6

            .= ( - 1) by SQUARE_1: 18;

            hence contradiction by A14, A15, A16, JGRAPH_2: 3, XCMPLX_1: 6;

          end;

          hence thesis by A1, A2, A3, A6, A12, Th18;

        end;

          suppose (q `1 ) > 0 ;

          then (f . p) = p by Th16;

          hence thesis by A6, Th16, JGRAPH_2: 3;

        end;

      end;

      

       A17: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

        let V be Subset of ( TOP-REAL 2);

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm11;

        assume that

         A18: (f . ( 0. ( TOP-REAL 2))) in V and

         A19: V is open;

        VV is open by A19, Lm11, PRE_TOPC: 30;

        then

        consider r be Real such that

         A20: r > 0 and

         A21: ( Ball (u0,r)) c= V by A3, A18, TOPMETR: 15;

        reconsider r as Real;

         the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

        then

        reconsider W1 = ( Ball (u0,r)) as Subset of ( TOP-REAL 2);

        

         A22: W1 is open by GOBOARD6: 3;

        

         A23: (f .: W1) c= W1

        proof

          let z be object;

          assume z in (f .: W1);

          then

          consider y be object such that

           A24: y in ( dom f) and

           A25: y in W1 and

           A26: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A24, A26, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          reconsider q = y as Point of ( TOP-REAL 2) by A24;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          ( dist (u0,qy)) < r by A25, METRIC_1: 11;

          then

           A27: |.(( 0. ( TOP-REAL 2)) - q).| < r by JGRAPH_1: 28;

          per cases by JGRAPH_2: 3;

            suppose (q `1 ) >= 0 ;

            hence thesis by A25, A26, Th16;

          end;

            suppose

             A28: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) / |.q.|) >= sn & (q `1 ) <= 0 ;

            then

             A29: (((q `2 ) / |.q.|) - sn) >= 0 by XREAL_1: 48;

             0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A30: (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            

             A31: (1 - sn) > 0 by A2, XREAL_1: 149;

             |.q.| <> 0 by A28, TOPRNS_1: 24;

            then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A30, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

            then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

            then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

            then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A31, XREAL_1: 72;

            then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A31, XCMPLX_1: 197;

            then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A31, A29, SQUARE_1: 49;

            then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A32: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A33: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th18;

            then

             A34: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by A26, EUCLID: 52;

            (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by A26, A33, EUCLID: 52;

            

            then

             A35: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A32, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A34, A35;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A36: |.qz.| = |.q.| by SQUARE_1: 22;

             |.( - q).| < r by A27, RLVECT_1: 4;

            then |.q.| < r by TOPRNS_1: 26;

            then |.( - qz).| < r by A36, TOPRNS_1: 26;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

            suppose

             A37: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) / |.q.|) < sn & (q `1 ) <= 0 ;

             0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A38: (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            

             A39: (1 + sn) > 0 by A1, XREAL_1: 148;

             |.q.| <> 0 by A37, TOPRNS_1: 24;

            then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A38, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

            then ( - ( - 1)) >= ( - ((q `2 ) / |.q.|)) by XREAL_1: 24;

            then (1 + sn) >= (( - ((q `2 ) / |.q.|)) + sn) by XREAL_1: 7;

            then

             A40: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A39, XREAL_1: 185;

            (sn - ((q `2 ) / |.q.|)) >= 0 by A37, XREAL_1: 48;

            then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A39;

            then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A40, SQUARE_1: 49;

            then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A41: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A42: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A37, Th18;

            then

             A43: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A26, EUCLID: 52;

            (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by A26, A42, EUCLID: 52;

            

            then

             A44: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A41, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A43, A44;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A45: |.qz.| = |.q.| by SQUARE_1: 22;

             |.( - q).| < r by A27, RLVECT_1: 4;

            then |.q.| < r by TOPRNS_1: 26;

            then |.( - qz).| < r by A45, TOPRNS_1: 26;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

        end;

        u0 in W1 by A20, GOBOARD6: 1;

        hence thesis by A21, A22, A23, XBOOLE_1: 1;

      end;

      

       A46: (D ` ) = {( 0. ( TOP-REAL 2))} by JGRAPH_3: 20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((sn -FanMorphW ) | D) & h is continuous by A1, A2, Th36;

      hence thesis by A3, A46, A4, A17, JGRAPH_3: 3;

    end;

    theorem :: JGRAPH_4:38

    

     Th38: for sn be Real st ( - 1) < sn & sn < 1 holds (sn -FanMorphW ) is one-to-one

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      for x1,x2 be object st x1 in ( dom (sn -FanMorphW )) & x2 in ( dom (sn -FanMorphW )) & ((sn -FanMorphW ) . x1) = ((sn -FanMorphW ) . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A3: x1 in ( dom (sn -FanMorphW )) and

         A4: x2 in ( dom (sn -FanMorphW )) and

         A5: ((sn -FanMorphW ) . x1) = ((sn -FanMorphW ) . x2);

        reconsider p2 = x2 as Point of ( TOP-REAL 2) by A4;

        reconsider p1 = x1 as Point of ( TOP-REAL 2) by A3;

        set q = p1, p = p2;

        

         A6: (1 - sn) > 0 by A2, XREAL_1: 149;

        now

          per cases by JGRAPH_2: 3;

            case

             A7: (q `1 ) >= 0 ;

            then

             A8: ((sn -FanMorphW ) . q) = q by Th16;

            now

              per cases by JGRAPH_2: 3;

                case (p `1 ) >= 0 ;

                hence thesis by A5, A8, Th16;

              end;

                case

                 A9: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 ;

                set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

                

                 A10: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then

                 A11: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A10, XREAL_1: 72;

                

                 A12: |.p.| > 0 by A9, Lm1;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A11, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `2 ) / |.p.|) by SQUARE_1: 51;

                then (1 - sn) >= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A13: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                

                 A14: (((p `2 ) / |.p.|) - sn) >= 0 by A9, XREAL_1: 48;

                

                 A15: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A2, A9, Th18;

                (((p `2 ) / |.p.|) - sn) >= 0 by A9, XREAL_1: 48;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A13, SQUARE_1: 49;

                then

                 A16: (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) & (q `1 ) = 0 by A5, A7, A8, A15, EUCLID: 52;

                then

                 A17: ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) = 0 by A5, A8, A15, A12, XCMPLX_1: 6;

                (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 - sn))) ^2 )) >= 0 by A16, XCMPLX_1: 187;

                then (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )) = 0 by A17, SQUARE_1: 24;

                then 1 = ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A6, A14, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - sn)) = (((p `2 ) / |.p.|) - sn) by A6, XCMPLX_1: 87;

                then (1 * |.p.|) = (p `2 ) by A12, XCMPLX_1: 87;

                then (p `1 ) = 0 by A10, XCMPLX_1: 6;

                hence thesis by A5, A8, Th16;

              end;

                case

                 A18: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) <= 0 ;

                then

                 A19: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A2, Th18;

                set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

                

                 A20: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then

                 A21: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A20, XREAL_1: 72;

                

                 A22: (1 + sn) > 0 by A1, XREAL_1: 148;

                

                 A23: (((p `2 ) / |.p.|) - sn) <= 0 by A18, XREAL_1: 47;

                then

                 A24: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by A22;

                

                 A25: |.p.| > 0 by A18, Lm1;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A21, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then (( - ((p `2 ) / |.p.|)) ^2 ) <= 1;

                then 1 >= ( - ((p `2 ) / |.p.|)) by SQUARE_1: 51;

                then (1 + sn) >= (( - ((p `2 ) / |.p.|)) + sn) by XREAL_1: 7;

                then (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) <= 1 by A22, XREAL_1: 185;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A24, SQUARE_1: 49;

                then

                 A26: (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) & (q `1 ) = 0 by A5, A7, A8, A19, EUCLID: 52;

                then

                 A27: ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) = 0 by A5, A8, A19, A25, XCMPLX_1: 6;

                (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) >= 0 by A26, XCMPLX_1: 187;

                then (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )) = 0 by A27, SQUARE_1: 24;

                then 1 = ( sqrt (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) by SQUARE_1: 18;

                then 1 = ( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by A22, A23, SQUARE_1: 22;

                then 1 = (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by XCMPLX_1: 187;

                then (1 * (1 + sn)) = ( - (((p `2 ) / |.p.|) - sn)) by A22, XCMPLX_1: 87;

                then ((1 + sn) - sn) = ( - ((p `2 ) / |.p.|));

                then 1 = (( - (p `2 )) / |.p.|) by XCMPLX_1: 187;

                then (1 * |.p.|) = ( - (p `2 )) by A25, XCMPLX_1: 87;

                then (((p `2 ) ^2 ) - ((p `2 ) ^2 )) = ((p `1 ) ^2 ) by A20, XCMPLX_1: 26;

                then (p `1 ) = 0 by XCMPLX_1: 6;

                hence thesis by A5, A8, Th16;

              end;

            end;

            hence thesis;

          end;

            case

             A28: ((q `2 ) / |.q.|) >= sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

            then |.q.| > 0 by Lm1;

            then

             A29: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

            

             A30: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

            

             A31: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th18;

            

             A32: (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

            now

              per cases by JGRAPH_2: 3;

                case

                 A33: (p `1 ) >= 0 ;

                

                 A34: (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                

                 A35: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

                then

                 A36: (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A35, XREAL_1: 72;

                

                 A37: ((sn -FanMorphW ) . p) = p by A33, Th16;

                

                 A38: (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                

                 A39: (1 - sn) > 0 by A2, XREAL_1: 149;

                

                 A40: |.q.| > 0 by A28, Lm1;

                then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A36, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

                then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A39, XREAL_1: 72;

                then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A39, XCMPLX_1: 197;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A39, A34, SQUARE_1: 49;

                then

                 A41: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p `1 ) = 0 by A5, A31, A33, A37, EUCLID: 52;

                then

                 A42: ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) = 0 by A5, A31, A32, A37, A40, XCMPLX_1: 6;

                (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by A41, XCMPLX_1: 187;

                then (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) = 0 by A42, SQUARE_1: 24;

                then 1 = ((((q `2 ) / |.q.|) - sn) / (1 - sn)) by A39, A38, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - sn)) = (((q `2 ) / |.q.|) - sn) by A39, XCMPLX_1: 87;

                then (1 * |.q.|) = (q `2 ) by A40, XCMPLX_1: 87;

                then (q `1 ) = 0 by A35, XCMPLX_1: 6;

                hence thesis by A5, A37, Th16;

              end;

                case

                 A43: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 ;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A29, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

                then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A44: ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A44, SQUARE_1: 49;

                then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A45: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

                

                then

                 A46: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

                .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A45, SQUARE_1:def 2;

                

                 A47: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

                ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.q.| ^2 ) by A47, A46;

                then ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

                then

                 A48: |.q4.| = |.q.| by SQUARE_1: 22;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then

                 A49: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

                

                 A50: |.p.| > 0 by A43, Lm1;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A49, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `2 ) / |.p.|) by SQUARE_1: 51;

                then (1 - sn) >= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A51: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

                

                 A52: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) by EUCLID: 52;

                (((p `2 ) / |.p.|) - sn) >= 0 by A43, XREAL_1: 48;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A51, SQUARE_1: 49;

                then (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A53: (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (p4 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

                

                then

                 A54: ((p4 `1 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

                .= (( |.p.| ^2 ) * (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) by A53, SQUARE_1:def 2;

                ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.p.| ^2 ) by A52, A54;

                then ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

                then

                 A55: |.p4.| = |.p.| by SQUARE_1: 22;

                

                 A56: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A2, A43, Th18;

                then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) = (( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) / |.p.|) by A5, A31, A30, A52, A50, XCMPLX_1: 89;

                then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) = ((((q `2 ) / |.q.|) - sn) / (1 - sn)) by A5, A31, A56, A48, A50, A55, XCMPLX_1: 89;

                then (((((p `2 ) / |.p.|) - sn) / (1 - sn)) * (1 - sn)) = (((q `2 ) / |.q.|) - sn) by A6, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) - sn) = (((q `2 ) / |.q.|) - sn) by A6, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) * |.p.|) = (q `2 ) by A5, A31, A56, A48, A50, A55, XCMPLX_1: 87;

                then

                 A57: (p `2 ) = (q `2 ) by A50, XCMPLX_1: 87;

                ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                then (( - (p `1 )) ^2 ) = ((q `1 ) ^2 ) by A5, A31, A56, A48, A55, A57;

                then ( - (p `1 )) = ( sqrt (( - (q `1 )) ^2 )) by A43, SQUARE_1: 22;

                then

                 A58: ( - ( - (p `1 ))) = ( - ( - (q `1 ))) by A28, SQUARE_1: 22;

                p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

                hence thesis by A57, A58, EUCLID: 53;

              end;

                case

                 A59: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) <= 0 ;

                then (((p `2 ) / |.p.|) - sn) < 0 by XREAL_1: 49;

                then

                 A60: ((((p `2 ) / |.p.|) - sn) / (1 + sn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

                set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

                

                 A61: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) & (((q `2 ) / |.q.|) - sn) >= 0 by A28, EUCLID: 52, XREAL_1: 48;

                

                 A62: (1 - sn) > 0 by A2, XREAL_1: 149;

                ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A2, A59, Th18;

                hence thesis by A5, A31, A30, A59, A60, A61, A62, Lm1, XREAL_1: 132;

              end;

            end;

            hence thesis;

          end;

            case

             A63: ((q `2 ) / |.q.|) < sn & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

            then

             A64: |.q.| > 0 by Lm1;

            then

             A65: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

            

             A66: (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

            

             A67: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

            

             A68: ((sn -FanMorphW ) . q) = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A63, Th18;

            per cases by JGRAPH_2: 3;

              suppose

               A69: (p `1 ) >= 0 ;

              

               A70: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

              

               A71: (1 + sn) > 0 by A1, XREAL_1: 148;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A70, XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A65, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then (( - ((q `2 ) / |.q.|)) ^2 ) <= 1;

              then 1 >= ( - ((q `2 ) / |.q.|)) by SQUARE_1: 51;

              then (1 + sn) >= (( - ((q `2 ) / |.q.|)) + sn) by XREAL_1: 7;

              then

               A72: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A71, XREAL_1: 185;

              

               A73: (((q `2 ) / |.q.|) - sn) <= 0 by A63, XREAL_1: 47;

              then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A71;

              then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A72, SQUARE_1: 49;

              then

               A74: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A75: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

              

               A76: ((sn -FanMorphW ) . p) = p by A69, Th16;

              ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by A74, SQUARE_1:def 2;

              then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

              then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

              then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

              then (p `1 ) = 0 by A5, A68, A69, A76, EUCLID: 52;

              then ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) = 0 by A5, A68, A66, A64, A76, XCMPLX_1: 6;

              then (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) = 0 by A75, SQUARE_1: 24;

              then 1 = ( sqrt (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) by SQUARE_1: 18;

              then 1 = ( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A71, A73, SQUARE_1: 22;

              then 1 = (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by XCMPLX_1: 187;

              then (1 * (1 + sn)) = ( - (((q `2 ) / |.q.|) - sn)) by A71, XCMPLX_1: 87;

              then ((1 + sn) - sn) = ( - ((q `2 ) / |.q.|));

              then 1 = (( - (q `2 )) / |.q.|) by XCMPLX_1: 187;

              then (1 * |.q.|) = ( - (q `2 )) by A64, XCMPLX_1: 87;

              then (((q `2 ) ^2 ) - ((q `2 ) ^2 )) = ((q `1 ) ^2 ) by A70, XCMPLX_1: 26;

              then (q `1 ) = 0 by XCMPLX_1: 6;

              hence thesis by A5, A76, Th16;

            end;

              suppose

               A77: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) <= 0 ;

              set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

              

               A78: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) & (1 - sn) > 0 by A2, EUCLID: 52, XREAL_1: 149;

              (((q `2 ) / |.q.|) - sn) < 0 by A63, XREAL_1: 49;

              then

               A79: ((((q `2 ) / |.q.|) - sn) / (1 + sn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

              ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| & (((p `2 ) / |.p.|) - sn) >= 0 by A1, A2, A77, Th18, XREAL_1: 48;

              hence thesis by A5, A63, A68, A67, A79, A78, Lm1, XREAL_1: 132;

            end;

              suppose

               A80: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) <= 0 ;

               0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

              then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then

               A81: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

              

               A82: (1 + sn) > 0 by A1, XREAL_1: 148;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A65, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

              then (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

              then ( - (( - 1) - sn)) >= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

              then

               A83: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A82, XREAL_1: 185;

              (((q `2 ) / |.q.|) - sn) <= 0 by A63, XREAL_1: 47;

              then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A82;

              then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A83, SQUARE_1: 49;

              then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A84: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

              (q4 `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

              

              then

               A85: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

              .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A84, SQUARE_1:def 2;

              

               A86: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

              set p4 = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

              

               A87: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by EUCLID: 52;

              ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.q.| ^2 ) by A86, A85;

              then ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

              then

               A88: |.q4.| = |.q.| by SQUARE_1: 22;

              (((p `2 ) / |.p.|) - sn) <= 0 by A80, XREAL_1: 47;

              then

               A89: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by A82;

              

               A90: |.p.| > 0 by A80, Lm1;

              then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

              then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A81, XCMPLX_1: 60;

              then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ( - 1) <= ((p `2 ) / |.p.|) by SQUARE_1: 51;

              then (( - 1) - sn) <= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

              then ( - (( - 1) - sn)) >= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

              then (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) <= 1 by A82, XREAL_1: 185;

              then ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A89, SQUARE_1: 49;

              then (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A91: (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

              (p4 `1 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) by EUCLID: 52;

              

              then

               A92: ((p4 `1 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

              .= (( |.p.| ^2 ) * (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) by A91, SQUARE_1:def 2;

              ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.p.| ^2 ) by A87, A92;

              then ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

              then

               A93: |.p4.| = |.p.| by SQUARE_1: 22;

              

               A94: ((sn -FanMorphW ) . p) = |[( |.p.| * ( - ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A2, A80, Th18;

              then ((((p `2 ) / |.p.|) - sn) / (1 + sn)) = (( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) / |.p.|) by A5, A68, A67, A87, A90, XCMPLX_1: 89;

              then ((((p `2 ) / |.p.|) - sn) / (1 + sn)) = ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A5, A68, A94, A88, A90, A93, XCMPLX_1: 89;

              then (((((p `2 ) / |.p.|) - sn) / (1 + sn)) * (1 + sn)) = (((q `2 ) / |.q.|) - sn) by A82, XCMPLX_1: 87;

              then (((p `2 ) / |.p.|) - sn) = (((q `2 ) / |.q.|) - sn) by A82, XCMPLX_1: 87;

              then (((p `2 ) / |.p.|) * |.p.|) = (q `2 ) by A5, A68, A94, A88, A90, A93, XCMPLX_1: 87;

              then

               A95: (p `2 ) = (q `2 ) by A90, XCMPLX_1: 87;

              ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

              then (( - (p `1 )) ^2 ) = ((q `1 ) ^2 ) by A5, A68, A94, A88, A93, A95;

              then ( - (p `1 )) = ( sqrt (( - (q `1 )) ^2 )) by A80, SQUARE_1: 22;

              then

               A96: ( - ( - (p `1 ))) = ( - ( - (q `1 ))) by A63, SQUARE_1: 22;

              p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

              hence thesis by A95, A96, EUCLID: 53;

            end;

          end;

        end;

        hence thesis;

      end;

      hence thesis by FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_4:39

    

     Th39: for sn be Real st ( - 1) < sn & sn < 1 holds (sn -FanMorphW ) is Function of ( TOP-REAL 2), ( TOP-REAL 2) & ( rng (sn -FanMorphW )) = the carrier of ( TOP-REAL 2)

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      thus (sn -FanMorphW ) is Function of ( TOP-REAL 2), ( TOP-REAL 2);

      for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (sn -FanMorphW ) holds ( rng (sn -FanMorphW )) = the carrier of ( TOP-REAL 2)

      proof

        let f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

        assume

         A3: f = (sn -FanMorphW );

        

         A4: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        the carrier of ( TOP-REAL 2) c= ( rng f)

        proof

          let y be object;

          assume y in the carrier of ( TOP-REAL 2);

          then

          reconsider p2 = y as Point of ( TOP-REAL 2);

          set q = p2;

          now

            per cases by JGRAPH_2: 3;

              case (q `1 ) >= 0 ;

              then y = ((sn -FanMorphW ) . q) by Th16;

              hence ex x be set st x in ( dom (sn -FanMorphW )) & y = ((sn -FanMorphW ) . x) by A3, A4;

            end;

              case

               A5: ((q `2 ) / |.q.|) >= 0 & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

              

               A6: ( - ( - (1 + sn))) > 0 by A1, XREAL_1: 148;

              

               A7: (1 - sn) >= 0 by A2, XREAL_1: 149;

              then (((q `2 ) / |.q.|) * (1 - sn)) >= 0 by A5;

              then ( - (1 + sn)) <= (((q `2 ) / |.q.|) * (1 - sn)) by A6;

              then

               A8: ((( - 1) - sn) + sn) <= ((((q `2 ) / |.q.|) * (1 - sn)) + sn) by XREAL_1: 7;

              set px = |[( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn))]|;

              

               A9: (px `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn)) by EUCLID: 52;

               |.q.| <> 0 by A5, TOPRNS_1: 24;

              then

               A10: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              

               A11: |.q.| > 0 by A5, Lm1;

              

               A12: ( dom (sn -FanMorphW )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A13: (1 - sn) > 0 by A2, XREAL_1: 149;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A10, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `2 ) / |.q.|) <= 1 by SQUARE_1: 51;

              then (((q `2 ) / |.q.|) * (1 - sn)) <= (1 * (1 - sn)) by A13, XREAL_1: 64;

              then (((((q `2 ) / |.q.|) * (1 - sn)) + sn) - sn) <= (1 - sn);

              then ((((q `2 ) / |.q.|) * (1 - sn)) + sn) <= 1 by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ) by A8, SQUARE_1: 49;

              then

               A14: (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A15: ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A16: (px `1 ) = ( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))))) ^2 ) + (( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn)) ^2 )) by A9, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )));

              

              then

               A17: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) by A14, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A18: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A19: px <> ( 0. ( TOP-REAL 2)) by A5, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `2 ) / |.q.|) * (1 - sn)) + sn) >= ( 0 + sn) by A5, A7, XREAL_1: 7;

              then ((px `2 ) / |.px.|) >= sn by A5, A9, A18, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A20: ((sn -FanMorphW ) . px) = |[( |.px.| * ( - ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 - sn)) ^2 ))))), ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 - sn)))]| by A1, A2, A16, A15, A19, Th18;

              

               A21: ( |.px.| * ( - ( sqrt (((q `1 ) / |.q.|) ^2 )))) = ( |.px.| * ( - ( - ((q `1 ) / |.q.|)))) by A5, SQUARE_1: 23

              .= (q `1 ) by A11, A18, XCMPLX_1: 87;

              

               A22: ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 - sn))) = ( |.q.| * ((((((q `2 ) / |.q.|) * (1 - sn)) + sn) - sn) / (1 - sn))) by A5, A9, A18, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `2 ) / |.q.|)) by A13, XCMPLX_1: 89

              .= (q `2 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( - ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 - sn)) ^2 ))))) = ( |.px.| * ( - ( sqrt (1 - (((q `2 ) / |.px.|) ^2 ))))) by A5, A18, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( - ( sqrt (1 - (((q `2 ) ^2 ) / ( |.px.| ^2 )))))) by XCMPLX_1: 76

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `2 ) ^2 ) / ( |.px.| ^2 )))))) by A10, A17, XCMPLX_1: 60

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) - ((q `2 ) ^2 )) / ( |.px.| ^2 ))))) by XCMPLX_1: 120

              .= ( |.px.| * ( - ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `2 ) ^2 )) / ( |.px.| ^2 ))))) by A17, JGRAPH_3: 1

              .= ( |.px.| * ( - ( sqrt (((q `1 ) / |.q.|) ^2 )))) by A18, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (sn -FanMorphW )) & y = ((sn -FanMorphW ) . x) by A20, A22, A21, A12, EUCLID: 53;

            end;

              case

               A23: ((q `2 ) / |.q.|) < 0 & (q `1 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

              

               A24: (1 + sn) >= 0 by A1, XREAL_1: 148;

              then (((q `2 ) / |.q.|) * (1 + sn)) <= 0 by A23;

              then (1 - sn) >= (((q `2 ) / |.q.|) * (1 + sn)) by A2, XREAL_1: 149;

              then

               A25: ((1 - sn) + sn) >= ((((q `2 ) / |.q.|) * (1 + sn)) + sn) by XREAL_1: 7;

              set px = |[( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn))]|;

              

               A26: (px `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn)) by EUCLID: 52;

               |.q.| <> 0 by A23, TOPRNS_1: 24;

              then

               A27: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              

               A28: |.q.| > 0 by A23, Lm1;

              

               A29: ( dom (sn -FanMorphW )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A30: (1 + sn) > 0 by A1, XREAL_1: 148;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A27, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `2 ) / |.q.|) >= ( - 1) by SQUARE_1: 51;

              then (((q `2 ) / |.q.|) * (1 + sn)) >= (( - 1) * (1 + sn)) by A30, XREAL_1: 64;

              then (((((q `2 ) / |.q.|) * (1 + sn)) + sn) - sn) >= (( - 1) - sn);

              then ((((q `2 ) / |.q.|) * (1 + sn)) + sn) >= ( - 1) by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ) by A25, SQUARE_1: 49;

              then

               A31: (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A32: ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A33: (px `1 ) = ( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( - ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))))) ^2 ) + (( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn)) ^2 )) by A26, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )));

              

              then

               A34: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) by A31, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A35: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A36: px <> ( 0. ( TOP-REAL 2)) by A23, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `2 ) / |.q.|) * (1 + sn)) + sn) <= ( 0 + sn) by A23, A24, XREAL_1: 7;

              then ((px `2 ) / |.px.|) <= sn by A23, A26, A35, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A37: ((sn -FanMorphW ) . px) = |[( |.px.| * ( - ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 + sn)) ^2 ))))), ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 + sn)))]| by A1, A2, A33, A32, A36, Th18;

              

               A38: ( |.px.| * ( - ( sqrt (((q `1 ) / |.q.|) ^2 )))) = ( |.px.| * ( - ( - ((q `1 ) / |.q.|)))) by A23, SQUARE_1: 23

              .= (q `1 ) by A28, A35, XCMPLX_1: 87;

              

               A39: ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 + sn))) = ( |.q.| * ((((((q `2 ) / |.q.|) * (1 + sn)) + sn) - sn) / (1 + sn))) by A23, A26, A35, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `2 ) / |.q.|)) by A30, XCMPLX_1: 89

              .= (q `2 ) by A23, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( - ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 + sn)) ^2 ))))) = ( |.px.| * ( - ( sqrt (1 - (((q `2 ) / |.px.|) ^2 ))))) by A23, A35, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( - ( sqrt (1 - (((q `2 ) ^2 ) / ( |.px.| ^2 )))))) by XCMPLX_1: 76

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `2 ) ^2 ) / ( |.px.| ^2 )))))) by A27, A34, XCMPLX_1: 60

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) - ((q `2 ) ^2 )) / ( |.px.| ^2 ))))) by XCMPLX_1: 120

              .= ( |.px.| * ( - ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `2 ) ^2 )) / ( |.px.| ^2 ))))) by A34, JGRAPH_3: 1

              .= ( |.px.| * ( - ( sqrt (((q `1 ) / |.q.|) ^2 )))) by A35, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (sn -FanMorphW )) & y = ((sn -FanMorphW ) . x) by A37, A39, A38, A29, EUCLID: 53;

            end;

          end;

          hence thesis by A3, FUNCT_1:def 3;

        end;

        hence thesis by A3, XBOOLE_0:def 10;

      end;

      hence thesis;

    end;

    

     Lm12: for q4,q,p2 be Point of ( TOP-REAL 2), O,u,uq be Point of ( Euclid 2) st u in ( cl_Ball (O,( |.p2.| + 1))) & q = uq & q4 = u & O = ( 0. ( TOP-REAL 2)) & |.q4.| = |.q.| holds q in ( cl_Ball (O,( |.p2.| + 1)))

    proof

      let q4,q,p2 be Point of ( TOP-REAL 2), O,u,uq be Point of ( Euclid 2);

      assume

       A1: u in ( cl_Ball (O,( |.p2.| + 1)));

      assume that

       A2: q = uq and

       A3: q4 = u and

       A4: O = ( 0. ( TOP-REAL 2));

      assume

       A5: |.q4.| = |.q.|;

      now

        assume not q in ( cl_Ball (O,( |.p2.| + 1)));

        then not ( dist (O,uq)) <= ( |.p2.| + 1) by A2, METRIC_1: 12;

        then |.(( 0. ( TOP-REAL 2)) - q).| > ( |.p2.| + 1) by A2, A4, JGRAPH_1: 28;

        then |.( - q).| > ( |.p2.| + 1) by RLVECT_1: 4;

        then |.q.| > ( |.p2.| + 1) by TOPRNS_1: 26;

        then |.( - q4).| > ( |.p2.| + 1) by A5, TOPRNS_1: 26;

        then |.(( 0. ( TOP-REAL 2)) - q4).| > ( |.p2.| + 1) by RLVECT_1: 4;

        then ( dist (O,u)) > ( |.p2.| + 1) by A3, A4, JGRAPH_1: 28;

        hence contradiction by A1, METRIC_1: 12;

      end;

      hence q in ( cl_Ball (O,( |.p2.| + 1)));

    end;

    theorem :: JGRAPH_4:40

    

     Th40: for sn be Real, p2 be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = ((sn -FanMorphW ) .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & ((sn -FanMorphW ) . p2) in V2

    proof

      reconsider O = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

      let sn be Real, p2 be Point of ( TOP-REAL 2);

      

       A1: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

       the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

      then

      reconsider V0 = ( Ball (O,( |.p2.| + 1))) as Subset of ( TOP-REAL 2);

      O in V0 & V0 c= ( cl_Ball (O,( |.p2.| + 1))) by GOBOARD6: 1, METRIC_1: 14;

      then

      reconsider K0 = ( cl_Ball (O,( |.p2.| + 1))) as non empty compact Subset of ( TOP-REAL 2) by A1, Th15;

      set q3 = ((sn -FanMorphW ) . p2);

      reconsider VV0 = V0 as Subset of ( TopSpaceMetr ( Euclid 2));

      reconsider u2 = p2 as Point of ( Euclid 2) by EUCLID: 67;

      reconsider u3 = q3 as Point of ( Euclid 2) by EUCLID: 67;

      

       A2: ((sn -FanMorphW ) .: K0) c= K0

      proof

        let y be object;

        assume y in ((sn -FanMorphW ) .: K0);

        then

        consider x be object such that

         A3: x in ( dom (sn -FanMorphW )) and

         A4: x in K0 and

         A5: y = ((sn -FanMorphW ) . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A3;

        reconsider uq = q as Point of ( Euclid 2) by EUCLID: 67;

        ( dist (O,uq)) <= ( |.p2.| + 1) by A4, METRIC_1: 12;

        then |.(( 0. ( TOP-REAL 2)) - q).| <= ( |.p2.| + 1) by JGRAPH_1: 28;

        then |.( - q).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then

         A6: |.q.| <= ( |.p2.| + 1) by TOPRNS_1: 26;

        

         A7: y in ( rng (sn -FanMorphW )) by A3, A5, FUNCT_1:def 3;

        then

        reconsider u = y as Point of ( Euclid 2) by EUCLID: 67;

        reconsider q4 = y as Point of ( TOP-REAL 2) by A7;

         |.q4.| = |.q.| by A5, Th33;

        then |.( - q4).| <= ( |.p2.| + 1) by A6, TOPRNS_1: 26;

        then |.(( 0. ( TOP-REAL 2)) - q4).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then ( dist (O,u)) <= ( |.p2.| + 1) by JGRAPH_1: 28;

        hence thesis by METRIC_1: 12;

      end;

      VV0 is open by TOPMETR: 14;

      then

       A8: V0 is open by Lm11, PRE_TOPC: 30;

      

       A9: |.p2.| < ( |.p2.| + 1) by XREAL_1: 29;

      then |.( - p2).| < ( |.p2.| + 1) by TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - p2).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u2)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A10: p2 in V0 by METRIC_1: 11;

       |.q3.| = |.p2.| by Th33;

      then |.( - q3).| < ( |.p2.| + 1) by A9, TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - q3).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u3)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A11: ((sn -FanMorphW ) . p2) in V0 by METRIC_1: 11;

      assume

       A12: ( - 1) < sn & sn < 1;

      K0 c= ((sn -FanMorphW ) .: K0)

      proof

        let y be object;

        assume

         A13: y in K0;

        then

        reconsider y as Point of ( Euclid 2);

        reconsider q4 = y as Point of ( TOP-REAL 2) by A13;

        the carrier of ( TOP-REAL 2) c= ( rng (sn -FanMorphW )) by A12, Th39;

        then q4 in ( rng (sn -FanMorphW ));

        then

        consider x be object such that

         A14: x in ( dom (sn -FanMorphW )) and

         A15: y = ((sn -FanMorphW ) . x) by FUNCT_1:def 3;

        reconsider x as Point of ( Euclid 2) by A14, Lm11;

        reconsider q = x as Point of ( TOP-REAL 2) by A14;

         |.q4.| = |.q.| by A15, Th33;

        then q in K0 by A13, Lm12;

        hence thesis by A14, A15, FUNCT_1:def 6;

      end;

      then K0 = ((sn -FanMorphW ) .: K0) by A2, XBOOLE_0:def 10;

      hence thesis by A10, A8, A11, METRIC_1: 14;

    end;

    theorem :: JGRAPH_4:41

    for sn be Real st ( - 1) < sn & sn < 1 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (sn -FanMorphW ) & f is being_homeomorphism

    proof

      let sn be Real;

      reconsider f = (sn -FanMorphW ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume

       A1: ( - 1) < sn & sn < 1;

      then

       A2: for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2 by Th40;

      ( rng (sn -FanMorphW )) = the carrier of ( TOP-REAL 2) & ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (sn -FanMorphW ) & h is continuous by A1, Th37, Th39;

      then f is being_homeomorphism by A1, A2, Th3, Th38;

      hence thesis;

    end;

     Lm13:

    now

      let q be Point of ( TOP-REAL 2), sn,t be Real;

      assume ((( - ((t / |.q.|) - sn)) / (1 - sn)) ^2 ) < (1 ^2 );

      then (1 - ((( - ((t / |.q.|) - sn)) / (1 - sn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - ((t / |.q.|) - sn)) / (1 - sn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - ((t / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - ((((t / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) > 0 ;

      then ( sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2 ))) > 0 by XCMPLX_1: 76;

      hence ( - ( sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2 )))) < ( - 0 ) by XREAL_1: 24;

    end;

    theorem :: JGRAPH_4:42

    

     Th42: for sn be Real, q be Point of ( TOP-REAL 2) st sn < 1 & (q `1 ) < 0 & ((q `2 ) / |.q.|) >= sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphW ) . q) holds (p `1 ) < 0 & (p `2 ) >= 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: sn < 1 and

       A2: (q `1 ) < 0 and

       A3: ((q `2 ) / |.q.|) >= sn;

      

       A4: (1 - sn) > 0 by A1, XREAL_1: 149;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((sn -FanMorphW ) . q);

      then

       A5: p = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A2, A3, Th16;

      then

       A6: (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

      

       A7: (((q `2 ) / |.q.|) - sn) >= 0 by A3, XREAL_1: 48;

      

       A8: |.q.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then

       A9: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A9, XREAL_1: 74;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A9, XCMPLX_1: 60;

      then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then 1 > ((q `2 ) / |.q.|) by SQUARE_1: 52;

      then (1 - sn) > (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

      then ( - (1 - sn)) < ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

      then (( - (1 - sn)) / (1 - sn)) < (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A4, XREAL_1: 74;

      then ( - 1) < (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A4, XCMPLX_1: 197;

      then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) < (1 ^2 ) by A4, A7, SQUARE_1: 50;

      hence thesis by A5, A8, A4, A6, A7, Lm13, EUCLID: 52, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:43

    

     Th43: for sn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < sn & (q `1 ) < 0 & ((q `2 ) / |.q.|) < sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphW ) . q) holds (p `1 ) < 0 & (p `2 ) < 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: (q `1 ) < 0 and

       A3: ((q `2 ) / |.q.|) < sn;

      

       A4: (1 + sn) > 0 by A1, XREAL_1: 148;

      

       A5: (((q `2 ) / |.q.|) - sn) < 0 by A3, XREAL_1: 49;

      then ( - (((q `2 ) / |.q.|) - sn)) > 0 by XREAL_1: 58;

      then (( - (1 + sn)) / (1 + sn)) < (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A4, XREAL_1: 74;

      then

       A6: ( - 1) < (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A4, XCMPLX_1: 197;

       |.q.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then

       A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, XREAL_1: 74;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

      then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then ( - 1) < ((q `2 ) / |.q.|) by SQUARE_1: 52;

      then (( - 1) - sn) < (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

      then ( - ( - (1 + sn))) > ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

      then (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) < 1 by A4, XREAL_1: 191;

      then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) < (1 ^2 ) by A6, SQUARE_1: 50;

      then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) > 0 ;

      then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) > 0 by XCMPLX_1: 76;

      then

       A8: ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) < ( - 0 ) by XREAL_1: 24;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((sn -FanMorphW ) . q);

      then p = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th17;

      then

       A9: (qz `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))) & (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

      ((((q `2 ) / |.q.|) - sn) / (1 + sn)) < 0 by A1, A5, XREAL_1: 141, XREAL_1: 148;

      hence thesis by A2, A9, A8, Lm1, JGRAPH_2: 3, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:44

    

     Th44: for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st sn < 1 & (q1 `1 ) < 0 & ((q1 `2 ) / |.q1.|) >= sn & (q2 `1 ) < 0 & ((q2 `2 ) / |.q2.|) >= sn & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphW ) . q1) & p2 = ((sn -FanMorphW ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: sn < 1 and

       A2: (q1 `1 ) < 0 and

       A3: ((q1 `2 ) / |.q1.|) >= sn and

       A4: (q2 `1 ) < 0 and

       A5: ((q2 `2 ) / |.q2.|) >= sn and

       A6: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      

       A7: (((q1 `2 ) / |.q1.|) - sn) < (((q2 `2 ) / |.q2.|) - sn) & (1 - sn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 149;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((sn -FanMorphW ) . q1) and

       A9: p2 = ((sn -FanMorphW ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th33;

      p2 = |[( |.q2.| * ( - ( sqrt (1 - (((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)) ^2 ))))), ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)))]| by A4, A5, A9, Th16;

      then

       A11: (p2 `2 ) = ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `2 ) / |.p2.|) = ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ( - ( sqrt (1 - (((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)) ^2 ))))), ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)))]| by A2, A3, A8, Th16;

      then

       A13: (p1 `2 ) = ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th33;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `2 ) / |.p1.|) = ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:45

    

     Th45: for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < sn & (q1 `1 ) < 0 & ((q1 `2 ) / |.q1.|) < sn & (q2 `1 ) < 0 & ((q2 `2 ) / |.q2.|) < sn & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphW ) . q1) & p2 = ((sn -FanMorphW ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: (q1 `1 ) < 0 and

       A3: ((q1 `2 ) / |.q1.|) < sn and

       A4: (q2 `1 ) < 0 and

       A5: ((q2 `2 ) / |.q2.|) < sn and

       A6: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      

       A7: (((q1 `2 ) / |.q1.|) - sn) < (((q2 `2 ) / |.q2.|) - sn) & (1 + sn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 148;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((sn -FanMorphW ) . q1) and

       A9: p2 = ((sn -FanMorphW ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th33;

      p2 = |[( |.q2.| * ( - ( sqrt (1 - (((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)) ^2 ))))), ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)))]| by A4, A5, A9, Th17;

      then

       A11: (p2 `2 ) = ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `2 ) / |.p2.|) = ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ( - ( sqrt (1 - (((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)) ^2 ))))), ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)))]| by A2, A3, A8, Th17;

      then

       A13: (p1 `2 ) = ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th33;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `2 ) / |.p1.|) = ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:46

    for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (q1 `1 ) < 0 & (q2 `1 ) < 0 & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphW ) . q1) & p2 = ((sn -FanMorphW ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1 and

       A3: (q1 `1 ) < 0 and

       A4: (q2 `1 ) < 0 and

       A5: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A6: p1 = ((sn -FanMorphW ) . q1) and

       A7: p2 = ((sn -FanMorphW ) . q2);

      now

        per cases ;

          case ((q1 `2 ) / |.q1.|) >= sn & ((q2 `2 ) / |.q2.|) >= sn;

          hence thesis by A2, A3, A4, A5, A6, A7, Th44;

        end;

          case ((q1 `2 ) / |.q1.|) >= sn & ((q2 `2 ) / |.q2.|) < sn;

          hence thesis by A5, XXREAL_0: 2;

        end;

          case

           A8: ((q1 `2 ) / |.q1.|) < sn & ((q2 `2 ) / |.q2.|) >= sn;

          then (p2 `2 ) >= 0 by A2, A4, A7, Th42;

          then

           A9: ((p2 `2 ) / |.p2.|) >= 0 ;

          (p1 `2 ) < 0 by A1, A3, A6, A8, Th43;

          hence thesis by A9, Lm1, JGRAPH_2: 3, XREAL_1: 141;

        end;

          case ((q1 `2 ) / |.q1.|) < sn & ((q2 `2 ) / |.q2.|) < sn;

          hence thesis by A1, A3, A4, A5, A6, A7, Th45;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_4:47

    for sn be Real, q be Point of ( TOP-REAL 2) st (q `1 ) < 0 & ((q `2 ) / |.q.|) = sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphW ) . q) holds (p `1 ) < 0 & (p `2 ) = 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: (q `1 ) < 0 and

       A2: ((q `2 ) / |.q.|) = sn;

      let p be Point of ( TOP-REAL 2);

      

       A3: |.q.| > 0 by A1, Lm1, JGRAPH_2: 3;

      assume p = ((sn -FanMorphW ) . q);

      then

       A4: p = |[( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, Th16;

      then (p `1 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))) by EUCLID: 52;

      hence thesis by A2, A4, A3, Lm13, EUCLID: 52, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:48

    for sn be Real holds ( 0. ( TOP-REAL 2)) = ((sn -FanMorphW ) . ( 0. ( TOP-REAL 2))) by Th16, JGRAPH_2: 3;

    begin

    definition

      let s be Real, q be Point of ( TOP-REAL 2);

      :: JGRAPH_4:def4

      func FanN (s,q) -> Point of ( TOP-REAL 2) equals

      : Def4: ( |.q.| * |[((((q `1 ) / |.q.|) - s) / (1 - s)), ( sqrt (1 - (((((q `1 ) / |.q.|) - s) / (1 - s)) ^2 )))]|) if ((q `1 ) / |.q.|) >= s & (q `2 ) > 0 ,

( |.q.| * |[((((q `1 ) / |.q.|) - s) / (1 + s)), ( sqrt (1 - (((((q `1 ) / |.q.|) - s) / (1 + s)) ^2 )))]|) if ((q `1 ) / |.q.|) < s & (q `2 ) > 0

      otherwise q;

      correctness ;

    end

    definition

      let c be Real;

      :: JGRAPH_4:def5

      func c -FanMorphN -> Function of ( TOP-REAL 2), ( TOP-REAL 2) means

      : Def5: for q be Point of ( TOP-REAL 2) holds (it . q) = ( FanN (c,q));

      existence

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanN (c,$1));

        thus ex IT be Function of ( TOP-REAL 2), ( TOP-REAL 2) st for q be Point of ( TOP-REAL 2) holds (IT . q) = F(q) from FUNCT_2:sch 4;

      end;

      uniqueness

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanN (c,$1));

        thus for a,b be Function of ( TOP-REAL 2), ( TOP-REAL 2) st (for q be Point of ( TOP-REAL 2) holds (a . q) = F(q)) & (for q be Point of ( TOP-REAL 2) holds (b . q) = F(q)) holds a = b from BINOP_2:sch 1;

      end;

    end

    theorem :: JGRAPH_4:49

    

     Th49: for cn be Real holds (((q `1 ) / |.q.|) >= cn & (q `2 ) > 0 implies ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|) & ((q `2 ) <= 0 implies ((cn -FanMorphN ) . q) = q)

    proof

      let cn be Real;

      hereby

        assume ((q `1 ) / |.q.|) >= cn & (q `2 ) > 0 ;

        

        then ( FanN (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 - cn)), ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))]|) by Def4

        .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by EUCLID: 58;

        hence ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by Def5;

      end;

      assume

       A1: (q `2 ) <= 0 ;

      ((cn -FanMorphN ) . q) = ( FanN (cn,q)) by Def5;

      hence thesis by A1, Def4;

    end;

    theorem :: JGRAPH_4:50

    

     Th50: for cn be Real holds (((q `1 ) / |.q.|) <= cn & (q `2 ) > 0 implies ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|)

    proof

      let cn be Real;

      assume that

       A1: ((q `1 ) / |.q.|) <= cn and

       A2: (q `2 ) > 0 ;

      per cases by A1, XXREAL_0: 1;

        suppose ((q `1 ) / |.q.|) < cn;

        

        then ( FanN (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 + cn)), ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))]|) by A2, Def4

        .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by EUCLID: 58;

        hence thesis by Def5;

      end;

        suppose

         A3: ((q `1 ) / |.q.|) = cn;

        then ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = 0 ;

        hence thesis by A2, A3, Th49;

      end;

    end;

    theorem :: JGRAPH_4:51

    

     Th51: for cn be Real st ( - 1) < cn & cn < 1 holds (((q `1 ) / |.q.|) >= cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|) & (((q `1 ) / |.q.|) <= cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|)

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      per cases ;

        suppose

         A3: ((q `1 ) / |.q.|) >= cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose

           A4: (q `2 ) > 0 ;

          

          then ( FanN (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 - cn)), ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))]|) by A3, Def4

          .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by EUCLID: 58;

          hence thesis by A4, Def5, Th50;

        end;

          suppose

           A5: (q `2 ) <= 0 ;

          then

           A6: ((cn -FanMorphN ) . q) = q by Th49;

          

           A7: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

          

           A8: (1 - cn) > 0 by A2, XREAL_1: 149;

          

           A9: (q `2 ) = 0 by A3, A5;

           |.q.| <> 0 by A3, TOPRNS_1: 24;

          then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          then (((q `1 ) ^2 ) / ( |.q.| ^2 )) = (1 ^2 ) by A7, A9, XCMPLX_1: 60;

          then (((q `1 ) / |.q.|) ^2 ) = (1 ^2 ) by XCMPLX_1: 76;

          then

           A10: ( sqrt (((q `1 ) / |.q.|) ^2 )) = 1 by SQUARE_1: 22;

           A11:

          now

            assume (q `1 ) < 0 ;

            then ( - ((q `1 ) / |.q.|)) = 1 by A10, SQUARE_1: 23;

            hence contradiction by A1, A3;

          end;

          ( sqrt ( |.q.| ^2 )) = |.q.| by SQUARE_1: 22;

          then

           A12: |.q.| = (q `1 ) by A7, A9, A11, SQUARE_1: 22;

          then 1 = ((q `1 ) / |.q.|) by A3, TOPRNS_1: 24, XCMPLX_1: 60;

          then ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = 1 by A8, XCMPLX_1: 60;

          hence thesis by A2, A6, A9, A12, EUCLID: 53, SQUARE_1: 17, TOPRNS_1: 24, XCMPLX_1: 60;

        end;

      end;

        suppose

         A13: ((q `1 ) / |.q.|) <= cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose (q `2 ) > 0 ;

          hence thesis by Th49, Th50;

        end;

          suppose

           A14: (q `2 ) <= 0 ;

          

           A15: (1 + cn) > 0 by A1, XREAL_1: 148;

          

           A16: |.q.| <> 0 by A13, TOPRNS_1: 24;

          

           A17: (q `2 ) = 0 by A13, A14;

           |.q.| > 0 & 1 > ((q `1 ) / |.q.|) by A2, A13, Lm1, XXREAL_0: 2;

          then (1 * |.q.|) > (((q `1 ) / |.q.|) * |.q.|) by XREAL_1: 68;

          then

           A18: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & |.q.| > (q `1 ) by A13, JGRAPH_3: 1, TOPRNS_1: 24, XCMPLX_1: 87;

          then

           A19: (q `1 ) = ( - |.q.|) by A17, SQUARE_1: 40;

          then ( - 1) = ((q `1 ) / |.q.|) by A13, TOPRNS_1: 24, XCMPLX_1: 197;

          

          then

           A20: ((((q `1 ) / |.q.|) - cn) / (1 + cn)) = (( - (1 + cn)) / (1 + cn))

          .= ( - 1) by A15, XCMPLX_1: 197;

           |.q.| = ( - (q `1 )) by A17, A18, SQUARE_1: 40;

          then |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| = q by A17, A20, EUCLID: 53, SQUARE_1: 17;

          hence thesis by A1, A14, A16, A19, Th49, XCMPLX_1: 197;

        end;

      end;

        suppose (q `2 ) < 0 or q = ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

    end;

    theorem :: JGRAPH_4:52

    

     Th52: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st cn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 - cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: cn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in K1 by A7, A8, A9, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        

         A11: (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        (f . r) = ( |.r.| * ((((r `1 ) / |.r.|) - cn) / (1 - cn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:53

    

     Th53: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < cn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 + cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: ( - 1) < cn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      (1 + cn) > 0 by A1, XREAL_1: 148;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A8: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in ( dom g3) by A7, A9;

        then x in K1 by A7, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        

         A11: (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        (f . r) = ( |.r.| * ((((r `1 ) / |.r.|) - cn) / (1 + cn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      ( dom f) = ( dom g3) by A7, FUNCT_2:def 1;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:54

    

     Th54: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st cn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 - cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: cn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)) and

       A6: g3 is continuous by A4, Th10;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        

         A9: (1 - cn) > 0 by A1, XREAL_1: 149;

        assume

         A10: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A11: |.r.| <> 0 by A3, A10, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `1 ) - |.r.|) * ((r `1 ) + |.r.|)) = ( - ((r `2 ) ^2 ));

        ((r `2 ) ^2 ) >= 0 by XREAL_1: 63;

        then (r `1 ) <= |.r.| by A12, XREAL_1: 93;

        then ((r `1 ) / |.r.|) <= ( |.r.| / |.r.|) by XREAL_1: 72;

        then ((r `1 ) / |.r.|) <= 1 by A11, XCMPLX_1: 60;

        then

         A13: (((r `1 ) / |.r.|) - cn) <= (1 - cn) by XREAL_1: 9;

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A10;

         A14:

        now

          assume ((1 - cn) ^2 ) = 0 ;

          then ((1 - cn) + cn) = ( 0 + cn) by XCMPLX_1: 6;

          hence contradiction by A1;

        end;

        (cn - ((r `1 ) / |.r.|)) <= 0 by A3, A10, XREAL_1: 47;

        then ( - (cn - ((r `1 ) / |.r.|))) >= ( - (1 - cn)) by A9, XREAL_1: 24;

        then ((1 - cn) ^2 ) >= 0 & ((((r `1 ) / |.r.|) - cn) ^2 ) <= ((1 - cn) ^2 ) by A13, SQUARE_1: 49, XREAL_1: 63;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 - cn) ^2 )) <= (((1 - cn) ^2 ) / ((1 - cn) ^2 )) by XREAL_1: 72;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 - cn) ^2 )) <= 1 by A14, XCMPLX_1: 60;

        then (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )).| = (1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )) by ABSVALUE:def 1;

        then

         A15: (f . r) = ( |.r.| * ( sqrt |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )).|)) by A2, A10;

        

         A16: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        hence thesis by A5, A15, A16;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:55

    

     Th55: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < cn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 + cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: ( - 1) < cn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2));

      

       A4: (1 + cn) > 0 by A1, XREAL_1: 148;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)) and

       A6: g3 is continuous by A4, Th10;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A9;

        

         A10: ((1 + cn) ^2 ) > 0 by A4, SQUARE_1: 12;

        

         A11: |.r.| <> 0 by A3, A9, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `1 ) - |.r.|) * ((r `1 ) + |.r.|)) = ( - ((r `2 ) ^2 ));

        ((r `2 ) ^2 ) >= 0 by XREAL_1: 63;

        then ( - |.r.|) <= (r `1 ) by A12, XREAL_1: 93;

        then ((r `1 ) / |.r.|) >= (( - |.r.|) / |.r.|) by XREAL_1: 72;

        then ((r `1 ) / |.r.|) >= ( - 1) by A11, XCMPLX_1: 197;

        then (((r `1 ) / |.r.|) - cn) >= (( - 1) - cn) by XREAL_1: 9;

        then

         A13: (((r `1 ) / |.r.|) - cn) >= ( - (1 + cn));

        (cn - ((r `1 ) / |.r.|)) >= 0 by A3, A9, XREAL_1: 48;

        then ( - (cn - ((r `1 ) / |.r.|))) <= ( - 0 );

        then ((((r `1 ) / |.r.|) - cn) ^2 ) <= ((1 + cn) ^2 ) by A4, A13, SQUARE_1: 49;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 + cn) ^2 )) <= (((1 + cn) ^2 ) / ((1 + cn) ^2 )) by A4, XREAL_1: 72;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 + cn) ^2 )) <= 1 by A10, XCMPLX_1: 60;

        then (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )).| = (1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )) by ABSVALUE:def 1;

        then

         A14: (f . r) = ( |.r.| * ( sqrt |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )).|)) by A2, A9;

        

         A15: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        hence thesis by A5, A14, A15;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:56

    

     Th56: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( sqrt (1 - (cn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then (sn ^2 ) = (1 - (cn ^2 )) by SQUARE_1:def 2;

      then

       A5: ((p0 `1 ) / |.p0.|) = cn by A2, EUCLID: 52, SQUARE_1: 18;

      (p0 `2 ) > 0 by A1, A4, SQUARE_1: 25;

      then

       A6: p0 in K0 by A3, A5, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A7: ( rng ( proj2 * ((cn -FanMorphN ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A8: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) / |.p8.|) >= cn & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A9: ( dom ((cn -FanMorphN ) | K1)) c= ( dom ( proj1 * ((cn -FanMorphN ) | K1)))

      proof

        let x be object;

        assume

         A10: x in ( dom ((cn -FanMorphN ) | K1));

        then x in (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphN )) by XBOOLE_0:def 4;

        then

         A11: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphN ) . x) in ( rng (cn -FanMorphN )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphN ) | K1) . x) = ((cn -FanMorphN ) . x) by A10, FUNCT_1: 47;

        hence thesis by A10, A11, FUNCT_1: 11;

      end;

      

       A12: ( rng ( proj1 * ((cn -FanMorphN ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj1 * ((cn -FanMorphN ) | K1))) c= ( dom ((cn -FanMorphN ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((cn -FanMorphN ) | K1))) = ( dom ((cn -FanMorphN ) | K1)) by A9, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj1 * ((cn -FanMorphN ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A12, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A13: ( dom ((cn -FanMorphN ) | K1)) = (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A14: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A15: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A14;

        then

         A16: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| by A3, Th51;

        (((cn -FanMorphN ) | K1) . p) = ((cn -FanMorphN ) . p) by A15, A14, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]|) by A15, A13, A14, A16, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A17: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)));

      

       A18: ( dom ((cn -FanMorphN ) | K1)) c= ( dom ( proj2 * ((cn -FanMorphN ) | K1)))

      proof

        let x be object;

        assume

         A19: x in ( dom ((cn -FanMorphN ) | K1));

        then x in (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphN )) by XBOOLE_0:def 4;

        then

         A20: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphN ) . x) in ( rng (cn -FanMorphN )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphN ) | K1) . x) = ((cn -FanMorphN ) . x) by A19, FUNCT_1: 47;

        hence thesis by A19, A20, FUNCT_1: 11;

      end;

      ( dom ( proj2 * ((cn -FanMorphN ) | K1))) c= ( dom ((cn -FanMorphN ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((cn -FanMorphN ) | K1))) = ( dom ((cn -FanMorphN ) | K1)) by A18, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj2 * ((cn -FanMorphN ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A7, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A21: ( dom ((cn -FanMorphN ) | K1)) = (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A22: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A23: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A22;

        then

         A24: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| by A3, Th51;

        (((cn -FanMorphN ) | K1) . p) = ((cn -FanMorphN ) . p) by A23, A22, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]|) by A23, A21, A22, A24, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A25: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A26: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A26;

        hence thesis;

      end;

      then

       A27: f1 is continuous by A3, A25, Th54;

      

       A28: for x,y,s,r be Real st |[x, y]| in K1 & s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|) holds (f . |[x, y]|) = |[s, r]|

      proof

        let x,y,s,r be Real;

        assume that

         A29: |[x, y]| in K1 and

         A30: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A31: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A29;

        

         A32: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A33: (f1 . p99) = ( |.p99.| * ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 - cn)) ^2 )))) by A25, A29;

        (((cn -FanMorphN ) | K0) . |[x, y]|) = ((cn -FanMorphN ) . |[x, y]|) by A29, FUNCT_1: 49

        .= |[( |.p99.| * ((((p99 `1 ) / |.p99.|) - cn) / (1 - cn))), ( |.p99.| * ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 - cn)) ^2 ))))]| by A3, A31, Th51

        .= |[s, r]| by A17, A29, A30, A32, A33;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A34: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A34;

        hence thesis;

      end;

      then f2 is continuous by A3, A17, Th52;

      hence thesis by A6, A8, A27, A28, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:57

    

     Th57: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( sqrt (1 - (cn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then (sn ^2 ) = (1 - (cn ^2 )) by SQUARE_1:def 2;

      then

       A5: ((p0 `1 ) / |.p0.|) = cn by A2, EUCLID: 52, SQUARE_1: 18;

      (p0 `2 ) > 0 by A1, A4, SQUARE_1: 25;

      then

       A6: p0 in K0 by A3, A5, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A7: ( rng ( proj2 * ((cn -FanMorphN ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A8: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) / |.p8.|) <= cn & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A9: ( dom ((cn -FanMorphN ) | K1)) c= ( dom ( proj1 * ((cn -FanMorphN ) | K1)))

      proof

        let x be object;

        assume

         A10: x in ( dom ((cn -FanMorphN ) | K1));

        then x in (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphN )) by XBOOLE_0:def 4;

        then

         A11: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphN ) . x) in ( rng (cn -FanMorphN )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphN ) | K1) . x) = ((cn -FanMorphN ) . x) by A10, FUNCT_1: 47;

        hence thesis by A10, A11, FUNCT_1: 11;

      end;

      

       A12: ( rng ( proj1 * ((cn -FanMorphN ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj1 * ((cn -FanMorphN ) | K1))) c= ( dom ((cn -FanMorphN ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((cn -FanMorphN ) | K1))) = ( dom ((cn -FanMorphN ) | K1)) by A9, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj1 * ((cn -FanMorphN ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A12, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A13: ( dom ((cn -FanMorphN ) | K1)) = (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A14: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A15: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A14;

        then

         A16: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| by A3, Th51;

        (((cn -FanMorphN ) | K1) . p) = ((cn -FanMorphN ) . p) by A15, A14, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]|) by A15, A13, A14, A16, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A17: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)));

      

       A18: ( dom ((cn -FanMorphN ) | K1)) c= ( dom ( proj2 * ((cn -FanMorphN ) | K1)))

      proof

        let x be object;

        assume

         A19: x in ( dom ((cn -FanMorphN ) | K1));

        then x in (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphN )) by XBOOLE_0:def 4;

        then

         A20: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphN ) . x) in ( rng (cn -FanMorphN )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphN ) | K1) . x) = ((cn -FanMorphN ) . x) by A19, FUNCT_1: 47;

        hence thesis by A19, A20, FUNCT_1: 11;

      end;

      ( dom ( proj2 * ((cn -FanMorphN ) | K1))) c= ( dom ((cn -FanMorphN ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((cn -FanMorphN ) | K1))) = ( dom ((cn -FanMorphN ) | K1)) by A18, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj2 * ((cn -FanMorphN ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A7, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A21: ( dom ((cn -FanMorphN ) | K1)) = (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A22: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A23: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A22;

        then

         A24: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| by A3, Th51;

        (((cn -FanMorphN ) | K1) . p) = ((cn -FanMorphN ) . p) by A23, A22, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]|) by A23, A21, A22, A24, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A25: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A26: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A26;

        hence thesis;

      end;

      then

       A27: f1 is continuous by A3, A25, Th55;

      

       A28: for x,y,s,r be Real st |[x, y]| in K1 & s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|) holds (f . |[x, y]|) = |[s, r]|

      proof

        let x,y,s,r be Real;

        assume that

         A29: |[x, y]| in K1 and

         A30: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A31: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A29;

        

         A32: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A33: (f1 . p99) = ( |.p99.| * ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 + cn)) ^2 )))) by A25, A29;

        (((cn -FanMorphN ) | K0) . |[x, y]|) = ((cn -FanMorphN ) . |[x, y]|) by A29, FUNCT_1: 49

        .= |[( |.p99.| * ((((p99 `1 ) / |.p99.|) - cn) / (1 + cn))), ( |.p99.| * ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 + cn)) ^2 ))))]| by A3, A31, Th51

        .= |[s, r]| by A17, A29, A30, A32, A33;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A34: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A34;

        hence thesis;

      end;

      then f2 is continuous by A3, A17, Th53;

      hence thesis by A6, A8, A27, A28, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:58

    

     Th58: for cn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `1 ) >= (cn * |.p.|) & (p `2 ) >= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `1 ) >= (sn * |.p.|) & (p `2 ) >= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) >= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm8, JORDAN6: 7;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:59

    

     Th59: for cn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `1 ) <= (cn * |.p.|) & (p `2 ) >= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `1 ) <= (sn * |.p.|) & (p `2 ) >= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) <= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm10, JORDAN6: 7;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:60

    

     Th60: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( sqrt (1 - (cn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      assume

       A2: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A3: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then

       A4: (p0 `2 ) > 0 by A1, SQUARE_1: 25;

      then p0 in K0 by A2, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      p0 <> ( 0. ( TOP-REAL 2)) by A1, A3, JGRAPH_2: 3, SQUARE_1: 25;

      then not p0 in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A2, XBOOLE_0:def 5;

      

       A5: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A6: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      

       A7: D <> {} ;

      (sn ^2 ) = (1 - (cn ^2 )) by A3, SQUARE_1:def 2;

      then

       A8: ((p0 `1 ) / |.p0.|) = cn by A6, EUCLID: 52, SQUARE_1: 18;

      then

       A9: p0 in { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A4, JGRAPH_2: 3;

      

       A10: { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A2;

      end;

      

       A11: { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A2;

      end;

      then

      reconsider K00 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A9, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A12: ( rng (f | K00)) c= D;

      p0 in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A4, A8, JGRAPH_2: 3;

      then

      reconsider K11 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A10, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A13: ( rng (f | K11)) c= D;

      the carrier of (( TOP-REAL 2) | B0) = the carrier of (( TOP-REAL 2) | D);

      

      then

       A14: ( dom f) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1

      .= K1 by PRE_TOPC: 8;

      

      then ( dom (f | K00)) = K00 by A11, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K00) by PRE_TOPC: 8;

      then

      reconsider f1 = (f | K00) as Function of ((( TOP-REAL 2) | K1) | K00), (( TOP-REAL 2) | D) by A12, FUNCT_2: 2;

      ( dom (f | K11)) = K11 by A10, A14, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K11) by PRE_TOPC: 8;

      then

      reconsider f2 = (f | K11) as Function of ((( TOP-REAL 2) | K1) | K11), (( TOP-REAL 2) | D) by A13, FUNCT_2: 2;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) / |.$1.|) >= cn & ($1 `2 ) >= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A15: ( dom f2) = the carrier of ((( TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1

      .= K11 by PRE_TOPC: 8;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K001 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A9;

      

       A16: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= (cn * |.$1.|) & ($1 `2 ) >= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K003 = { p : (p `1 ) >= (cn * |.p.|) & (p `2 ) >= 0 } as Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) / |.$1.|) <= cn & ($1 `2 ) >= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A17: { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      

       A18: ( rng ((cn -FanMorphN ) | K001)) c= K1

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphN ) | K001));

        then

        consider x be object such that

         A19: x in ( dom ((cn -FanMorphN ) | K001)) and

         A20: y = (((cn -FanMorphN ) | K001) . x) by FUNCT_1:def 3;

        x in ( dom (cn -FanMorphN )) by A19, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A21: y = ((cn -FanMorphN ) . q) by A19, A20, FUNCT_1: 47;

        ( dom ((cn -FanMorphN ) | K001)) = (( dom (cn -FanMorphN )) /\ K001) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K001) by FUNCT_2:def 1

        .= K001 by XBOOLE_1: 28;

        then

         A22: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `1 ) / |.p2.|) >= cn & (p2 `2 ) >= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A19;

        then

         A23: (((q `1 ) / |.q.|) - cn) >= 0 by XREAL_1: 48;

         |.q.| <> 0 by A22, TOPRNS_1: 24;

        then

         A24: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|;

        

         A25: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

        

         A26: (1 - cn) > 0 by A2, XREAL_1: 149;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then ((q `1 ) ^2 ) <= ( |.q.| ^2 ) by JGRAPH_3: 1;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A24, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

        then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A26, XREAL_1: 72;

        then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A26, XCMPLX_1: 197;

        then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A26, A23, SQUARE_1: 49;

        then

         A27: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

        then

         A28: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

        ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by A27, SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

        then

         A29: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A30: (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

        

        then

         A31: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A28, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A25, A31;

        then

         A32: q4 <> ( 0. ( TOP-REAL 2)) by A24, TOPRNS_1: 23;

        ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A2, A22, Th51;

        hence thesis by A2, A21, A30, A29, A32;

      end;

      

       A33: ( dom (cn -FanMorphN )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((cn -FanMorphN ) | K001)) = K001 by RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K001) by PRE_TOPC: 8;

      then

      reconsider f3 = ((cn -FanMorphN ) | K001) as Function of (( TOP-REAL 2) | K001), (( TOP-REAL 2) | K1) by A5, A18, FUNCT_2: 2;

      

       A34: K003 is closed by Th58;

      K1 c= D

      proof

        let x be object;

        assume

         A35: x in K1;

        then ex p6 be Point of ( TOP-REAL 2) st p6 = x & (p6 `2 ) >= 0 & p6 <> ( 0. ( TOP-REAL 2)) by A2;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A2, A35, XBOOLE_0:def 5;

      end;

      then D = (K1 \/ D) by XBOOLE_1: 12;

      then

       A36: (( TOP-REAL 2) | K1) is SubSpace of (( TOP-REAL 2) | D) by TOPMETR: 4;

      p0 in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A4, A8, JGRAPH_2: 3;

      then

      reconsider K111 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A17;

      

       A37: ( rng ((cn -FanMorphN ) | K111)) c= K1

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphN ) | K111));

        then

        consider x be object such that

         A38: x in ( dom ((cn -FanMorphN ) | K111)) and

         A39: y = (((cn -FanMorphN ) | K111) . x) by FUNCT_1:def 3;

        x in ( dom (cn -FanMorphN )) by A38, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A40: y = ((cn -FanMorphN ) . q) by A38, A39, FUNCT_1: 47;

        ( dom ((cn -FanMorphN ) | K111)) = (( dom (cn -FanMorphN )) /\ K111) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K111) by FUNCT_2:def 1

        .= K111 by XBOOLE_1: 28;

        then

         A41: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `1 ) / |.p2.|) <= cn & (p2 `2 ) >= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A38;

        then

         A42: (((q `1 ) / |.q.|) - cn) <= 0 by XREAL_1: 47;

         |.q.| <> 0 by A41, TOPRNS_1: 24;

        then

         A43: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|;

        

         A44: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

        

         A45: (1 + cn) > 0 by A2, XREAL_1: 148;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A43, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        then (( - (1 + cn)) / (1 + cn)) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A45, XREAL_1: 72;

        then ( - 1) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A45, XCMPLX_1: 197;

        then

         A46: (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) <= (1 ^2 ) by A45, A42, SQUARE_1: 49;

        then

         A47: (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

        (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by A46, XREAL_1: 48;

        then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XCMPLX_1: 187;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

        then

         A48: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A49: (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

        

        then

         A50: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A47, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A44, A50;

        then

         A51: q4 <> ( 0. ( TOP-REAL 2)) by A43, TOPRNS_1: 23;

        ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A2, A41, Th51;

        hence thesis by A2, A40, A49, A48, A51;

      end;

      ( dom ((cn -FanMorphN ) | K111)) = K111 by A33, RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K111) by PRE_TOPC: 8;

      then

      reconsider f4 = ((cn -FanMorphN ) | K111) as Function of (( TOP-REAL 2) | K111), (( TOP-REAL 2) | K1) by A16, A37, FUNCT_2: 2;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K11) = (( TOP-REAL 2) | K111) & f2 = f4 by A2, FUNCT_1: 51, GOBOARD9: 2;

      then

       A52: f2 is continuous by A2, A36, Th57, PRE_TOPC: 26;

      

       A53: the carrier of (( TOP-REAL 2) | K1) = K0 by PRE_TOPC: 8;

      set T1 = ((( TOP-REAL 2) | K1) | K00), T2 = ((( TOP-REAL 2) | K1) | K11);

      

       A54: ( [#] ((( TOP-REAL 2) | K1) | K11)) = K11 by PRE_TOPC:def 5;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= (cn * |.$1.|) & ($1 `2 ) >= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K004 = { p : (p `1 ) <= (cn * |.p.|) & (p `2 ) >= 0 } as Subset of ( TOP-REAL 2);

      

       A55: (K004 /\ K1) c= K11

      proof

        let x be object;

        assume

         A56: x in (K004 /\ K1);

        then x in K004 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A57: q1 = x and

         A58: (q1 `1 ) <= (cn * |.q1.|) and (q1 `2 ) >= 0 ;

        x in K1 by A56, XBOOLE_0:def 4;

        then

         A59: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `2 ) >= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A2;

        ((q1 `1 ) / |.q1.|) <= ((cn * |.q1.|) / |.q1.|) by A58, XREAL_1: 72;

        then ((q1 `1 ) / |.q1.|) <= cn by A57, A59, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A57, A59;

      end;

      

       A60: K004 is closed by Th59;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K00) = (( TOP-REAL 2) | K001) & f1 = f3 by A2, FUNCT_1: 51, GOBOARD9: 2;

      then

       A61: f1 is continuous by A2, A36, Th56, PRE_TOPC: 26;

      

       A62: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

      K11 c= (K004 /\ K1)

      proof

        let x be object;

        assume x in K11;

        then

        consider p such that

         A63: p = x and

         A64: ((p `1 ) / |.p.|) <= cn and

         A65: (p `2 ) >= 0 and

         A66: p <> ( 0. ( TOP-REAL 2));

        (((p `1 ) / |.p.|) * |.p.|) <= (cn * |.p.|) by A64, XREAL_1: 64;

        then (p `1 ) <= (cn * |.p.|) by A66, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A67: x in K004 by A63, A65;

        x in K1 by A2, A63, A65, A66;

        hence thesis by A67, XBOOLE_0:def 4;

      end;

      then K11 = (K004 /\ ( [#] (( TOP-REAL 2) | K1))) by A62, A55, XBOOLE_0:def 10;

      then

       A68: K11 is closed by A60, PRE_TOPC: 13;

      

       A69: (K003 /\ K1) c= K00

      proof

        let x be object;

        assume

         A70: x in (K003 /\ K1);

        then x in K003 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A71: q1 = x and

         A72: (q1 `1 ) >= (cn * |.q1.|) and (q1 `2 ) >= 0 ;

        x in K1 by A70, XBOOLE_0:def 4;

        then

         A73: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `2 ) >= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A2;

        ((q1 `1 ) / |.q1.|) >= ((cn * |.q1.|) / |.q1.|) by A72, XREAL_1: 72;

        then ((q1 `1 ) / |.q1.|) >= cn by A71, A73, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A71, A73;

      end;

      K00 c= (K003 /\ K1)

      proof

        let x be object;

        assume x in K00;

        then

        consider p such that

         A74: p = x and

         A75: ((p `1 ) / |.p.|) >= cn and

         A76: (p `2 ) >= 0 and

         A77: p <> ( 0. ( TOP-REAL 2));

        (((p `1 ) / |.p.|) * |.p.|) >= (cn * |.p.|) by A75, XREAL_1: 64;

        then (p `1 ) >= (cn * |.p.|) by A77, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A78: x in K003 by A74, A76;

        x in K1 by A2, A74, A76, A77;

        hence thesis by A78, XBOOLE_0:def 4;

      end;

      then K00 = (K003 /\ ( [#] (( TOP-REAL 2) | K1))) by A62, A69, XBOOLE_0:def 10;

      then

       A79: K00 is closed by A34, PRE_TOPC: 13;

      

       A80: ( [#] ((( TOP-REAL 2) | K1) | K00)) = K00 by PRE_TOPC:def 5;

      

       A81: for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f1 . p) = (f2 . p)

      proof

        let p be object;

        assume

         A82: p in (( [#] T1) /\ ( [#] T2));

        then p in K00 by A80, XBOOLE_0:def 4;

        

        hence (f1 . p) = (f . p) by FUNCT_1: 49

        .= (f2 . p) by A54, A82, FUNCT_1: 49;

      end;

      

       A83: K1 c= (K00 \/ K11)

      proof

        let x be object;

        assume x in K1;

        then

        consider p such that

         A84: p = x & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) by A2;

        per cases ;

          suppose ((p `1 ) / |.p.|) >= cn;

          then x in K00 by A84;

          hence thesis by XBOOLE_0:def 3;

        end;

          suppose ((p `1 ) / |.p.|) < cn;

          then x in K11 by A84;

          hence thesis by XBOOLE_0:def 3;

        end;

      end;

      then (( [#] ((( TOP-REAL 2) | K1) | K00)) \/ ( [#] ((( TOP-REAL 2) | K1) | K11))) = ( [#] (( TOP-REAL 2) | K1)) by A80, A54, A62, XBOOLE_0:def 10;

      then

      consider h be Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D) such that

       A85: h = (f1 +* f2) and

       A86: h is continuous by A80, A54, A79, A68, A61, A52, A81, JGRAPH_2: 1;

      

       A87: ( dom h) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A88: ( dom f1) = the carrier of ((( TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1

      .= K00 by PRE_TOPC: 8;

      

       A89: for y be object st y in ( dom h) holds (h . y) = (f . y)

      proof

        let y be object;

        assume

         A90: y in ( dom h);

        per cases by A83, A87, A53, A90, XBOOLE_0:def 3;

          suppose

           A91: y in K00 & not y in K11;

          then y in (( dom f1) \/ ( dom f2)) by A88, XBOOLE_0:def 3;

          

          hence (h . y) = (f1 . y) by A15, A85, A91, FUNCT_4:def 1

          .= (f . y) by A91, FUNCT_1: 49;

        end;

          suppose

           A92: y in K11;

          then y in (( dom f1) \/ ( dom f2)) by A15, XBOOLE_0:def 3;

          

          hence (h . y) = (f2 . y) by A15, A85, A92, FUNCT_4:def 1

          .= (f . y) by A92, FUNCT_1: 49;

        end;

      end;

      K0 = the carrier of (( TOP-REAL 2) | K0) by PRE_TOPC: 8

      .= ( dom f) by A7, FUNCT_2:def 1;

      hence thesis by A86, A87, A89, FUNCT_1: 2, PRE_TOPC: 8;

    end;

    theorem :: JGRAPH_4:61

    

     Th61: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( sqrt (1 - (cn ^2 )));

      set p0 = |[cn, ( - sn)]|;

      assume

       A1: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then (p0 `2 ) = ( - sn) & ( - ( - sn)) > 0 by EUCLID: 52, SQUARE_1: 25;

      then

       A2: (p0 `2 ) < 0 ;

      then p0 in K0 by A1, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

       not p0 in {( 0. ( TOP-REAL 2))} by A2, JGRAPH_2: 3, TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A1, XBOOLE_0:def 5;

      

       A3: K1 c= D

      proof

        let x be object;

        assume x in K1;

        then

        consider p2 be Point of ( TOP-REAL 2) such that

         A4: p2 = x and (p2 `2 ) <= 0 and

         A5: p2 <> ( 0. ( TOP-REAL 2)) by A1;

         not p2 in {( 0. ( TOP-REAL 2))} by A5, TARSKI:def 1;

        hence thesis by A1, A4, XBOOLE_0:def 5;

      end;

      for p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D) st (f . p) in V & V is open holds ex W be Subset of (( TOP-REAL 2) | K1) st p in W & W is open & (f .: W) c= V

      proof

        let p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D);

        assume that

         A6: (f . p) in V and

         A7: V is open;

        consider V2 be Subset of ( TOP-REAL 2) such that

         A8: V2 is open and

         A9: (V2 /\ ( [#] (( TOP-REAL 2) | D))) = V by A7, TOPS_2: 24;

        reconsider W2 = (V2 /\ ( [#] (( TOP-REAL 2) | K1))) as Subset of (( TOP-REAL 2) | K1);

        

         A10: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

        then

         A11: (f . p) = ((cn -FanMorphN ) . p) by A1, FUNCT_1: 49;

        

         A12: (f .: W2) c= V

        proof

          let y be object;

          assume y in (f .: W2);

          then

          consider x be object such that

           A13: x in ( dom f) and

           A14: x in W2 and

           A15: y = (f . x) by FUNCT_1:def 6;

          f is Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D);

          then ( dom f) = K1 by A10, FUNCT_2:def 1;

          then

          consider p4 be Point of ( TOP-REAL 2) such that

           A16: x = p4 and

           A17: (p4 `2 ) <= 0 and p4 <> ( 0. ( TOP-REAL 2)) by A1, A13;

          

           A18: p4 in V2 by A14, A16, XBOOLE_0:def 4;

          p4 in ( [#] (( TOP-REAL 2) | K1)) by A13, A16;

          then p4 in D by A3, A10;

          then

           A19: p4 in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

          (f . p4) = ((cn -FanMorphN ) . p4) by A1, A10, A13, A16, FUNCT_1: 49

          .= p4 by A17, Th49;

          hence thesis by A9, A15, A16, A18, A19, XBOOLE_0:def 4;

        end;

        p in the carrier of (( TOP-REAL 2) | K1);

        then

        consider q be Point of ( TOP-REAL 2) such that

         A20: q = p and

         A21: (q `2 ) <= 0 and q <> ( 0. ( TOP-REAL 2)) by A1, A10;

        ((cn -FanMorphN ) . q) = q by A21, Th49;

        then p in V2 by A6, A9, A11, A20, XBOOLE_0:def 4;

        then

         A22: p in W2 by XBOOLE_0:def 4;

        W2 is open by A8, TOPS_2: 24;

        hence thesis by A22, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JGRAPH_4:62

    

     Th62: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0) st B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds K0 is closed

    proof

      set J0 = ( NonZero ( TOP-REAL 2));

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= 0 ;

      set I1 = { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      let B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0);

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A1: I1 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ J0) from JGRAPH_3:sch 2;

      assume B0 = J0 & K0 = I1;

      then K1 is closed & K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, JORDAN6: 7, PRE_TOPC:def 5;

      hence thesis by PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_4:63

    

     Th63: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0) st B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds K0 is closed

    proof

      set J0 = ( NonZero ( TOP-REAL 2));

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= 0 ;

      set I1 = { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      let B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0);

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A1: I1 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ J0) from JGRAPH_3:sch 2;

      assume B0 = J0 & K0 = I1;

      then K1 is closed & K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, JORDAN6: 8, PRE_TOPC:def 5;

      hence thesis by PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_4:64

    

     Th64: for cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th60;

    end;

    theorem :: JGRAPH_4:65

    

     Th65: for cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphN ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th61;

    end;

    theorem :: JGRAPH_4:66

    

     Th66: for cn be Real, p be Point of ( TOP-REAL 2) holds |.((cn -FanMorphN ) . p).| = |.p.|

    proof

      let cn be Real, p be Point of ( TOP-REAL 2);

      set f = (cn -FanMorphN );

      set z = (f . p);

      reconsider q = p as Point of ( TOP-REAL 2);

      reconsider qz = z as Point of ( TOP-REAL 2);

      per cases ;

        suppose

         A1: ((q `1 ) / |.q.|) >= cn & (q `2 ) > 0 ;

        then

         A2: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by Th49;

        then

         A3: (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

        

         A4: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by A2, EUCLID: 52;

        

         A5: (((q `1 ) / |.q.|) - cn) >= 0 by A1, XREAL_1: 48;

        

         A6: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         |.q.| <> 0 by A1, JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A6, XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A7, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then

         A8: (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        per cases ;

          suppose

           A9: (1 - cn) = 0 ;

          

           A10: ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = ((((q `1 ) / |.q.|) - cn) * ((1 - cn) " )) by XCMPLX_0:def 9

          .= ((((q `1 ) / |.q.|) - cn) * 0 ) by A9

          .= 0 ;

          then (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) = 1;

          

          then ((cn -FanMorphN ) . q) = |[( |.q.| * 0 ), ( |.q.| * 1)]| by A1, A10, Th49, SQUARE_1: 18

          .= |[ 0 , |.q.|]|;

          then (((cn -FanMorphN ) . q) `2 ) = |.q.| & (((cn -FanMorphN ) . q) `1 ) = 0 by EUCLID: 52;

          

          then |.((cn -FanMorphN ) . p).| = ( sqrt (( |.q.| ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A11: (1 - cn) <> 0 ;

          per cases by A11;

            suppose

             A12: (1 - cn) > 0 ;

            ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by A8, XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A12, XREAL_1: 72;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A12, XCMPLX_1: 197;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A5, A12, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A13: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A14: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 )) by A3

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A13, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A4, A14;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A15: (1 - cn) < 0 ;

            ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A1, SQUARE_1: 12, XREAL_1: 8;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, A6, XREAL_1: 74;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then

             A16: 1 > ((q `1 ) / |.p.|) by SQUARE_1: 52;

            (((q `1 ) / |.q.|) - cn) >= 0 by A1, XREAL_1: 48;

            hence thesis by A15, A16, XREAL_1: 9;

          end;

        end;

      end;

        suppose

         A17: ((q `1 ) / |.q.|) < cn & (q `2 ) > 0 ;

        then |.q.| <> 0 by JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A18: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

        

         A19: (((q `1 ) / |.q.|) - cn) < 0 by A17, XREAL_1: 49;

        

         A20: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A20, XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A18, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then

         A21: (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        

         A22: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A17, Th50;

        then

         A23: (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

        

         A24: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A22, EUCLID: 52;

        per cases ;

          suppose

           A25: (1 + cn) = 0 ;

          ((((q `1 ) / |.q.|) - cn) / (1 + cn)) = ((((q `1 ) / |.q.|) - cn) * ((1 + cn) " )) by XCMPLX_0:def 9

          .= ((((q `1 ) / |.q.|) - cn) * 0 ) by A25

          .= 0 ;

          then (((cn -FanMorphN ) . q) `2 ) = |.q.| & (((cn -FanMorphN ) . q) `1 ) = 0 by A22, EUCLID: 52, SQUARE_1: 18;

          

          then |.((cn -FanMorphN ) . p).| = ( sqrt (( |.q.| ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A26: (1 + cn) <> 0 ;

          per cases by A26;

            suppose

             A27: (1 + cn) > 0 ;

            then (( - (1 + cn)) / (1 + cn)) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A21, XREAL_1: 72;

            then ( - 1) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A27, XCMPLX_1: 197;

            then (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) <= (1 ^2 ) by A19, A27, SQUARE_1: 49;

            then

             A28: (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            

             A29: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 )) by A23

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A28, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A24, A29;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A30: (1 + cn) < 0 ;

            ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A17, SQUARE_1: 12, XREAL_1: 8;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A18, A20, XREAL_1: 74;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A18, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then ( - 1) < ((q `1 ) / |.p.|) by SQUARE_1: 52;

            then

             A31: (((q `1 ) / |.q.|) - cn) > (( - 1) - cn) by XREAL_1: 9;

            ( - (1 + cn)) > ( - 0 ) by A30, XREAL_1: 24;

            hence thesis by A17, A31, XREAL_1: 49;

          end;

        end;

      end;

        suppose (q `2 ) <= 0 ;

        hence thesis by Th49;

      end;

    end;

    theorem :: JGRAPH_4:67

    

     Th67: for cn be Real, x,K0 be set st ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((cn -FanMorphN ) . x) in K0

    proof

      let cn be Real, x,K0 be set;

      assume

       A1: ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then

      consider p such that

       A2: p = x and

       A3: (p `2 ) >= 0 and

       A4: p <> ( 0. ( TOP-REAL 2));

       A5:

      now

        assume |.p.| <= 0 ;

        then |.p.| = 0 ;

        hence contradiction by A4, TOPRNS_1: 24;

      end;

      then

       A6: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

      per cases ;

        suppose

         A7: ((p `1 ) / |.p.|) <= cn;

        reconsider p9 = ((cn -FanMorphN ) . p) as Point of ( TOP-REAL 2);

        ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A3, A4, A7, Th51;

        then

         A8: (p9 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

        

         A9: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A10: (1 + cn) > 0 by A1, XREAL_1: 148;

        per cases ;

          suppose (p `2 ) = 0 ;

          hence thesis by A1, A2, Th49;

        end;

          suppose (p `2 ) <> 0 ;

          then ( 0 + ((p `1 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A9, XREAL_1: 74;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `1 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ( - 1) < ((p `1 ) / |.p.|) by SQUARE_1: 52;

          then (( - 1) - cn) < (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

          then ((( - 1) * (1 + cn)) / (1 + cn)) < ((((p `1 ) / |.p.|) - cn) / (1 + cn)) by A10, XREAL_1: 74;

          then

           A11: ( - 1) < ((((p `1 ) / |.p.|) - cn) / (1 + cn)) by A10, XCMPLX_1: 89;

          (((p `1 ) / |.p.|) - cn) <= 0 by A7, XREAL_1: 47;

          then (1 ^2 ) > (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ) by A10, A11, SQUARE_1: 50;

          then (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )) > 0 by XREAL_1: 50;

          then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) > 0 by SQUARE_1: 25;

          then ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) > 0 by A5, XREAL_1: 129;

          hence thesis by A1, A2, A8, JGRAPH_2: 3;

        end;

      end;

        suppose

         A12: ((p `1 ) / |.p.|) > cn;

        reconsider p9 = ((cn -FanMorphN ) . p) as Point of ( TOP-REAL 2);

        ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A3, A4, A12, Th51;

        then

         A13: (p9 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

        

         A14: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A15: (1 - cn) > 0 by A1, XREAL_1: 149;

        per cases ;

          suppose (p `2 ) = 0 ;

          hence thesis by A1, A2, Th49;

        end;

          suppose (p `2 ) <> 0 ;

          then ( 0 + ((p `1 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A14, XREAL_1: 74;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `1 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ((p `1 ) / |.p.|) < 1 by SQUARE_1: 52;

          then (((p `1 ) / |.p.|) - cn) < (1 - cn) by XREAL_1: 9;

          then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) < ((1 - cn) / (1 - cn)) by A15, XREAL_1: 74;

          then

           A16: ((((p `1 ) / |.p.|) - cn) / (1 - cn)) < 1 by A15, XCMPLX_1: 60;

          ( - (1 - cn)) < ( - 0 ) & (((p `1 ) / |.p.|) - cn) >= (cn - cn) by A12, A15, XREAL_1: 9, XREAL_1: 24;

          then ((( - 1) * (1 - cn)) / (1 - cn)) < ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A15, XREAL_1: 74;

          then ( - 1) < ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A15, XCMPLX_1: 89;

          then (1 ^2 ) > (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ) by A16, SQUARE_1: 50;

          then (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )) > 0 by XREAL_1: 50;

          then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) > 0 by SQUARE_1: 25;

          then (p9 `2 ) > 0 by A5, A13, XREAL_1: 129;

          hence thesis by A1, A2, JGRAPH_2: 3;

        end;

      end;

    end;

    theorem :: JGRAPH_4:68

    

     Th68: for cn be Real, x,K0 be set st ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((cn -FanMorphN ) . x) in K0

    proof

      let cn be Real, x,K0 be set;

      assume

       A1: ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then ex p st p = x & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

      hence thesis by A1, Th49;

    end;

    theorem :: JGRAPH_4:69

    

     Th69: for cn be Real, D be non empty Subset of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (D ` ) = {( 0. ( TOP-REAL 2))} holds ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((cn -FanMorphN ) | D) & h is continuous

    proof

      ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

      then

       A1: |[ 0 , 1]| in { p where p be Point of ( TOP-REAL 2) : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      set Y1 = |[ 0 , ( - 1)]|;

      reconsider B0 = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= 0 ;

      let cn be Real, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A2: ( - 1) < cn & cn < 1 and

       A3: (D ` ) = {( 0. ( TOP-REAL 2))};

      

       A4: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A5: D = (B0 ` ) by A3

      .= ( NonZero ( TOP-REAL 2)) by SUBSET_1:def 4;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A1;

      

       A6: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      

       A7: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A8: ( rng ((cn -FanMorphN ) | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphN ) | K0));

        then

        consider x be object such that

         A9: x in ( dom ((cn -FanMorphN ) | K0)) and

         A10: y = (((cn -FanMorphN ) | K0) . x) by FUNCT_1:def 3;

        x in (( dom (cn -FanMorphN )) /\ K0) by A9, RELAT_1: 61;

        then

         A11: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A11;

        ((cn -FanMorphN ) . p) = y by A10, A11, FUNCT_1: 49;

        then y in K0 by A2, A11, Th67;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A12: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      (Y1 `2 ) = ( - 1) & ( 0. ( TOP-REAL 2)) <> Y1 by EUCLID: 52, JGRAPH_2: 3;

      then

       A13: Y1 in { p where p be Point of ( TOP-REAL 2) : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      

       A14: the carrier of (( TOP-REAL 2) | D) = ( NonZero ( TOP-REAL 2)) by A5, PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= 0 ;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K1 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A13;

      

       A15: K0 is closed & K1 is closed by A5, Th62, Th63;

      ( dom ((cn -FanMorphN ) | K0)) = (( dom (cn -FanMorphN )) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by FUNCT_2:def 1

      .= K0 by A12, XBOOLE_1: 28;

      then

      reconsider f = ((cn -FanMorphN ) | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A6, A8, FUNCT_2: 2, XBOOLE_1: 1;

      

       A16: K1 = the carrier of ((( TOP-REAL 2) | D) | K1) by PRE_TOPC: 8;

      

       A17: ( rng ((cn -FanMorphN ) | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphN ) | K1));

        then

        consider x be object such that

         A18: x in ( dom ((cn -FanMorphN ) | K1)) and

         A19: y = (((cn -FanMorphN ) | K1) . x) by FUNCT_1:def 3;

        x in (( dom (cn -FanMorphN )) /\ K1) by A18, RELAT_1: 61;

        then

         A20: x in K1 by XBOOLE_0:def 4;

        K1 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A20;

        ((cn -FanMorphN ) . p) = y by A19, A20, FUNCT_1: 49;

        then y in K1 by A2, A20, Th68;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A21: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ((cn -FanMorphN ) | K1)) = (( dom (cn -FanMorphN )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by A21, XBOOLE_1: 28;

      then

      reconsider g = ((cn -FanMorphN ) | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A16, A17, FUNCT_2: 2, XBOOLE_1: 1;

      

       A22: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A23: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A24: x in D;

        then

        reconsider px = x as Point of ( TOP-REAL 2);

         not x in {( 0. ( TOP-REAL 2))} by A5, A24, XBOOLE_0:def 5;

        then (px `2 ) >= 0 & px <> ( 0. ( TOP-REAL 2)) or (px `2 ) <= 0 & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        then x in K0 or x in K1;

        hence thesis by XBOOLE_0:def 3;

      end;

      

       A25: ( dom f) = K0 by A6, FUNCT_2:def 1;

      

       A26: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A27: for x be object st x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1))) holds (f . x) = (g . x)

      proof

        let x be object;

        assume

         A28: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A26, XBOOLE_0:def 4;

        then (f . x) = ((cn -FanMorphN ) . x) by FUNCT_1: 49;

        hence thesis by A22, A28, FUNCT_1: 49;

      end;

      D = ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

      then

       A29: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A26, A22, A23, XBOOLE_0:def 10;

      

       A30: f is continuous & g is continuous by A2, A5, Th64, Th65;

      then

      consider h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) such that

       A31: h = (f +* g) and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2: 1;

      

       A32: ( dom h) = the carrier of (( TOP-REAL 2) | D) by FUNCT_2:def 1;

      

       A33: ( dom g) = K1 by A16, FUNCT_2:def 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A34: f tolerates g by A27, A25, A33, PARTFUN1:def 4;

      

       A35: for x be object st x in ( dom h) holds (h . x) = (((cn -FanMorphN ) | D) . x)

      proof

        let x be object;

        assume

         A36: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A14, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A14, A36, XBOOLE_0:def 5;

        then

         A37: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        

         A38: x in ((D ` ) ` ) by A32, A36, PRE_TOPC: 8;

        now

          per cases ;

            case

             A39: x in K0;

            

             A40: (((cn -FanMorphN ) | D) . p) = ((cn -FanMorphN ) . p) by A38, FUNCT_1: 49

            .= (f . p) by A39, FUNCT_1: 49;

            (h . p) = ((g +* f) . p) by A31, A34, FUNCT_4: 34

            .= (f . p) by A25, A39, FUNCT_4: 13;

            hence thesis by A40;

          end;

            case not x in K0;

            then not (p `2 ) >= 0 by A37;

            then

             A41: x in K1 by A37;

            (((cn -FanMorphN ) | D) . p) = ((cn -FanMorphN ) . p) by A38, FUNCT_1: 49

            .= (g . p) by A41, FUNCT_1: 49;

            hence thesis by A31, A33, A41, FUNCT_4: 13;

          end;

        end;

        hence thesis;

      end;

      ( dom (cn -FanMorphN )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((cn -FanMorphN ) | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A4, XBOOLE_1: 28;

      then (f +* g) = ((cn -FanMorphN ) | D) by A31, A32, A35, FUNCT_1: 2;

      hence thesis by A26, A22, A29, A30, A15, A27, JGRAPH_2: 1;

    end;

    theorem :: JGRAPH_4:70

    

     Th70: for cn be Real st ( - 1) < cn & cn < 1 holds ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (cn -FanMorphN ) & h is continuous

    proof

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      reconsider f = (cn -FanMorphN ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A3: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Th49, JGRAPH_2: 3;

      

       A4: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

        let p be Point of (( TOP-REAL 2) | D);

        

         A5: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A5, XBOOLE_0:def 5;

        then

         A6: not p = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        now

          per cases ;

            case

             A7: ((q `1 ) / |.q.|) >= cn & (q `2 ) >= 0 ;

            set q9 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|;

            

             A8: (q9 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

            now

              assume

               A9: q9 = ( 0. ( TOP-REAL 2));

              

               A10: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

              

              then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) = ( sqrt (1 - 0 )) by A8, A9, JGRAPH_2: 3, XCMPLX_1: 6

              .= 1 by SQUARE_1: 18;

              hence contradiction by A9, A10, EUCLID: 52, JGRAPH_2: 3;

            end;

            hence thesis by A1, A2, A3, A6, A7, Th51;

          end;

            case

             A11: ((q `1 ) / |.q.|) < cn & (q `2 ) >= 0 ;

            set q9 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|;

            

             A12: (q9 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

            now

              assume

               A13: q9 = ( 0. ( TOP-REAL 2));

              

               A14: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

              

              then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) = ( sqrt (1 - 0 )) by A12, A13, JGRAPH_2: 3, XCMPLX_1: 6

              .= 1 by SQUARE_1: 18;

              hence contradiction by A13, A14, EUCLID: 52, JGRAPH_2: 3;

            end;

            hence thesis by A1, A2, A3, A6, A11, Th51;

          end;

            case (q `2 ) < 0 ;

            then (f . p) = p by Th49;

            hence thesis by A6, Th49, JGRAPH_2: 3;

          end;

        end;

        hence thesis;

      end;

      

       A15: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

        let V be Subset of ( TOP-REAL 2);

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm11;

        assume that

         A16: (f . ( 0. ( TOP-REAL 2))) in V and

         A17: V is open;

        VV is open by A17, Lm11, PRE_TOPC: 30;

        then

        consider r be Real such that

         A18: r > 0 and

         A19: ( Ball (u0,r)) c= V by A3, A16, TOPMETR: 15;

        reconsider r as Real;

         the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

        then

        reconsider W1 = ( Ball (u0,r)) as Subset of ( TOP-REAL 2);

        

         A20: W1 is open by GOBOARD6: 3;

        

         A21: (f .: W1) c= W1

        proof

          let z be object;

          assume z in (f .: W1);

          then

          consider y be object such that

           A22: y in ( dom f) and

           A23: y in W1 and

           A24: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A22, A24, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider q = y as Point of ( TOP-REAL 2) by A22;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          ( dist (u0,qy)) < r by A23, METRIC_1: 11;

          then

           A25: |.(( 0. ( TOP-REAL 2)) - q).| < r by JGRAPH_1: 28;

          now

            per cases by JGRAPH_2: 3;

              case (q `2 ) <= 0 ;

              hence thesis by A23, A24, Th49;

            end;

              case

               A26: q <> ( 0. ( TOP-REAL 2)) & ((q `1 ) / |.q.|) >= cn & (q `2 ) >= 0 ;

              then

               A27: (((q `1 ) / |.q.|) - cn) >= 0 by XREAL_1: 48;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then

               A28: (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              

               A29: (1 - cn) > 0 by A2, XREAL_1: 149;

               |.q.| <> 0 by A26, TOPRNS_1: 24;

              then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A28, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

              then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

              then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

              then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A29, XREAL_1: 72;

              then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A29, XCMPLX_1: 197;

              then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A29, A27, SQUARE_1: 49;

              then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A30: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

              

               A31: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, A26, Th51;

              then

               A32: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by A24, EUCLID: 52;

              (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by A24, A31, EUCLID: 52;

              

              then

               A33: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

              .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A30, SQUARE_1:def 2;

              ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.q.| ^2 ) by A32, A33;

              then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

              then

               A34: |.qz.| = |.q.| by SQUARE_1: 22;

               |.( - q).| < r by A25, RLVECT_1: 4;

              then |.q.| < r by TOPRNS_1: 26;

              then |.( - qz).| < r by A34, TOPRNS_1: 26;

              then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

              then ( dist (u0,pz)) < r by JGRAPH_1: 28;

              hence thesis by METRIC_1: 11;

            end;

              case

               A35: q <> ( 0. ( TOP-REAL 2)) & ((q `1 ) / |.q.|) < cn & (q `2 ) >= 0 ;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then

               A36: (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              

               A37: (1 + cn) > 0 by A1, XREAL_1: 148;

               |.q.| <> 0 by A35, TOPRNS_1: 24;

              then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A36, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

              then ( - ( - 1)) >= ( - ((q `1 ) / |.q.|)) by XREAL_1: 24;

              then (1 + cn) >= (( - ((q `1 ) / |.q.|)) + cn) by XREAL_1: 7;

              then

               A38: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A37, XREAL_1: 185;

              (cn - ((q `1 ) / |.q.|)) >= 0 by A35, XREAL_1: 48;

              then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A37;

              then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A38, SQUARE_1: 49;

              then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A39: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

              

               A40: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A2, A35, Th51;

              then

               A41: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A24, EUCLID: 52;

              (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) by A24, A40, EUCLID: 52;

              

              then

               A42: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

              .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A39, SQUARE_1:def 2;

              ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.q.| ^2 ) by A41, A42;

              then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

              then

               A43: |.qz.| = |.q.| by SQUARE_1: 22;

               |.( - q).| < r by A25, RLVECT_1: 4;

              then |.q.| < r by TOPRNS_1: 26;

              then |.( - qz).| < r by A43, TOPRNS_1: 26;

              then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

              then ( dist (u0,pz)) < r by JGRAPH_1: 28;

              hence thesis by METRIC_1: 11;

            end;

          end;

          hence thesis;

        end;

        u0 in W1 by A18, GOBOARD6: 1;

        hence thesis by A19, A20, A21, XBOOLE_1: 1;

      end;

      

       A44: (D ` ) = {( 0. ( TOP-REAL 2))} by JGRAPH_3: 20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((cn -FanMorphN ) | D) & h is continuous by A1, A2, Th69;

      hence thesis by A3, A44, A4, A15, JGRAPH_3: 3;

    end;

    theorem :: JGRAPH_4:71

    

     Th71: for cn be Real st ( - 1) < cn & cn < 1 holds (cn -FanMorphN ) is one-to-one

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      for x1,x2 be object st x1 in ( dom (cn -FanMorphN )) & x2 in ( dom (cn -FanMorphN )) & ((cn -FanMorphN ) . x1) = ((cn -FanMorphN ) . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A3: x1 in ( dom (cn -FanMorphN )) and

         A4: x2 in ( dom (cn -FanMorphN )) and

         A5: ((cn -FanMorphN ) . x1) = ((cn -FanMorphN ) . x2);

        reconsider p2 = x2 as Point of ( TOP-REAL 2) by A4;

        reconsider p1 = x1 as Point of ( TOP-REAL 2) by A3;

        set q = p1, p = p2;

        

         A6: (1 - cn) > 0 by A2, XREAL_1: 149;

        now

          per cases by JGRAPH_2: 3;

            case

             A7: (q `2 ) <= 0 ;

            then

             A8: ((cn -FanMorphN ) . q) = q by Th49;

            now

              per cases by JGRAPH_2: 3;

                case (p `2 ) <= 0 ;

                hence thesis by A5, A8, Th49;

              end;

                case

                 A9: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 ;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]|;

                

                 A10: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                 0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then

                 A11: (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A10, XREAL_1: 72;

                

                 A12: |.p.| > 0 by A9, Lm1;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A11, XCMPLX_1: 60;

                then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `1 ) / |.p.|) by SQUARE_1: 51;

                then (1 - cn) >= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

                then ( - (1 - cn)) <= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

                then (( - (1 - cn)) / (1 - cn)) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

                then

                 A13: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

                

                 A14: (((p `1 ) / |.p.|) - cn) >= 0 by A9, XREAL_1: 48;

                

                 A15: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, A9, Th51;

                (((p `1 ) / |.p.|) - cn) >= 0 by A9, XREAL_1: 48;

                then ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A13, SQUARE_1: 49;

                then

                 A16: (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) & (q `2 ) = 0 by A5, A7, A8, A15, EUCLID: 52;

                then

                 A17: ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) = 0 by A5, A8, A15, A12, XCMPLX_1: 6;

                (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 - cn))) ^2 )) >= 0 by A16, XCMPLX_1: 187;

                then (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )) = 0 by A17, SQUARE_1: 24;

                then 1 = ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A6, A14, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - cn)) = (((p `1 ) / |.p.|) - cn) by A6, XCMPLX_1: 87;

                then (1 * |.p.|) = (p `1 ) by A12, XCMPLX_1: 87;

                then (p `2 ) = 0 by A10, XCMPLX_1: 6;

                hence thesis by A5, A8, Th49;

              end;

                case

                 A18: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) >= 0 ;

                then

                 A19: |.p.| <> 0 by TOPRNS_1: 24;

                then

                 A20: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]|;

                

                 A21: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A22: (1 + cn) > 0 by A1, XREAL_1: 148;

                

                 A23: (((p `1 ) / |.p.|) - cn) <= 0 by A18, XREAL_1: 47;

                then

                 A24: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by A22;

                

                 A25: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A2, A18, Th51;

                 0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A21, XREAL_1: 72;

                then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A20, XCMPLX_1: 60;

                then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then (( - ((p `1 ) / |.p.|)) ^2 ) <= 1;

                then 1 >= ( - ((p `1 ) / |.p.|)) by SQUARE_1: 51;

                then (1 + cn) >= (( - ((p `1 ) / |.p.|)) + cn) by XREAL_1: 7;

                then (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) <= 1 by A22, XREAL_1: 185;

                then ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A24, SQUARE_1: 49;

                then

                 A26: (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) & (q `2 ) = 0 by A5, A7, A8, A25, EUCLID: 52;

                then

                 A27: ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) = 0 by A5, A8, A25, A19, XCMPLX_1: 6;

                (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 )) >= 0 by A26, XCMPLX_1: 187;

                then (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )) = 0 by A27, SQUARE_1: 24;

                then 1 = ( sqrt (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 )) by SQUARE_1: 18;

                then 1 = ( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by A22, A23, SQUARE_1: 22;

                then 1 = (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by XCMPLX_1: 187;

                then (1 * (1 + cn)) = ( - (((p `1 ) / |.p.|) - cn)) by A22, XCMPLX_1: 87;

                then ((1 + cn) - cn) = ( - ((p `1 ) / |.p.|));

                then 1 = (( - (p `1 )) / |.p.|) by XCMPLX_1: 187;

                then (1 * |.p.|) = ( - (p `1 )) by A18, TOPRNS_1: 24, XCMPLX_1: 87;

                then (((p `1 ) ^2 ) - ((p `1 ) ^2 )) = ((p `2 ) ^2 ) by A21, XCMPLX_1: 26;

                then (p `2 ) = 0 by XCMPLX_1: 6;

                hence thesis by A5, A8, Th49;

              end;

            end;

            hence thesis;

          end;

            case

             A28: ((q `1 ) / |.q.|) >= cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

            then |.q.| <> 0 by TOPRNS_1: 24;

            then

             A29: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|;

            

             A30: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

            

             A31: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, A28, Th51;

            

             A32: (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

            now

              per cases by JGRAPH_2: 3;

                case

                 A33: (p `2 ) <= 0 ;

                then

                 A34: ((cn -FanMorphN ) . p) = p by Th49;

                

                 A35: |.q.| <> 0 by A28, TOPRNS_1: 24;

                then

                 A36: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

                

                 A37: (((q `1 ) / |.q.|) - cn) >= 0 by A28, XREAL_1: 48;

                

                 A38: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A39: (((q `1 ) / |.q.|) - cn) >= 0 by A28, XREAL_1: 48;

                

                 A40: (1 - cn) > 0 by A2, XREAL_1: 149;

                 0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A38, XREAL_1: 72;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A36, XCMPLX_1: 60;

                then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

                then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

                then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

                then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A40, XREAL_1: 72;

                then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A40, XCMPLX_1: 197;

                then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A40, A37, SQUARE_1: 49;

                then

                 A41: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p `2 ) = 0 by A5, A31, A33, A34, EUCLID: 52;

                then

                 A42: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) = 0 by A5, A31, A32, A34, A35, XCMPLX_1: 6;

                (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by A41, XCMPLX_1: 187;

                then (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) = 0 by A42, SQUARE_1: 24;

                then 1 = ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by A40, A39, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - cn)) = (((q `1 ) / |.q.|) - cn) by A40, XCMPLX_1: 87;

                then (1 * |.q.|) = (q `1 ) by A28, TOPRNS_1: 24, XCMPLX_1: 87;

                then (q `2 ) = 0 by A38, XCMPLX_1: 6;

                hence thesis by A5, A34, Th49;

              end;

                case

                 A43: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 ;

                 0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

                then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A29, XCMPLX_1: 60;

                then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

                then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

                then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

                then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

                then

                 A44: ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

                (((q `1 ) / |.q.|) - cn) >= 0 by A28, XREAL_1: 48;

                then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A44, SQUARE_1: 49;

                then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A45: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

                

                then

                 A46: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

                .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A45, SQUARE_1:def 2;

                

                 A47: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

                ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.q.| ^2 ) by A47, A46;

                then

                 A48: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

                then

                 A49: |.q4.| = |.q.| by SQUARE_1: 22;

                 0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

                then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then

                 A50: (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

                 |.p.| <> 0 by A43, TOPRNS_1: 24;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A50, XCMPLX_1: 60;

                then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `1 ) / |.p.|) by SQUARE_1: 51;

                then (1 - cn) >= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

                then ( - (1 - cn)) <= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

                then (( - (1 - cn)) / (1 - cn)) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

                then

                 A51: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

                (((p `1 ) / |.p.|) - cn) >= 0 by A43, XREAL_1: 48;

                then ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A51, SQUARE_1: 49;

                then (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A52: (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]|;

                

                 A53: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) by EUCLID: 52;

                (p4 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

                

                then

                 A54: ((p4 `2 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

                .= (( |.p.| ^2 ) * (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) by A52, SQUARE_1:def 2;

                ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.p.| ^2 ) by A53, A54;

                then

                 A55: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

                then

                 A56: |.p4.| = |.p.| by SQUARE_1: 22;

                

                 A57: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, A43, Th51;

                then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) = (( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) / |.p.|) by A5, A31, A30, A43, A53, TOPRNS_1: 24, XCMPLX_1: 89;

                then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) = ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by A5, A31, A43, A57, A48, A55, TOPRNS_1: 24, XCMPLX_1: 89;

                then (((((p `1 ) / |.p.|) - cn) / (1 - cn)) * (1 - cn)) = (((q `1 ) / |.q.|) - cn) by A6, XCMPLX_1: 87;

                then (((p `1 ) / |.p.|) - cn) = (((q `1 ) / |.q.|) - cn) by A6, XCMPLX_1: 87;

                then (((p `1 ) / |.p.|) * |.p.|) = (q `1 ) by A5, A31, A43, A57, A49, A56, TOPRNS_1: 24, XCMPLX_1: 87;

                then

                 A58: (p `1 ) = (q `1 ) by A43, TOPRNS_1: 24, XCMPLX_1: 87;

                

                 A59: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

                ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                then (p `2 ) = ( sqrt ((q `2 ) ^2 )) by A5, A31, A43, A57, A49, A56, A58, SQUARE_1: 22;

                then (p `2 ) = (q `2 ) by A28, SQUARE_1: 22;

                hence thesis by A58, A59, EUCLID: 53;

              end;

                case

                 A60: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) >= 0 ;

                then (((p `1 ) / |.p.|) - cn) < 0 by XREAL_1: 49;

                then

                 A61: ((((p `1 ) / |.p.|) - cn) / (1 + cn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]|;

                

                 A62: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) & (((q `1 ) / |.q.|) - cn) >= 0 by A28, EUCLID: 52, XREAL_1: 48;

                

                 A63: (1 - cn) > 0 by A2, XREAL_1: 149;

                ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| & |.p.| <> 0 by A1, A2, A60, Th51, TOPRNS_1: 24;

                hence thesis by A5, A31, A30, A61, A62, A63, XREAL_1: 132;

              end;

            end;

            hence thesis;

          end;

            case

             A64: ((q `1 ) / |.q.|) < cn & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

            then

             A65: |.q.| <> 0 by TOPRNS_1: 24;

            then

             A66: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|;

            

             A67: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

            

             A68: ((cn -FanMorphN ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A2, A64, Th51;

            

             A69: (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

            now

              per cases by JGRAPH_2: 3;

                case

                 A70: (p `2 ) <= 0 ;

                

                 A71: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A72: (1 + cn) > 0 by A1, XREAL_1: 148;

                 0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A71, XREAL_1: 72;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A66, XCMPLX_1: 60;

                then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then (( - ((q `1 ) / |.q.|)) ^2 ) <= 1;

                then 1 >= ( - ((q `1 ) / |.q.|)) by SQUARE_1: 51;

                then (1 + cn) >= (( - ((q `1 ) / |.q.|)) + cn) by XREAL_1: 7;

                then

                 A73: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A72, XREAL_1: 185;

                

                 A74: (((q `1 ) / |.q.|) - cn) <= 0 by A64, XREAL_1: 47;

                then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A72;

                then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A73, SQUARE_1: 49;

                then

                 A75: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A76: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

                

                 A77: ((cn -FanMorphN ) . p) = p by A70, Th49;

                ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by A75, SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p `2 ) = 0 by A5, A68, A70, A77, EUCLID: 52;

                then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) = 0 by A5, A68, A69, A65, A77, XCMPLX_1: 6;

                then (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) = 0 by A76, SQUARE_1: 24;

                then 1 = ( sqrt (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) by SQUARE_1: 18;

                then 1 = ( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A72, A74, SQUARE_1: 22;

                then 1 = (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by XCMPLX_1: 187;

                then (1 * (1 + cn)) = ( - (((q `1 ) / |.q.|) - cn)) by A72, XCMPLX_1: 87;

                then ((1 + cn) - cn) = ( - ((q `1 ) / |.q.|));

                then 1 = (( - (q `1 )) / |.q.|) by XCMPLX_1: 187;

                then (1 * |.q.|) = ( - (q `1 )) by A64, TOPRNS_1: 24, XCMPLX_1: 87;

                then (((q `1 ) ^2 ) - ((q `1 ) ^2 )) = ((q `2 ) ^2 ) by A71, XCMPLX_1: 26;

                then (q `2 ) = 0 by XCMPLX_1: 6;

                hence thesis by A5, A77, Th49;

              end;

                case

                 A78: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) >= 0 ;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]|;

                

                 A79: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) & |.q.| <> 0 by A64, EUCLID: 52, TOPRNS_1: 24;

                (((q `1 ) / |.q.|) - cn) < 0 by A64, XREAL_1: 49;

                then

                 A80: ((((q `1 ) / |.q.|) - cn) / (1 + cn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

                

                 A81: (1 - cn) > 0 by A2, XREAL_1: 149;

                ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))]| & (((p `1 ) / |.p.|) - cn) >= 0 by A1, A2, A78, Th51, XREAL_1: 48;

                hence thesis by A5, A68, A67, A80, A79, A81, XREAL_1: 132;

              end;

                case

                 A82: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) >= 0 ;

                 0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

                then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then

                 A83: (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

                

                 A84: (1 + cn) > 0 by A1, XREAL_1: 148;

                 0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

                then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

                then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A66, XCMPLX_1: 60;

                then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

                then (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

                then ( - (( - 1) - cn)) >= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

                then

                 A85: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A84, XREAL_1: 185;

                (((q `1 ) / |.q.|) - cn) <= 0 by A64, XREAL_1: 47;

                then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A84;

                then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A85, SQUARE_1: 49;

                then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A86: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (q4 `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

                

                then

                 A87: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

                .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A86, SQUARE_1:def 2;

                

                 A88: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

                set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]|;

                

                 A89: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by EUCLID: 52;

                 |.p.| <> 0 by A82, TOPRNS_1: 24;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A83, XCMPLX_1: 60;

                then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then ( - 1) <= ((p `1 ) / |.p.|) by SQUARE_1: 51;

                then (( - 1) - cn) <= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

                then ( - (( - 1) - cn)) >= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

                then

                 A90: (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) <= 1 by A84, XREAL_1: 185;

                (((p `1 ) / |.p.|) - cn) <= 0 by A82, XREAL_1: 47;

                then ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by A84;

                then ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A90, SQUARE_1: 49;

                then (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A91: (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (p4 `2 ) = ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) by EUCLID: 52;

                

                then

                 A92: ((p4 `2 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

                .= (( |.p.| ^2 ) * (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) by A91, SQUARE_1:def 2;

                ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.p.| ^2 ) by A89, A92;

                then

                 A93: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

                then

                 A94: |.p4.| = |.p.| by SQUARE_1: 22;

                ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.q.| ^2 ) by A88, A87;

                then

                 A95: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

                then

                 A96: |.q4.| = |.q.| by SQUARE_1: 22;

                

                 A97: ((cn -FanMorphN ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A2, A82, Th51;

                then ((((p `1 ) / |.p.|) - cn) / (1 + cn)) = (( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) / |.p.|) by A5, A68, A67, A82, A89, TOPRNS_1: 24, XCMPLX_1: 89;

                then ((((p `1 ) / |.p.|) - cn) / (1 + cn)) = ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A5, A68, A82, A97, A95, A93, TOPRNS_1: 24, XCMPLX_1: 89;

                then (((((p `1 ) / |.p.|) - cn) / (1 + cn)) * (1 + cn)) = (((q `1 ) / |.q.|) - cn) by A84, XCMPLX_1: 87;

                then (((p `1 ) / |.p.|) - cn) = (((q `1 ) / |.q.|) - cn) by A84, XCMPLX_1: 87;

                then (((p `1 ) / |.p.|) * |.p.|) = (q `1 ) by A5, A68, A82, A97, A96, A94, TOPRNS_1: 24, XCMPLX_1: 87;

                then

                 A98: (p `1 ) = (q `1 ) by A82, TOPRNS_1: 24, XCMPLX_1: 87;

                

                 A99: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

                ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                then (p `2 ) = ( sqrt ((q `2 ) ^2 )) by A5, A68, A82, A97, A96, A94, A98, SQUARE_1: 22;

                then (p `2 ) = (q `2 ) by A64, SQUARE_1: 22;

                hence thesis by A98, A99, EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      hence thesis by FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_4:72

    

     Th72: for cn be Real st ( - 1) < cn & cn < 1 holds (cn -FanMorphN ) is Function of ( TOP-REAL 2), ( TOP-REAL 2) & ( rng (cn -FanMorphN )) = the carrier of ( TOP-REAL 2)

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      thus (cn -FanMorphN ) is Function of ( TOP-REAL 2), ( TOP-REAL 2);

      for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (cn -FanMorphN ) holds ( rng (cn -FanMorphN )) = the carrier of ( TOP-REAL 2)

      proof

        let f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

        assume

         A3: f = (cn -FanMorphN );

        

         A4: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        the carrier of ( TOP-REAL 2) c= ( rng f)

        proof

          let y be object;

          assume y in the carrier of ( TOP-REAL 2);

          then

          reconsider p2 = y as Point of ( TOP-REAL 2);

          set q = p2;

          now

            per cases by JGRAPH_2: 3;

              case (q `2 ) <= 0 ;

              then y = ((cn -FanMorphN ) . q) by Th49;

              hence ex x be set st x in ( dom (cn -FanMorphN )) & y = ((cn -FanMorphN ) . x) by A3, A4;

            end;

              case

               A5: ((q `1 ) / |.q.|) >= 0 & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

              ( - ( - (1 + cn))) > 0 by A1, XREAL_1: 148;

              then

               A6: ( - (( - 1) - cn)) > 0 ;

              

               A7: (1 - cn) >= 0 by A2, XREAL_1: 149;

              then (((q `1 ) / |.q.|) * (1 - cn)) >= 0 by A5;

              then (( - 1) - cn) <= (((q `1 ) / |.q.|) * (1 - cn)) by A6;

              then

               A8: ((( - 1) - cn) + cn) <= ((((q `1 ) / |.q.|) * (1 - cn)) + cn) by XREAL_1: 7;

              set px = |[( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))))]|;

              

               A9: (px `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)) by EUCLID: 52;

               |.q.| <> 0 by A5, TOPRNS_1: 24;

              then

               A10: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              

               A11: ( dom (cn -FanMorphN )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A12: (1 - cn) > 0 by A2, XREAL_1: 149;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A10, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `1 ) / |.q.|) <= 1 by SQUARE_1: 51;

              then (((q `1 ) / |.q.|) * (1 - cn)) <= (1 * (1 - cn)) by A12, XREAL_1: 64;

              then (((((q `1 ) / |.q.|) * (1 - cn)) + cn) - cn) <= (1 - cn);

              then ((((q `1 ) / |.q.|) * (1 - cn)) + cn) <= 1 by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ) by A8, SQUARE_1: 49;

              then

               A13: (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A14: ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A15: (px `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )))) ^2 ) + (( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)) ^2 )) by A9, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )));

              

              then

               A16: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) by A13, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A17: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A18: px <> ( 0. ( TOP-REAL 2)) by A5, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `1 ) / |.q.|) * (1 - cn)) + cn) >= ( 0 + cn) by A5, A7, XREAL_1: 7;

              then ((px `1 ) / |.px.|) >= cn by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A19: ((cn -FanMorphN ) . px) = |[( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 - cn))), ( |.px.| * ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, A15, A14, A18, Th51;

              

               A20: ( |.px.| * ( sqrt (((q `2 ) / |.q.|) ^2 ))) = ( |.q.| * ((q `2 ) / |.q.|)) by A5, A17, SQUARE_1: 22

              .= (q `2 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

               A21: ( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 - cn))) = ( |.q.| * ((((((q `1 ) / |.q.|) * (1 - cn)) + cn) - cn) / (1 - cn))) by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `1 ) / |.q.|)) by A12, XCMPLX_1: 89

              .= (q `1 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 - cn)) ^2 )))) = ( |.px.| * ( sqrt (1 - (((q `1 ) / |.px.|) ^2 )))) by A5, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( sqrt (1 - (((q `1 ) ^2 ) / ( |.px.| ^2 ))))) by XCMPLX_1: 76

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `1 ) ^2 ) / ( |.px.| ^2 ))))) by A10, A16, XCMPLX_1: 60

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) - ((q `1 ) ^2 )) / ( |.px.| ^2 )))) by XCMPLX_1: 120

              .= ( |.px.| * ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `1 ) ^2 )) / ( |.px.| ^2 )))) by A16, JGRAPH_3: 1

              .= ( |.px.| * ( sqrt (((q `2 ) / |.q.|) ^2 ))) by A17, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (cn -FanMorphN )) & y = ((cn -FanMorphN ) . x) by A19, A21, A20, A11, EUCLID: 53;

            end;

              case

               A22: ((q `1 ) / |.q.|) < 0 & (q `2 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

              

               A23: (1 + cn) >= 0 by A1, XREAL_1: 148;

              (1 - cn) > 0 by A2, XREAL_1: 149;

              then

               A24: ((1 - cn) + cn) >= ((((q `1 ) / |.q.|) * (1 + cn)) + cn) by A22, A23, XREAL_1: 7;

              

               A25: (1 + cn) > 0 by A1, XREAL_1: 148;

               |.q.| <> 0 by A22, TOPRNS_1: 24;

              then

               A26: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A26, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `1 ) / |.q.|) >= ( - 1) by SQUARE_1: 51;

              then (((q `1 ) / |.q.|) * (1 + cn)) >= (( - 1) * (1 + cn)) by A25, XREAL_1: 64;

              then (((((q `1 ) / |.q.|) * (1 + cn)) + cn) - cn) >= (( - 1) - cn);

              then ((((q `1 ) / |.q.|) * (1 + cn)) + cn) >= ( - 1) by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ) by A24, SQUARE_1: 49;

              then

               A27: (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A28: ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A29: ( dom (cn -FanMorphN )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              set px = |[( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))))]|;

              

               A30: (px `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)) by EUCLID: 52;

              

               A31: (px `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )))) ^2 ) + (( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)) ^2 )) by A30, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )));

              

              then

               A32: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) by A27, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A33: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A34: px <> ( 0. ( TOP-REAL 2)) by A22, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `1 ) / |.q.|) * (1 + cn)) + cn) <= ( 0 + cn) by A22, A23, XREAL_1: 7;

              then ((px `1 ) / |.px.|) <= cn by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A35: ((cn -FanMorphN ) . px) = |[( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 + cn))), ( |.px.| * ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 + cn)) ^2 ))))]| by A1, A2, A31, A28, A34, Th51;

              

               A36: ( |.px.| * ( sqrt (((q `2 ) / |.q.|) ^2 ))) = ( |.q.| * ((q `2 ) / |.q.|)) by A22, A33, SQUARE_1: 22

              .= (q `2 ) by A22, TOPRNS_1: 24, XCMPLX_1: 87;

              

               A37: ( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 + cn))) = ( |.q.| * ((((((q `1 ) / |.q.|) * (1 + cn)) + cn) - cn) / (1 + cn))) by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `1 ) / |.q.|)) by A25, XCMPLX_1: 89

              .= (q `1 ) by A22, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 + cn)) ^2 )))) = ( |.px.| * ( sqrt (1 - (((q `1 ) / |.px.|) ^2 )))) by A22, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( sqrt (1 - (((q `1 ) ^2 ) / ( |.px.| ^2 ))))) by XCMPLX_1: 76

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `1 ) ^2 ) / ( |.px.| ^2 ))))) by A26, A32, XCMPLX_1: 60

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) - ((q `1 ) ^2 )) / ( |.px.| ^2 )))) by XCMPLX_1: 120

              .= ( |.px.| * ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `1 ) ^2 )) / ( |.px.| ^2 )))) by A32, JGRAPH_3: 1

              .= ( |.px.| * ( sqrt (((q `2 ) / |.q.|) ^2 ))) by A33, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (cn -FanMorphN )) & y = ((cn -FanMorphN ) . x) by A35, A37, A36, A29, EUCLID: 53;

            end;

          end;

          hence thesis by A3, FUNCT_1:def 3;

        end;

        hence thesis by A3, XBOOLE_0:def 10;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_4:73

    

     Th73: for cn be Real, p2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = ((cn -FanMorphN ) .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & ((cn -FanMorphN ) . p2) in V2

    proof

      reconsider O = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

      let cn be Real, p2 be Point of ( TOP-REAL 2);

      

       A1: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

       the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

      then

      reconsider V0 = ( Ball (O,( |.p2.| + 1))) as Subset of ( TOP-REAL 2);

      O in V0 & V0 c= ( cl_Ball (O,( |.p2.| + 1))) by GOBOARD6: 1, METRIC_1: 14;

      then

      reconsider K0 = ( cl_Ball (O,( |.p2.| + 1))) as non empty compact Subset of ( TOP-REAL 2) by A1, Th15;

      set q3 = ((cn -FanMorphN ) . p2);

      reconsider VV0 = V0 as Subset of ( TopSpaceMetr ( Euclid 2));

      reconsider u2 = p2 as Point of ( Euclid 2) by EUCLID: 67;

      reconsider u3 = q3 as Point of ( Euclid 2) by EUCLID: 67;

      

       A2: ((cn -FanMorphN ) .: K0) c= K0

      proof

        let y be object;

        assume y in ((cn -FanMorphN ) .: K0);

        then

        consider x be object such that

         A3: x in ( dom (cn -FanMorphN )) and

         A4: x in K0 and

         A5: y = ((cn -FanMorphN ) . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A3;

        reconsider uq = q as Point of ( Euclid 2) by EUCLID: 67;

        ( dist (O,uq)) <= ( |.p2.| + 1) by A4, METRIC_1: 12;

        then |.(( 0. ( TOP-REAL 2)) - q).| <= ( |.p2.| + 1) by JGRAPH_1: 28;

        then |.( - q).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then

         A6: |.q.| <= ( |.p2.| + 1) by TOPRNS_1: 26;

        

         A7: y in ( rng (cn -FanMorphN )) by A3, A5, FUNCT_1:def 3;

        then

        reconsider u = y as Point of ( Euclid 2) by EUCLID: 67;

        reconsider q4 = y as Point of ( TOP-REAL 2) by A7;

         |.q4.| = |.q.| by A5, Th66;

        then |.( - q4).| <= ( |.p2.| + 1) by A6, TOPRNS_1: 26;

        then |.(( 0. ( TOP-REAL 2)) - q4).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then ( dist (O,u)) <= ( |.p2.| + 1) by JGRAPH_1: 28;

        hence thesis by METRIC_1: 12;

      end;

      VV0 is open by TOPMETR: 14;

      then

       A8: V0 is open by Lm11, PRE_TOPC: 30;

      

       A9: |.p2.| < ( |.p2.| + 1) by XREAL_1: 29;

      then |.( - p2).| < ( |.p2.| + 1) by TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - p2).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u2)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A10: p2 in V0 by METRIC_1: 11;

       |.q3.| = |.p2.| by Th66;

      then |.( - q3).| < ( |.p2.| + 1) by A9, TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - q3).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u3)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A11: ((cn -FanMorphN ) . p2) in V0 by METRIC_1: 11;

      assume

       A12: ( - 1) < cn & cn < 1;

      K0 c= ((cn -FanMorphN ) .: K0)

      proof

        let y be object;

        assume

         A13: y in K0;

        then

        reconsider q4 = y as Point of ( TOP-REAL 2);

        reconsider y as Point of ( Euclid 2) by A13;

        the carrier of ( TOP-REAL 2) c= ( rng (cn -FanMorphN )) by A12, Th72;

        then q4 in ( rng (cn -FanMorphN ));

        then

        consider x be object such that

         A14: x in ( dom (cn -FanMorphN )) and

         A15: y = ((cn -FanMorphN ) . x) by FUNCT_1:def 3;

        reconsider x as Point of ( Euclid 2) by A14, Lm11;

        reconsider q = x as Point of ( TOP-REAL 2) by A14;

         |.q4.| = |.q.| by A15, Th66;

        then q in K0 by A13, Lm12;

        hence thesis by A14, A15, FUNCT_1:def 6;

      end;

      then K0 = ((cn -FanMorphN ) .: K0) by A2, XBOOLE_0:def 10;

      hence thesis by A10, A8, A11, METRIC_1: 14;

    end;

    theorem :: JGRAPH_4:74

    for cn be Real st ( - 1) < cn & cn < 1 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (cn -FanMorphN ) & f is being_homeomorphism

    proof

      let cn be Real;

      reconsider f = (cn -FanMorphN ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume

       A1: ( - 1) < cn & cn < 1;

      then

       A2: for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2 by Th73;

      ( rng (cn -FanMorphN )) = the carrier of ( TOP-REAL 2) & ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (cn -FanMorphN ) & h is continuous by A1, Th70, Th72;

      then f is being_homeomorphism by A1, A2, Th3, Th71;

      hence thesis;

    end;

    theorem :: JGRAPH_4:75

    

     Th75: for cn be Real, q be Point of ( TOP-REAL 2) st cn < 1 & (q `2 ) > 0 & ((q `1 ) / |.q.|) >= cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphN ) . q) holds (p `2 ) > 0 & (p `1 ) >= 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: cn < 1 and

       A2: (q `2 ) > 0 and

       A3: ((q `1 ) / |.q.|) >= cn;

      

       A4: (((q `1 ) / |.q.|) - cn) >= 0 by A3, XREAL_1: 48;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      

       A5: (1 - cn) > 0 by A1, XREAL_1: 149;

      

       A6: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, XREAL_1: 74;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

      then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then 1 > ((q `1 ) / |.q.|) by SQUARE_1: 52;

      then (1 - cn) > (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

      then ( - (1 - cn)) < ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

      then (( - (1 - cn)) / (1 - cn)) < (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A5, XREAL_1: 74;

      then ( - 1) < (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A5, XCMPLX_1: 197;

      then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) < (1 ^2 ) by A5, A4, SQUARE_1: 50;

      then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 )))) > 0 ;

      then

       A8: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) > 0 by XCMPLX_1: 76;

      assume p = ((cn -FanMorphN ) . q);

      then

       A9: p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A2, A3, Th49;

      then (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

      hence thesis by A9, A6, A5, A4, A8, EUCLID: 52, XREAL_1: 129;

    end;

    theorem :: JGRAPH_4:76

    

     Th76: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & (q `2 ) > 0 & ((q `1 ) / |.q.|) < cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphN ) . q) holds (p `2 ) > 0 & (p `1 ) < 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: (q `2 ) > 0 and

       A3: ((q `1 ) / |.q.|) < cn;

      

       A4: (1 + cn) > 0 by A1, XREAL_1: 148;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((cn -FanMorphN ) . q);

      then p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A2, A3, Th50;

      then

       A5: (qz `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) & (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

      

       A6: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      

       A8: (((q `1 ) / |.q.|) - cn) < 0 by A3, XREAL_1: 49;

      then ( - (((q `1 ) / |.q.|) - cn)) > 0 by XREAL_1: 58;

      then (( - (1 + cn)) / (1 + cn)) < (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A4, XREAL_1: 74;

      then

       A9: ( - 1) < (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A4, XCMPLX_1: 197;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, XREAL_1: 74;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

      then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then ( - 1) < ((q `1 ) / |.q.|) by SQUARE_1: 52;

      then (( - 1) - cn) < (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

      then ( - ( - (1 + cn))) > ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

      then (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) < 1 by A4, XREAL_1: 191;

      then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) < (1 ^2 ) by A9, SQUARE_1: 50;

      then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) > 0 ;

      then

       A10: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) > 0 by XCMPLX_1: 76;

      ((((q `1 ) / |.q.|) - cn) / (1 + cn)) < 0 by A1, A8, XREAL_1: 141, XREAL_1: 148;

      hence thesis by A6, A5, A10, XREAL_1: 129, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:77

    

     Th77: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st cn < 1 & (q1 `2 ) > 0 & ((q1 `1 ) / |.q1.|) >= cn & (q2 `2 ) > 0 & ((q2 `1 ) / |.q2.|) >= cn & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphN ) . q1) & p2 = ((cn -FanMorphN ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: cn < 1 and

       A2: (q1 `2 ) > 0 and

       A3: ((q1 `1 ) / |.q1.|) >= cn and

       A4: (q2 `2 ) > 0 and

       A5: ((q2 `1 ) / |.q2.|) >= cn and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      

       A7: (((q1 `1 ) / |.q1.|) - cn) < (((q2 `1 ) / |.q2.|) - cn) & (1 - cn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 149;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((cn -FanMorphN ) . q1) and

       A9: p2 = ((cn -FanMorphN ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th66;

      p2 = |[( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn))), ( |.q2.| * ( sqrt (1 - (((((q2 `1 ) / |.q2.|) - cn) / (1 - cn)) ^2 ))))]| by A4, A5, A9, Th49;

      then

       A11: (p2 `1 ) = ( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `1 ) / |.p2.|) = ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn))), ( |.q1.| * ( sqrt (1 - (((((q1 `1 ) / |.q1.|) - cn) / (1 - cn)) ^2 ))))]| by A2, A3, A8, Th49;

      then

       A13: (p1 `1 ) = ( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th66;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `1 ) / |.p1.|) = ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:78

    

     Th78: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & (q1 `2 ) > 0 & ((q1 `1 ) / |.q1.|) < cn & (q2 `2 ) > 0 & ((q2 `1 ) / |.q2.|) < cn & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphN ) . q1) & p2 = ((cn -FanMorphN ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: (q1 `2 ) > 0 and

       A3: ((q1 `1 ) / |.q1.|) < cn and

       A4: (q2 `2 ) > 0 and

       A5: ((q2 `1 ) / |.q2.|) < cn and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      

       A7: (((q1 `1 ) / |.q1.|) - cn) < (((q2 `1 ) / |.q2.|) - cn) & (1 + cn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 148;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((cn -FanMorphN ) . q1) and

       A9: p2 = ((cn -FanMorphN ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th66;

      p2 = |[( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn))), ( |.q2.| * ( sqrt (1 - (((((q2 `1 ) / |.q2.|) - cn) / (1 + cn)) ^2 ))))]| by A4, A5, A9, Th50;

      then

       A11: (p2 `1 ) = ( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `1 ) / |.p2.|) = ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn))), ( |.q1.| * ( sqrt (1 - (((((q1 `1 ) / |.q1.|) - cn) / (1 + cn)) ^2 ))))]| by A2, A3, A8, Th50;

      then

       A13: (p1 `1 ) = ( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th66;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `1 ) / |.p1.|) = ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:79

    for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q1 `2 ) > 0 & (q2 `2 ) > 0 & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphN ) . q1) & p2 = ((cn -FanMorphN ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q1 `2 ) > 0 and

       A4: (q2 `2 ) > 0 and

       A5: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A6: p1 = ((cn -FanMorphN ) . q1) and

       A7: p2 = ((cn -FanMorphN ) . q2);

      per cases ;

        suppose ((q1 `1 ) / |.q1.|) >= cn & ((q2 `1 ) / |.q2.|) >= cn;

        hence thesis by A2, A3, A4, A5, A6, A7, Th77;

      end;

        suppose ((q1 `1 ) / |.q1.|) >= cn & ((q2 `1 ) / |.q2.|) < cn;

        hence thesis by A5, XXREAL_0: 2;

      end;

        suppose

         A8: ((q1 `1 ) / |.q1.|) < cn & ((q2 `1 ) / |.q2.|) >= cn;

        then (p2 `1 ) >= 0 by A2, A4, A7, Th75;

        then

         A9: ((p2 `1 ) / |.p2.|) >= 0 ;

        (p1 `1 ) < 0 by A1, A3, A6, A8, Th76;

        hence thesis by A9, Lm1, JGRAPH_2: 3, XREAL_1: 141;

      end;

        suppose ((q1 `1 ) / |.q1.|) < cn & ((q2 `1 ) / |.q2.|) < cn;

        hence thesis by A1, A3, A4, A5, A6, A7, Th78;

      end;

    end;

    theorem :: JGRAPH_4:80

    for cn be Real, q be Point of ( TOP-REAL 2) st (q `2 ) > 0 & ((q `1 ) / |.q.|) = cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphN ) . q) holds (p `2 ) > 0 & (p `1 ) = 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: (q `2 ) > 0 and

       A2: ((q `1 ) / |.q.|) = cn;

      

       A3: |.q.| <> 0 & ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) > 0 by A1, A2, JGRAPH_2: 3, SQUARE_1: 25, TOPRNS_1: 24;

      let p be Point of ( TOP-REAL 2);

      assume p = ((cn -FanMorphN ) . q);

      then

       A4: p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A1, A2, Th49;

      then (p `2 ) = ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) by EUCLID: 52;

      hence thesis by A2, A4, A3, EUCLID: 52, XREAL_1: 129;

    end;

    theorem :: JGRAPH_4:81

    for cn be Real holds ( 0. ( TOP-REAL 2)) = ((cn -FanMorphN ) . ( 0. ( TOP-REAL 2))) by Th49, JGRAPH_2: 3;

    begin

    definition

      let s be Real, q be Point of ( TOP-REAL 2);

      :: JGRAPH_4:def6

      func FanE (s,q) -> Point of ( TOP-REAL 2) equals

      : Def6: ( |.q.| * |[( sqrt (1 - (((((q `2 ) / |.q.|) - s) / (1 - s)) ^2 ))), ((((q `2 ) / |.q.|) - s) / (1 - s))]|) if ((q `2 ) / |.q.|) >= s & (q `1 ) > 0 ,

( |.q.| * |[( sqrt (1 - (((((q `2 ) / |.q.|) - s) / (1 + s)) ^2 ))), ((((q `2 ) / |.q.|) - s) / (1 + s))]|) if ((q `2 ) / |.q.|) < s & (q `1 ) > 0

      otherwise q;

      correctness ;

    end

    definition

      let s be Real;

      :: JGRAPH_4:def7

      func s -FanMorphE -> Function of ( TOP-REAL 2), ( TOP-REAL 2) means

      : Def7: for q be Point of ( TOP-REAL 2) holds (it . q) = ( FanE (s,q));

      existence

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanE (s,$1));

        thus ex IT be Function of ( TOP-REAL 2), ( TOP-REAL 2) st for q be Point of ( TOP-REAL 2) holds (IT . q) = F(q) from FUNCT_2:sch 4;

      end;

      uniqueness

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanE (s,$1));

        thus for a,b be Function of ( TOP-REAL 2), ( TOP-REAL 2) st (for q be Point of ( TOP-REAL 2) holds (a . q) = F(q)) & (for q be Point of ( TOP-REAL 2) holds (b . q) = F(q)) holds a = b from BINOP_2:sch 1;

      end;

    end

    theorem :: JGRAPH_4:82

    

     Th82: for sn be Real holds (((q `2 ) / |.q.|) >= sn & (q `1 ) > 0 implies ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|) & ((q `1 ) <= 0 implies ((sn -FanMorphE ) . q) = q)

    proof

      let sn be Real;

      hereby

        assume ((q `2 ) / |.q.|) >= sn & (q `1 ) > 0 ;

        

        then ( FanE (sn,q)) = ( |.q.| * |[( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))), ((((q `2 ) / |.q.|) - sn) / (1 - sn))]|) by Def6

        .= |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by EUCLID: 58;

        hence ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by Def7;

      end;

      assume

       A1: (q `1 ) <= 0 ;

      ((sn -FanMorphE ) . q) = ( FanE (sn,q)) by Def7;

      hence thesis by A1, Def6;

    end;

    theorem :: JGRAPH_4:83

    

     Th83: for sn be Real holds (((q `2 ) / |.q.|) <= sn & (q `1 ) > 0 implies ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|)

    proof

      let sn be Real;

      assume that

       A1: ((q `2 ) / |.q.|) <= sn and

       A2: (q `1 ) > 0 ;

      now

        per cases by A1, XXREAL_0: 1;

          case ((q `2 ) / |.q.|) < sn;

          

          then ( FanE (sn,q)) = ( |.q.| * |[( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))), ((((q `2 ) / |.q.|) - sn) / (1 + sn))]|) by A2, Def6

          .= |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by EUCLID: 58;

          hence thesis by Def7;

        end;

          case

           A3: ((q `2 ) / |.q.|) = sn;

          then ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = 0 ;

          hence thesis by A2, A3, Th82;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_4:84

    

     Th84: for sn be Real st ( - 1) < sn & sn < 1 holds (((q `2 ) / |.q.|) >= sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|) & (((q `2 ) / |.q.|) <= sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|)

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      per cases ;

        suppose

         A3: ((q `2 ) / |.q.|) >= sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose

           A4: (q `1 ) > 0 ;

          

          then ( FanE (sn,q)) = ( |.q.| * |[( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))), ((((q `2 ) / |.q.|) - sn) / (1 - sn))]|) by A3, Def6

          .= |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by EUCLID: 58;

          hence thesis by A4, Def7, Th83;

        end;

          suppose

           A5: (q `1 ) <= 0 ;

          then

           A6: ((sn -FanMorphE ) . q) = q by Th82;

          

           A7: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

          

           A8: (1 - sn) > 0 by A2, XREAL_1: 149;

          

           A9: (q `1 ) = 0 by A3, A5;

           |.q.| <> 0 by A3, TOPRNS_1: 24;

          then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          then (((q `2 ) ^2 ) / ( |.q.| ^2 )) = (1 ^2 ) by A7, A9, XCMPLX_1: 60;

          then (((q `2 ) / |.q.|) ^2 ) = (1 ^2 ) by XCMPLX_1: 76;

          then

           A10: ( sqrt (((q `2 ) / |.q.|) ^2 )) = 1 by SQUARE_1: 22;

           A11:

          now

            assume (q `2 ) < 0 ;

            then ( - ((q `2 ) / |.q.|)) = 1 by A10, SQUARE_1: 23;

            hence contradiction by A1, A3;

          end;

          ( sqrt ( |.q.| ^2 )) = |.q.| by SQUARE_1: 22;

          then

           A12: |.q.| = (q `2 ) by A7, A9, A11, SQUARE_1: 22;

          then 1 = ((q `2 ) / |.q.|) by A3, TOPRNS_1: 24, XCMPLX_1: 60;

          then ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = 1 by A8, XCMPLX_1: 60;

          hence thesis by A2, A6, A9, A12, EUCLID: 53, SQUARE_1: 17, TOPRNS_1: 24, XCMPLX_1: 60;

        end;

      end;

        suppose

         A13: ((q `2 ) / |.q.|) <= sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose (q `1 ) > 0 ;

          hence thesis by Th82, Th83;

        end;

          suppose

           A14: (q `1 ) <= 0 ;

          then

           A15: (q `1 ) = 0 by A13;

          

           A16: (1 + sn) > 0 by A1, XREAL_1: 148;

          

           A17: |.q.| <> 0 by A13, TOPRNS_1: 24;

          1 > ((q `2 ) / |.q.|) by A2, A13, XXREAL_0: 2;

          then (1 * |.q.|) > (((q `2 ) / |.q.|) * |.q.|) by A17, XREAL_1: 68;

          then

           A18: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & |.q.| > (q `2 ) by A13, JGRAPH_3: 1, TOPRNS_1: 24, XCMPLX_1: 87;

          then

           A19: |.q.| = ( - (q `2 )) by A15, SQUARE_1: 40;

          

           A20: (q `2 ) = ( - |.q.|) by A15, A18, SQUARE_1: 40;

          then ( - 1) = ((q `2 ) / |.q.|) by A13, TOPRNS_1: 24, XCMPLX_1: 197;

          

          then ((((q `2 ) / |.q.|) - sn) / (1 + sn)) = (( - (1 + sn)) / (1 + sn))

          .= ( - 1) by A16, XCMPLX_1: 197;

          then |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| = q by A15, A19, EUCLID: 53, SQUARE_1: 17;

          hence thesis by A1, A14, A17, A20, Th82, XCMPLX_1: 197;

        end;

      end;

        suppose (q `1 ) < 0 or q = ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

    end;

    theorem :: JGRAPH_4:85

    

     Th85: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st sn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 - sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: sn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in K1 by A7, A8, A9, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        

         A11: (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        (f . r) = ( |.r.| * ((((r `2 ) / |.r.|) - sn) / (1 - sn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:86

    

     Th86: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < sn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 + sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: ( - 1) < sn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      (1 + sn) > 0 by A1, XREAL_1: 148;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A8: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in ( dom g3) by A7, A9;

        then x in K1 by A7, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        

         A11: (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        (f . r) = ( |.r.| * ((((r `2 ) / |.r.|) - sn) / (1 + sn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      ( dom f) = ( dom g3) by A7, FUNCT_2:def 1;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:87

    

     Th87: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st sn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 - sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: sn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)) and

       A6: g3 is continuous by A4, Th10;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        

         A9: (1 - sn) > 0 by A1, XREAL_1: 149;

        assume

         A10: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A11: |.r.| <> 0 by A3, A10, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `2 ) - |.r.|) * ((r `2 ) + |.r.|)) = ( - ((r `1 ) ^2 ));

        ((r `1 ) ^2 ) >= 0 by XREAL_1: 63;

        then (r `2 ) <= |.r.| by A12, XREAL_1: 93;

        then ((r `2 ) / |.r.|) <= ( |.r.| / |.r.|) by XREAL_1: 72;

        then ((r `2 ) / |.r.|) <= 1 by A11, XCMPLX_1: 60;

        then

         A13: (((r `2 ) / |.r.|) - sn) <= (1 - sn) by XREAL_1: 9;

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A10;

         A14:

        now

          assume ((1 - sn) ^2 ) = 0 ;

          then ((1 - sn) + sn) = ( 0 + sn) by XCMPLX_1: 6;

          hence contradiction by A1;

        end;

        (sn - ((r `2 ) / |.r.|)) <= 0 by A3, A10, XREAL_1: 47;

        then ( - (sn - ((r `2 ) / |.r.|))) >= ( - (1 - sn)) by A9, XREAL_1: 24;

        then ((1 - sn) ^2 ) >= 0 & ((((r `2 ) / |.r.|) - sn) ^2 ) <= ((1 - sn) ^2 ) by A13, SQUARE_1: 49, XREAL_1: 63;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 - sn) ^2 )) <= (((1 - sn) ^2 ) / ((1 - sn) ^2 )) by XREAL_1: 72;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 - sn) ^2 )) <= 1 by A14, XCMPLX_1: 60;

        then (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )).| = (1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )) by ABSVALUE:def 1;

        then

         A15: (f . r) = ( |.r.| * ( sqrt |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 - sn)) ^2 )).|)) by A2, A10;

        

         A16: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        hence thesis by A5, A15, A16;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:88

    

     Th88: for sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < sn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let sn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = sn, b = (1 + sn);

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm3;

      assume that

       A1: ( - 1) < sn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2));

      

       A4: (1 + sn) > 0 by A1, XREAL_1: 148;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|)) and

       A6: g3 is continuous by A4, Th10;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A9;

        

         A10: ((1 + sn) ^2 ) > 0 by A4, SQUARE_1: 12;

        

         A11: |.r.| <> 0 by A3, A9, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `2 ) - |.r.|) * ((r `2 ) + |.r.|)) = ( - ((r `1 ) ^2 ));

        ((r `1 ) ^2 ) >= 0 by XREAL_1: 63;

        then ( - |.r.|) <= (r `2 ) by A12, XREAL_1: 93;

        then ((r `2 ) / |.r.|) >= (( - |.r.|) / |.r.|) by XREAL_1: 72;

        then ((r `2 ) / |.r.|) >= ( - 1) by A11, XCMPLX_1: 197;

        then (((r `2 ) / |.r.|) - sn) >= (( - 1) - sn) by XREAL_1: 9;

        then

         A13: (((r `2 ) / |.r.|) - sn) >= ( - (1 + sn));

        (sn - ((r `2 ) / |.r.|)) >= 0 by A3, A9, XREAL_1: 48;

        then ( - (sn - ((r `2 ) / |.r.|))) <= ( - 0 );

        then ((((r `2 ) / |.r.|) - sn) ^2 ) <= ((1 + sn) ^2 ) by A4, A13, SQUARE_1: 49;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 + sn) ^2 )) <= (((1 + sn) ^2 ) / ((1 + sn) ^2 )) by A4, XREAL_1: 72;

        then (((((r `2 ) / |.r.|) - sn) ^2 ) / ((1 + sn) ^2 )) <= 1 by A10, XCMPLX_1: 60;

        then (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )).| = (1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )) by ABSVALUE:def 1;

        then

         A14: (f . r) = ( |.r.| * ( sqrt |.(1 - (((((r `2 ) / |.r.|) - sn) / (1 + sn)) ^2 )).|)) by A2, A9;

        

         A15: ( proj2 . r) = (r `2 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 6;

        (g2 . s) = ( proj2 . s) & (g1 . s) = ((2 NormF ) . s) by Lm3, Lm5;

        hence thesis by A5, A14, A15;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:89

    

     Th89: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `1 ) = cn by EUCLID: 52;

      (p0 `2 ) = sn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((cn ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - ( - cn)) > 0 by SQUARE_1: 25;

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A6: p0 in K0 by A3, A1, A5, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A7: ( rng ( proj1 * ((sn -FanMorphE ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A8: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) / |.p8.|) >= sn & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A9: ( dom ((sn -FanMorphE ) | K1)) c= ( dom ( proj2 * ((sn -FanMorphE ) | K1)))

      proof

        let x be object;

        assume

         A10: x in ( dom ((sn -FanMorphE ) | K1));

        then x in (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphE )) by XBOOLE_0:def 4;

        then

         A11: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphE ) . x) in ( rng (sn -FanMorphE )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphE ) | K1) . x) = ((sn -FanMorphE ) . x) by A10, FUNCT_1: 47;

        hence thesis by A10, A11, FUNCT_1: 11;

      end;

      

       A12: ( rng ( proj2 * ((sn -FanMorphE ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj2 * ((sn -FanMorphE ) | K1))) c= ( dom ((sn -FanMorphE ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((sn -FanMorphE ) | K1))) = ( dom ((sn -FanMorphE ) | K1)) by A9, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj2 * ((sn -FanMorphE ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A12, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A13: ( dom ((sn -FanMorphE ) | K1)) = (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A14: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A15: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A14;

        then

         A16: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A3, Th84;

        (((sn -FanMorphE ) | K1) . p) = ((sn -FanMorphE ) . p) by A15, A14, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|) by A15, A13, A14, A16, FUNCT_1: 13

        .= ( |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A17: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)));

      

       A18: ( dom ((sn -FanMorphE ) | K1)) c= ( dom ( proj1 * ((sn -FanMorphE ) | K1)))

      proof

        let x be object;

        assume

         A19: x in ( dom ((sn -FanMorphE ) | K1));

        then x in (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphE )) by XBOOLE_0:def 4;

        then

         A20: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphE ) . x) in ( rng (sn -FanMorphE )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphE ) | K1) . x) = ((sn -FanMorphE ) . x) by A19, FUNCT_1: 47;

        hence thesis by A19, A20, FUNCT_1: 11;

      end;

      ( dom ( proj1 * ((sn -FanMorphE ) | K1))) c= ( dom ((sn -FanMorphE ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((sn -FanMorphE ) | K1))) = ( dom ((sn -FanMorphE ) | K1)) by A18, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj1 * ((sn -FanMorphE ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A7, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A21: ( dom ((sn -FanMorphE ) | K1)) = (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A22: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A23: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A22;

        then

         A24: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A3, Th84;

        (((sn -FanMorphE ) | K1) . p) = ((sn -FanMorphE ) . p) by A23, A22, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|) by A23, A21, A22, A24, FUNCT_1: 13

        .= ( |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A25: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) >= sn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A26: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A26;

        hence thesis;

      end;

      then

       A27: f1 is continuous by A3, A25, Th87;

      

       A28: for x,y,r,s be Real st |[x, y]| in K1 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that

         A29: |[x, y]| in K1 and

         A30: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A31: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A29;

        

         A32: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A33: (f1 . p99) = ( |.p99.| * ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)) ^2 )))) by A25, A29;

        (((sn -FanMorphE ) | K0) . |[x, y]|) = ((sn -FanMorphE ) . |[x, y]|) by A29, FUNCT_1: 49

        .= |[( |.p99.| * ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)) ^2 )))), ( |.p99.| * ((((p99 `2 ) / |.p99.|) - sn) / (1 - sn)))]| by A3, A31, Th84

        .= |[r, s]| by A17, A29, A30, A32, A33;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A34: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) >= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A34;

        hence thesis;

      end;

      then f2 is continuous by A3, A17, Th85;

      hence thesis by A6, A8, A27, A28, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:90

    

     Th90: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `1 ) = cn by EUCLID: 52;

      (p0 `2 ) = sn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((cn ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - ( - cn)) > 0 by SQUARE_1: 25;

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A6: p0 in K0 by A3, A1, A5, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A7: ( rng ( proj1 * ((sn -FanMorphE ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A8: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) / |.p8.|) <= sn & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A9: ( dom ((sn -FanMorphE ) | K1)) c= ( dom ( proj2 * ((sn -FanMorphE ) | K1)))

      proof

        let x be object;

        assume

         A10: x in ( dom ((sn -FanMorphE ) | K1));

        then x in (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphE )) by XBOOLE_0:def 4;

        then

         A11: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphE ) . x) in ( rng (sn -FanMorphE )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphE ) | K1) . x) = ((sn -FanMorphE ) . x) by A10, FUNCT_1: 47;

        hence thesis by A10, A11, FUNCT_1: 11;

      end;

      

       A12: ( rng ( proj2 * ((sn -FanMorphE ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj2 * ((sn -FanMorphE ) | K1))) c= ( dom ((sn -FanMorphE ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((sn -FanMorphE ) | K1))) = ( dom ((sn -FanMorphE ) | K1)) by A9, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj2 * ((sn -FanMorphE ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A12, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A13: ( dom ((sn -FanMorphE ) | K1)) = (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A14: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A15: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A14;

        then

         A16: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A3, Th84;

        (((sn -FanMorphE ) | K1) . p) = ((sn -FanMorphE ) . p) by A15, A14, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|) by A15, A13, A14, A16, FUNCT_1: 13

        .= ( |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A17: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)));

      

       A18: ( dom ((sn -FanMorphE ) | K1)) c= ( dom ( proj1 * ((sn -FanMorphE ) | K1)))

      proof

        let x be object;

        assume

         A19: x in ( dom ((sn -FanMorphE ) | K1));

        then x in (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61;

        then x in ( dom (sn -FanMorphE )) by XBOOLE_0:def 4;

        then

         A20: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((sn -FanMorphE ) . x) in ( rng (sn -FanMorphE )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((sn -FanMorphE ) | K1) . x) = ((sn -FanMorphE ) . x) by A19, FUNCT_1: 47;

        hence thesis by A19, A20, FUNCT_1: 11;

      end;

      ( dom ( proj1 * ((sn -FanMorphE ) | K1))) c= ( dom ((sn -FanMorphE ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((sn -FanMorphE ) | K1))) = ( dom ((sn -FanMorphE ) | K1)) by A18, XBOOLE_0:def 10

      .= (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj1 * ((sn -FanMorphE ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A7, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A21: ( dom ((sn -FanMorphE ) | K1)) = (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A22: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A23: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A22;

        then

         A24: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A3, Th84;

        (((sn -FanMorphE ) | K1) . p) = ((sn -FanMorphE ) . p) by A23, A22, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|) by A23, A21, A22, A24, FUNCT_1: 13

        .= ( |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A25: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & ((q `2 ) / |.q.|) <= sn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A26: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A26;

        hence thesis;

      end;

      then

       A27: f1 is continuous by A3, A25, Th88;

      

       A28: for x,y,r,s be Real st |[x, y]| in K1 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that

         A29: |[x, y]| in K1 and

         A30: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A31: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A29;

        

         A32: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A33: (f1 . p99) = ( |.p99.| * ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)) ^2 )))) by A25, A29;

        (((sn -FanMorphE ) | K0) . |[x, y]|) = ((sn -FanMorphE ) . |[x, y]|) by A29, FUNCT_1: 49

        .= |[( |.p99.| * ( sqrt (1 - (((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)) ^2 )))), ( |.p99.| * ((((p99 `2 ) / |.p99.|) - sn) / (1 + sn)))]| by A3, A31, Th84

        .= |[r, s]| by A17, A29, A30, A32, A33;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A34: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) / |.p3.|) <= sn & (p3 `1 ) >= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A34;

        hence thesis;

      end;

      then f2 is continuous by A3, A17, Th86;

      hence thesis by A6, A8, A27, A28, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:91

    

     Th91: for sn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) >= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) >= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) >= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm7, JORDAN6: 4;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:92

    

     Th92: for sn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `2 ) <= (sn * |.p.|) & (p `1 ) >= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `2 ) <= (sn * |.p.|) & (p `1 ) >= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) <= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm9, JORDAN6: 4;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:93

    

     Th93: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `1 ) = cn by EUCLID: 52;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) / |.$1.|) >= sn & ($1 `1 ) >= 0 & $1 <> ( 0. ( TOP-REAL 2));

      (p0 `2 ) = sn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((cn ^2 ) + (sn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: (p0 `1 ) > 0 by A1, SQUARE_1: 25;

      then p0 in K0 by A3, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      (cn ^2 ) = (1 - (sn ^2 )) by A4, SQUARE_1:def 2;

      then

       A6: ((p0 `2 ) / |.p0.|) = sn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A7: p0 in { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A5, JGRAPH_2: 3;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K001 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A7;

      

       A8: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= (sn * |.$1.|) & ($1 `1 ) >= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K003 = { p : (p `2 ) >= (sn * |.p.|) & (p `1 ) >= 0 } as Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `2 ) / |.$1.|) <= sn & ($1 `1 ) >= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A9: { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      

       A10: ( - ( - cn)) > 0 by A4, SQUARE_1: 25;

      then p0 in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, JGRAPH_2: 3;

      then

      reconsider K111 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A9;

      

       A11: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

      

       A12: ( rng ((sn -FanMorphE ) | K001)) c= K1

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphE ) | K001));

        then

        consider x be object such that

         A13: x in ( dom ((sn -FanMorphE ) | K001)) and

         A14: y = (((sn -FanMorphE ) | K001) . x) by FUNCT_1:def 3;

        x in ( dom (sn -FanMorphE )) by A13, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A15: y = ((sn -FanMorphE ) . q) by A13, A14, FUNCT_1: 47;

        ( dom ((sn -FanMorphE ) | K001)) = (( dom (sn -FanMorphE )) /\ K001) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K001) by FUNCT_2:def 1

        .= K001 by XBOOLE_1: 28;

        then

         A16: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `2 ) / |.p2.|) >= sn & (p2 `1 ) >= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A13;

        then

         A17: (((q `2 ) / |.q.|) - sn) >= 0 by XREAL_1: 48;

         |.q.| <> 0 by A16, TOPRNS_1: 24;

        then

         A18: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

        

         A19: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

        

         A20: (1 - sn) > 0 by A3, XREAL_1: 149;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then ((q `2 ) ^2 ) <= ( |.q.| ^2 ) by JGRAPH_3: 1;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A18, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

        then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A20, XREAL_1: 72;

        then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A20, XCMPLX_1: 197;

        then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A20, A17, SQUARE_1: 49;

        then

         A21: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

        then

         A22: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

        ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by A21, SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

        then

         A23: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A24: (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

        

        then

         A25: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A22, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A19, A25;

        then

         A26: q4 <> ( 0. ( TOP-REAL 2)) by A18, TOPRNS_1: 23;

        ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A3, A16, Th84;

        hence thesis by A3, A15, A24, A23, A26;

      end;

      

       A27: { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      p0 <> ( 0. ( TOP-REAL 2)) by A1, A4, JGRAPH_2: 3, SQUARE_1: 25;

      then not p0 in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A3, XBOOLE_0:def 5;

      K1 c= D

      proof

        let x be object;

        assume

         A28: x in K1;

        then ex p6 be Point of ( TOP-REAL 2) st p6 = x & (p6 `1 ) >= 0 & p6 <> ( 0. ( TOP-REAL 2)) by A3;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A3, A28, XBOOLE_0:def 5;

      end;

      then D = (K1 \/ D) by XBOOLE_1: 12;

      then

       A29: (( TOP-REAL 2) | K1) is SubSpace of (( TOP-REAL 2) | D) by TOPMETR: 4;

      

       A30: { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      then

      reconsider K00 = { p : ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A7, PRE_TOPC: 8;

      

       A31: K003 is closed by Th91;

      

       A32: ( rng ((sn -FanMorphE ) | K111)) c= K1

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphE ) | K111));

        then

        consider x be object such that

         A33: x in ( dom ((sn -FanMorphE ) | K111)) and

         A34: y = (((sn -FanMorphE ) | K111) . x) by FUNCT_1:def 3;

        x in ( dom (sn -FanMorphE )) by A33, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A35: y = ((sn -FanMorphE ) . q) by A33, A34, FUNCT_1: 47;

        ( dom ((sn -FanMorphE ) | K111)) = (( dom (sn -FanMorphE )) /\ K111) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K111) by FUNCT_2:def 1

        .= K111 by XBOOLE_1: 28;

        then

         A36: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `2 ) / |.p2.|) <= sn & (p2 `1 ) >= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A33;

        then

         A37: (((q `2 ) / |.q.|) - sn) <= 0 by XREAL_1: 47;

         |.q.| <> 0 by A36, TOPRNS_1: 24;

        then

         A38: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

        

         A39: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

        

         A40: (1 + sn) > 0 by A3, XREAL_1: 148;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A38, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        then (( - (1 + sn)) / (1 + sn)) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A40, XREAL_1: 72;

        then ( - 1) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A40, XCMPLX_1: 197;

        then

         A41: (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) <= (1 ^2 ) by A40, A37, SQUARE_1: 49;

        then

         A42: (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

        (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by A41, XREAL_1: 48;

        then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XCMPLX_1: 187;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

        then

         A43: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A44: (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

        

        then

         A45: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A42, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A39, A45;

        then

         A46: q4 <> ( 0. ( TOP-REAL 2)) by A38, TOPRNS_1: 23;

        ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A3, A36, Th84;

        hence thesis by A3, A35, A44, A43, A46;

      end;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A47: ( rng (f | K00)) c= D;

      the carrier of (( TOP-REAL 2) | B0) = the carrier of (( TOP-REAL 2) | D);

      

      then

       A48: ( dom f) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1

      .= K1 by PRE_TOPC: 8;

      

      then ( dom (f | K00)) = K00 by A30, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K00) by PRE_TOPC: 8;

      then

      reconsider f1 = (f | K00) as Function of ((( TOP-REAL 2) | K1) | K00), (( TOP-REAL 2) | D) by A47, FUNCT_2: 2;

      

       A49: the carrier of (( TOP-REAL 2) | K1) = K0 by PRE_TOPC: 8;

      p0 in { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A10, A6, JGRAPH_2: 3;

      then

      reconsider K11 = { p : ((p `2 ) / |.p.|) <= sn & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A27, PRE_TOPC: 8;

      

       A50: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      

       A51: ( dom (sn -FanMorphE )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((sn -FanMorphE ) | K001)) = K001 by RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K001) by PRE_TOPC: 8;

      then

      reconsider f3 = ((sn -FanMorphE ) | K001) as Function of (( TOP-REAL 2) | K001), (( TOP-REAL 2) | K1) by A8, A12, FUNCT_2: 2;

      

       A52: D <> {} ;

      ( dom ((sn -FanMorphE ) | K111)) = K111 by A51, RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K111) by PRE_TOPC: 8;

      then

      reconsider f4 = ((sn -FanMorphE ) | K111) as Function of (( TOP-REAL 2) | K111), (( TOP-REAL 2) | K1) by A50, A32, FUNCT_2: 2;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A53: ( rng (f | K11)) c= D;

      ( dom (f | K11)) = K11 by A27, A48, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K11) by PRE_TOPC: 8;

      then

      reconsider f2 = (f | K11) as Function of ((( TOP-REAL 2) | K1) | K11), (( TOP-REAL 2) | D) by A53, FUNCT_2: 2;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K11) = (( TOP-REAL 2) | K111) & f2 = f4 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A54: f2 is continuous by A3, A29, Th90, PRE_TOPC: 26;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K00) = (( TOP-REAL 2) | K001) & f1 = f3 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A55: f1 is continuous by A3, A29, Th89, PRE_TOPC: 26;

      

       A56: ( dom f2) = the carrier of ((( TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1

      .= K11 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= (sn * |.$1.|) & ($1 `1 ) >= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K004 = { p : (p `2 ) <= (sn * |.p.|) & (p `1 ) >= 0 } as Subset of ( TOP-REAL 2);

      

       A57: (K004 /\ K1) c= K11

      proof

        let x be object;

        assume

         A58: x in (K004 /\ K1);

        then x in K004 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A59: q1 = x and

         A60: (q1 `2 ) <= (sn * |.q1.|) and (q1 `1 ) >= 0 ;

        x in K1 by A58, XBOOLE_0:def 4;

        then

         A61: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `1 ) >= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `2 ) / |.q1.|) <= ((sn * |.q1.|) / |.q1.|) by A60, XREAL_1: 72;

        then ((q1 `2 ) / |.q1.|) <= sn by A59, A61, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A59, A61;

      end;

      

       A62: K004 is closed by Th92;

      K11 c= (K004 /\ K1)

      proof

        let x be object;

        assume x in K11;

        then

        consider p such that

         A63: p = x and

         A64: ((p `2 ) / |.p.|) <= sn and

         A65: (p `1 ) >= 0 and

         A66: p <> ( 0. ( TOP-REAL 2));

        (((p `2 ) / |.p.|) * |.p.|) <= (sn * |.p.|) by A64, XREAL_1: 64;

        then (p `2 ) <= (sn * |.p.|) by A66, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A67: x in K004 by A63, A65;

        x in K1 by A3, A63, A65, A66;

        hence thesis by A67, XBOOLE_0:def 4;

      end;

      then K11 = (K004 /\ ( [#] (( TOP-REAL 2) | K1))) by A11, A57, XBOOLE_0:def 10;

      then

       A68: K11 is closed by A62, PRE_TOPC: 13;

      

       A69: (K003 /\ K1) c= K00

      proof

        let x be object;

        assume

         A70: x in (K003 /\ K1);

        then x in K003 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A71: q1 = x and

         A72: (q1 `2 ) >= (sn * |.q1.|) and (q1 `1 ) >= 0 ;

        x in K1 by A70, XBOOLE_0:def 4;

        then

         A73: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `1 ) >= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `2 ) / |.q1.|) >= ((sn * |.q1.|) / |.q1.|) by A72, XREAL_1: 72;

        then ((q1 `2 ) / |.q1.|) >= sn by A71, A73, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A71, A73;

      end;

      K00 c= (K003 /\ K1)

      proof

        let x be object;

        assume x in K00;

        then

        consider p such that

         A74: p = x and

         A75: ((p `2 ) / |.p.|) >= sn and

         A76: (p `1 ) >= 0 and

         A77: p <> ( 0. ( TOP-REAL 2));

        (((p `2 ) / |.p.|) * |.p.|) >= (sn * |.p.|) by A75, XREAL_1: 64;

        then (p `2 ) >= (sn * |.p.|) by A77, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A78: x in K003 by A74, A76;

        x in K1 by A3, A74, A76, A77;

        hence thesis by A78, XBOOLE_0:def 4;

      end;

      then K00 = (K003 /\ ( [#] (( TOP-REAL 2) | K1))) by A11, A69, XBOOLE_0:def 10;

      then

       A79: K00 is closed by A31, PRE_TOPC: 13;

      set T1 = ((( TOP-REAL 2) | K1) | K00), T2 = ((( TOP-REAL 2) | K1) | K11);

      

       A80: ( [#] ((( TOP-REAL 2) | K1) | K11)) = K11 by PRE_TOPC:def 5;

      

       A81: ( [#] ((( TOP-REAL 2) | K1) | K00)) = K00 by PRE_TOPC:def 5;

      

       A82: for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f1 . p) = (f2 . p)

      proof

        let p be object;

        assume

         A83: p in (( [#] T1) /\ ( [#] T2));

        then p in K00 by A81, XBOOLE_0:def 4;

        

        hence (f1 . p) = (f . p) by FUNCT_1: 49

        .= (f2 . p) by A80, A83, FUNCT_1: 49;

      end;

      

       A84: K1 c= (K00 \/ K11)

      proof

        let x be object;

        assume x in K1;

        then

        consider p such that

         A85: p = x & (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) by A3;

        per cases ;

          suppose ((p `2 ) / |.p.|) >= sn;

          then x in K00 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

          suppose ((p `2 ) / |.p.|) < sn;

          then x in K11 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

      end;

      then (( [#] ((( TOP-REAL 2) | K1) | K00)) \/ ( [#] ((( TOP-REAL 2) | K1) | K11))) = ( [#] (( TOP-REAL 2) | K1)) by A81, A80, A11, XBOOLE_0:def 10;

      then

      consider h be Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D) such that

       A86: h = (f1 +* f2) and

       A87: h is continuous by A81, A80, A79, A68, A55, A54, A82, JGRAPH_2: 1;

      

       A88: ( dom h) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A89: ( dom f1) = the carrier of ((( TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1

      .= K00 by PRE_TOPC: 8;

      

       A90: for y be object st y in ( dom h) holds (h . y) = (f . y)

      proof

        let y be object;

        assume

         A91: y in ( dom h);

        now

          per cases by A84, A88, A49, A91, XBOOLE_0:def 3;

            suppose

             A92: y in K00 & not y in K11;

            then y in (( dom f1) \/ ( dom f2)) by A89, XBOOLE_0:def 3;

            

            hence (h . y) = (f1 . y) by A56, A86, A92, FUNCT_4:def 1

            .= (f . y) by A92, FUNCT_1: 49;

          end;

            suppose

             A93: y in K11;

            then y in (( dom f1) \/ ( dom f2)) by A56, XBOOLE_0:def 3;

            

            hence (h . y) = (f2 . y) by A56, A86, A93, FUNCT_4:def 1

            .= (f . y) by A93, FUNCT_1: 49;

          end;

        end;

        hence thesis;

      end;

      K0 = the carrier of (( TOP-REAL 2) | K0) by PRE_TOPC: 8

      .= ( dom f) by A52, FUNCT_2:def 1;

      hence thesis by A87, A88, A90, FUNCT_1: 2, PRE_TOPC: 8;

    end;

    theorem :: JGRAPH_4:94

    

     Th94: for sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set cn = ( sqrt (1 - (sn ^2 )));

      set p0 = |[( - cn), ( - sn)]|;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (sn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then (1 - (sn ^2 )) > 0 by XREAL_1: 50;

      then ( - ( - cn)) > 0 by SQUARE_1: 25;

      then

       A2: (p0 `1 ) = ( - cn) & ( - cn) < 0 by EUCLID: 52;

      then p0 in K0 by A1, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

       not p0 in {( 0. ( TOP-REAL 2))} by A2, JGRAPH_2: 3, TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A1, XBOOLE_0:def 5;

      

       A3: K1 c= D

      proof

        let x be object;

        assume x in K1;

        then

        consider p2 be Point of ( TOP-REAL 2) such that

         A4: p2 = x and (p2 `1 ) <= 0 and

         A5: p2 <> ( 0. ( TOP-REAL 2)) by A1;

         not p2 in {( 0. ( TOP-REAL 2))} by A5, TARSKI:def 1;

        hence thesis by A1, A4, XBOOLE_0:def 5;

      end;

      for p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D) st (f . p) in V & V is open holds ex W be Subset of (( TOP-REAL 2) | K1) st p in W & W is open & (f .: W) c= V

      proof

        let p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D);

        assume that

         A6: (f . p) in V and

         A7: V is open;

        consider V2 be Subset of ( TOP-REAL 2) such that

         A8: V2 is open and

         A9: (V2 /\ ( [#] (( TOP-REAL 2) | D))) = V by A7, TOPS_2: 24;

        reconsider W2 = (V2 /\ ( [#] (( TOP-REAL 2) | K1))) as Subset of (( TOP-REAL 2) | K1);

        

         A10: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

        then

         A11: (f . p) = ((sn -FanMorphE ) . p) by A1, FUNCT_1: 49;

        

         A12: (f .: W2) c= V

        proof

          let y be object;

          assume y in (f .: W2);

          then

          consider x be object such that

           A13: x in ( dom f) and

           A14: x in W2 and

           A15: y = (f . x) by FUNCT_1:def 6;

          f is Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D);

          then ( dom f) = K1 by A10, FUNCT_2:def 1;

          then

          consider p4 be Point of ( TOP-REAL 2) such that

           A16: x = p4 and

           A17: (p4 `1 ) <= 0 and p4 <> ( 0. ( TOP-REAL 2)) by A1, A13;

          

           A18: p4 in V2 by A14, A16, XBOOLE_0:def 4;

          p4 in ( [#] (( TOP-REAL 2) | K1)) by A13, A16;

          then p4 in D by A3, A10;

          then

           A19: p4 in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

          (f . p4) = ((sn -FanMorphE ) . p4) by A1, A10, A13, A16, FUNCT_1: 49

          .= p4 by A17, Th82;

          hence thesis by A9, A15, A16, A18, A19, XBOOLE_0:def 4;

        end;

        p in the carrier of (( TOP-REAL 2) | K1);

        then

        consider q be Point of ( TOP-REAL 2) such that

         A20: q = p and

         A21: (q `1 ) <= 0 and q <> ( 0. ( TOP-REAL 2)) by A1, A10;

        ((sn -FanMorphE ) . q) = q by A21, Th82;

        then p in V2 by A6, A9, A11, A20, XBOOLE_0:def 4;

        then

         A22: p in W2 by XBOOLE_0:def 4;

        W2 is open by A8, TOPS_2: 24;

        hence thesis by A22, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JGRAPH_4:95

    

     Th95: for sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th93;

    end;

    theorem :: JGRAPH_4:96

    

     Th96: for sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let sn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < sn & sn < 1 & f = ((sn -FanMorphE ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th94;

    end;

    theorem :: JGRAPH_4:97

    

     Th97: for sn be Real, p be Point of ( TOP-REAL 2) holds |.((sn -FanMorphE ) . p).| = |.p.|

    proof

      let sn be Real, p be Point of ( TOP-REAL 2);

      set f = (sn -FanMorphE );

      set z = (f . p);

      reconsider q = p as Point of ( TOP-REAL 2);

      reconsider qz = z as Point of ( TOP-REAL 2);

      per cases ;

        suppose

         A1: ((q `2 ) / |.q.|) >= sn & (q `1 ) > 0 ;

        then

         A2: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by Th82;

        then

         A3: (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

        

         A4: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by A2, EUCLID: 52;

        

         A5: (((q `2 ) / |.q.|) - sn) >= 0 by A1, XREAL_1: 48;

        

         A6: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         |.q.| <> 0 by A1, JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A6, XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A7, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then

         A8: (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        per cases ;

          suppose

           A9: (1 - sn) = 0 ;

          

           A10: ((((q `2 ) / |.q.|) - sn) / (1 - sn)) = ((((q `2 ) / |.q.|) - sn) * ((1 - sn) " )) by XCMPLX_0:def 9

          .= ((((q `2 ) / |.q.|) - sn) * 0 ) by A9

          .= 0 ;

          then (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) = 1;

          

          then ((sn -FanMorphE ) . q) = |[( |.q.| * 1), ( |.q.| * 0 )]| by A1, A10, Th82, SQUARE_1: 18

          .= |[ |.q.|, 0 ]|;

          then (((sn -FanMorphE ) . q) `1 ) = |.q.| & (((sn -FanMorphE ) . q) `2 ) = 0 by EUCLID: 52;

          

          then |.((sn -FanMorphE ) . p).| = ( sqrt (( |.q.| ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A11: (1 - sn) <> 0 ;

          per cases by A11;

            suppose

             A12: (1 - sn) > 0 ;

            ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by A8, XREAL_1: 24;

            then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A12, XREAL_1: 72;

            then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A12, XCMPLX_1: 197;

            then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A5, A12, SQUARE_1: 49;

            then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A13: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A14: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 )) by A3

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A13, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A4, A14;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A15: (1 - sn) < 0 ;

            ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A1, SQUARE_1: 12, XREAL_1: 8;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, A6, XREAL_1: 74;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then

             A16: 1 > ((q `2 ) / |.p.|) by SQUARE_1: 52;

            (((q `2 ) / |.q.|) - sn) >= 0 by A1, XREAL_1: 48;

            hence thesis by A15, A16, XREAL_1: 9;

          end;

        end;

      end;

        suppose

         A17: ((q `2 ) / |.q.|) < sn & (q `1 ) > 0 ;

        then |.q.| <> 0 by JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A18: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

        

         A19: (((q `2 ) / |.q.|) - sn) < 0 by A17, XREAL_1: 49;

        

         A20: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A20, XREAL_1: 72;

        then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A18, XCMPLX_1: 60;

        then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

        then

         A21: (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

        

         A22: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A17, Th83;

        then

         A23: (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

        

         A24: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A22, EUCLID: 52;

        per cases ;

          suppose

           A25: (1 + sn) = 0 ;

          ((((q `2 ) / |.q.|) - sn) / (1 + sn)) = ((((q `2 ) / |.q.|) - sn) * ((1 + sn) " )) by XCMPLX_0:def 9

          .= ((((q `2 ) / |.q.|) - sn) * 0 ) by A25

          .= 0 ;

          then (((sn -FanMorphE ) . q) `1 ) = |.q.| & (((sn -FanMorphE ) . q) `2 ) = 0 by A22, EUCLID: 52, SQUARE_1: 18;

          

          then |.((sn -FanMorphE ) . p).| = ( sqrt (( |.q.| ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A26: (1 + sn) <> 0 ;

          per cases by A26;

            suppose

             A27: (1 + sn) > 0 ;

            then (( - (1 + sn)) / (1 + sn)) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A21, XREAL_1: 72;

            then ( - 1) <= ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A27, XCMPLX_1: 197;

            then (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) <= (1 ^2 ) by A19, A27, SQUARE_1: 49;

            then

             A28: (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

            

             A29: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 )) by A23

            .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A28, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A24, A29;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A30: (1 + sn) < 0 ;

            ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A17, SQUARE_1: 12, XREAL_1: 8;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A18, A20, XREAL_1: 74;

            then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A18, XCMPLX_1: 60;

            then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then ( - 1) < ((q `2 ) / |.p.|) by SQUARE_1: 52;

            then

             A31: (((q `2 ) / |.q.|) - sn) > (( - 1) - sn) by XREAL_1: 9;

            ( - (1 + sn)) > ( - 0 ) by A30, XREAL_1: 24;

            hence thesis by A17, A31, XREAL_1: 49;

          end;

        end;

      end;

        suppose (q `1 ) <= 0 ;

        hence thesis by Th82;

      end;

    end;

    theorem :: JGRAPH_4:98

    

     Th98: for sn be Real, x,K0 be set st ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((sn -FanMorphE ) . x) in K0

    proof

      let sn be Real, x,K0 be set;

      assume

       A1: ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then

      consider p such that

       A2: p = x and

       A3: (p `1 ) >= 0 and

       A4: p <> ( 0. ( TOP-REAL 2));

       A5:

      now

        assume |.p.| <= 0 ;

        then |.p.| = 0 ;

        hence contradiction by A4, TOPRNS_1: 24;

      end;

      then

       A6: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

      per cases ;

        suppose

         A7: ((p `2 ) / |.p.|) <= sn;

        reconsider p9 = ((sn -FanMorphE ) . p) as Point of ( TOP-REAL 2);

        ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A3, A4, A7, Th84;

        then

         A8: (p9 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

        

         A9: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A10: (1 + sn) > 0 by A1, XREAL_1: 148;

        per cases ;

          suppose (p `1 ) = 0 ;

          hence thesis by A1, A2, Th82;

        end;

          suppose (p `1 ) <> 0 ;

          then ( 0 + ((p `2 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A9, XREAL_1: 74;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `2 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ( - 1) < ((p `2 ) / |.p.|) by SQUARE_1: 52;

          then (( - 1) - sn) < (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

          then ((( - 1) * (1 + sn)) / (1 + sn)) < ((((p `2 ) / |.p.|) - sn) / (1 + sn)) by A10, XREAL_1: 74;

          then

           A11: ( - 1) < ((((p `2 ) / |.p.|) - sn) / (1 + sn)) by A10, XCMPLX_1: 89;

          (((p `2 ) / |.p.|) - sn) <= 0 by A7, XREAL_1: 47;

          then (1 ^2 ) > (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ) by A10, A11, SQUARE_1: 50;

          then (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )) > 0 by XREAL_1: 50;

          then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) > 0 by SQUARE_1: 25;

          then ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) > 0 by A5, XREAL_1: 129;

          hence thesis by A1, A2, A8, JGRAPH_2: 3;

        end;

      end;

        suppose

         A12: ((p `2 ) / |.p.|) > sn;

        reconsider p9 = ((sn -FanMorphE ) . p) as Point of ( TOP-REAL 2);

        ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A3, A4, A12, Th84;

        then

         A13: (p9 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

        

         A14: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A15: (1 - sn) > 0 by A1, XREAL_1: 149;

        per cases ;

          suppose (p `1 ) = 0 ;

          hence thesis by A1, A2, Th82;

        end;

          suppose (p `1 ) <> 0 ;

          then ( 0 + ((p `2 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A14, XREAL_1: 74;

          then (((p `2 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `2 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ((p `2 ) / |.p.|) < 1 by SQUARE_1: 52;

          then (((p `2 ) / |.p.|) - sn) < (1 - sn) by XREAL_1: 9;

          then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) < ((1 - sn) / (1 - sn)) by A15, XREAL_1: 74;

          then

           A16: ((((p `2 ) / |.p.|) - sn) / (1 - sn)) < 1 by A15, XCMPLX_1: 60;

          ( - (1 - sn)) < ( - 0 ) & (((p `2 ) / |.p.|) - sn) >= (sn - sn) by A12, A15, XREAL_1: 9, XREAL_1: 24;

          then ((( - 1) * (1 - sn)) / (1 - sn)) < ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A15, XREAL_1: 74;

          then ( - 1) < ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A15, XCMPLX_1: 89;

          then (1 ^2 ) > (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ) by A16, SQUARE_1: 50;

          then (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )) > 0 by XREAL_1: 50;

          then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) > 0 by SQUARE_1: 25;

          then (p9 `1 ) > 0 by A5, A13, XREAL_1: 129;

          hence thesis by A1, A2, JGRAPH_2: 3;

        end;

      end;

    end;

    theorem :: JGRAPH_4:99

    

     Th99: for sn be Real, x,K0 be set st ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((sn -FanMorphE ) . x) in K0

    proof

      let sn be Real, x,K0 be set;

      assume

       A1: ( - 1) < sn & sn < 1 & x in K0 & K0 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then ex p st p = x & (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

      hence thesis by A1, Th82;

    end;

    theorem :: JGRAPH_4:100

    

     Th100: for sn be Real, D be non empty Subset of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (D ` ) = {( 0. ( TOP-REAL 2))} holds ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((sn -FanMorphE ) | D) & h is continuous

    proof

      ( |[ 0 , 1]| `1 ) = 0 & ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

      then

       A1: |[ 0 , 1]| in { p where p be Point of ( TOP-REAL 2) : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      set Y1 = |[ 0 , 1]|;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= 0 ;

      reconsider B0 = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

      let sn be Real, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A2: ( - 1) < sn & sn < 1 and

       A3: (D ` ) = {( 0. ( TOP-REAL 2))};

      

       A4: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A5: D = (B0 ` ) by A3

      .= ( NonZero ( TOP-REAL 2)) by SUBSET_1:def 4;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K0 = { p : (p `1 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A1;

      

       A6: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      

       A7: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A8: ( rng ((sn -FanMorphE ) | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphE ) | K0));

        then

        consider x be object such that

         A9: x in ( dom ((sn -FanMorphE ) | K0)) and

         A10: y = (((sn -FanMorphE ) | K0) . x) by FUNCT_1:def 3;

        x in (( dom (sn -FanMorphE )) /\ K0) by A9, RELAT_1: 61;

        then

         A11: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A11;

        ((sn -FanMorphE ) . p) = y by A10, A11, FUNCT_1: 49;

        then y in K0 by A2, A11, Th98;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A12: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `1 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      (Y1 `1 ) = 0 & (Y1 `2 ) = 1 by EUCLID: 52;

      then

       A13: Y1 in { p where p be Point of ( TOP-REAL 2) : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      

       A14: the carrier of (( TOP-REAL 2) | D) = ( NonZero ( TOP-REAL 2)) by A5, PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= 0 ;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A5);

      then

      reconsider K1 = { p : (p `1 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A13;

      

       A15: K0 is closed & K1 is closed by A5, Th29, Th31;

      ( dom ((sn -FanMorphE ) | K0)) = (( dom (sn -FanMorphE )) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by FUNCT_2:def 1

      .= K0 by A12, XBOOLE_1: 28;

      then

      reconsider f = ((sn -FanMorphE ) | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A6, A8, FUNCT_2: 2, XBOOLE_1: 1;

      

       A16: K1 = the carrier of ((( TOP-REAL 2) | D) | K1) by PRE_TOPC: 8;

      

       A17: ( rng ((sn -FanMorphE ) | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        let y be object;

        assume y in ( rng ((sn -FanMorphE ) | K1));

        then

        consider x be object such that

         A18: x in ( dom ((sn -FanMorphE ) | K1)) and

         A19: y = (((sn -FanMorphE ) | K1) . x) by FUNCT_1:def 3;

        x in (( dom (sn -FanMorphE )) /\ K1) by A18, RELAT_1: 61;

        then

         A20: x in K1 by XBOOLE_0:def 4;

        K1 c= the carrier of ( TOP-REAL 2) by A7, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A20;

        ((sn -FanMorphE ) . p) = y by A19, A20, FUNCT_1: 49;

        then y in K1 by A2, A20, Th99;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A21: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `1 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ((sn -FanMorphE ) | K1)) = (( dom (sn -FanMorphE )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by A21, XBOOLE_1: 28;

      then

      reconsider g = ((sn -FanMorphE ) | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A16, A17, FUNCT_2: 2, XBOOLE_1: 1;

      

       A22: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A23: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A24: x in D;

        then

        reconsider px = x as Point of ( TOP-REAL 2);

         not x in {( 0. ( TOP-REAL 2))} by A5, A24, XBOOLE_0:def 5;

        then (px `1 ) >= 0 & px <> ( 0. ( TOP-REAL 2)) or (px `1 ) <= 0 & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        then x in K0 or x in K1;

        hence thesis by XBOOLE_0:def 3;

      end;

      

       A25: ( dom f) = K0 by A6, FUNCT_2:def 1;

      

       A26: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A27: for x be object st x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1))) holds (f . x) = (g . x)

      proof

        let x be object;

        assume

         A28: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A26, XBOOLE_0:def 4;

        then (f . x) = ((sn -FanMorphE ) . x) by FUNCT_1: 49;

        hence thesis by A22, A28, FUNCT_1: 49;

      end;

      D = ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

      then

       A29: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A26, A22, A23, XBOOLE_0:def 10;

      

       A30: f is continuous & g is continuous by A2, A5, Th95, Th96;

      then

      consider h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) such that

       A31: h = (f +* g) and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2: 1;

      

       A32: ( dom h) = the carrier of (( TOP-REAL 2) | D) by FUNCT_2:def 1;

      

       A33: ( dom g) = K1 by A16, FUNCT_2:def 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A34: f tolerates g by A27, A25, A33, PARTFUN1:def 4;

      

       A35: for x be object st x in ( dom h) holds (h . x) = (((sn -FanMorphE ) | D) . x)

      proof

        let x be object;

        assume

         A36: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A14, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A14, A36, XBOOLE_0:def 5;

        then

         A37: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        

         A38: x in ((D ` ) ` ) by A32, A36, PRE_TOPC: 8;

        now

          per cases ;

            case

             A39: x in K0;

            

             A40: (((sn -FanMorphE ) | D) . p) = ((sn -FanMorphE ) . p) by A38, FUNCT_1: 49

            .= (f . p) by A39, FUNCT_1: 49;

            (h . p) = ((g +* f) . p) by A31, A34, FUNCT_4: 34

            .= (f . p) by A25, A39, FUNCT_4: 13;

            hence thesis by A40;

          end;

            case not x in K0;

            then not (p `1 ) >= 0 by A37;

            then

             A41: x in K1 by A37;

            (((sn -FanMorphE ) | D) . p) = ((sn -FanMorphE ) . p) by A38, FUNCT_1: 49

            .= (g . p) by A41, FUNCT_1: 49;

            hence thesis by A31, A33, A41, FUNCT_4: 13;

          end;

        end;

        hence thesis;

      end;

      ( dom (sn -FanMorphE )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((sn -FanMorphE ) | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A4, XBOOLE_1: 28;

      then (f +* g) = ((sn -FanMorphE ) | D) by A31, A32, A35, FUNCT_1: 2;

      hence thesis by A26, A22, A29, A30, A15, A27, JGRAPH_2: 1;

    end;

    theorem :: JGRAPH_4:101

    

     Th101: for sn be Real st ( - 1) < sn & sn < 1 holds ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (sn -FanMorphE ) & h is continuous

    proof

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      reconsider f = (sn -FanMorphE ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A3: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Th82, JGRAPH_2: 3;

      

       A4: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

        let p be Point of (( TOP-REAL 2) | D);

        

         A5: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A5, XBOOLE_0:def 5;

        then

         A6: not p = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        now

          per cases ;

            case

             A7: ((q `2 ) / |.q.|) >= sn & (q `1 ) >= 0 ;

            set q9 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

            

             A8: (q9 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

            now

              assume

               A9: q9 = ( 0. ( TOP-REAL 2));

              

               A10: |.q.| <> 0 by A6, TOPRNS_1: 24;

              

              then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) = ( sqrt (1 - ( 0 ^2 ))) by A8, A9, JGRAPH_2: 3, XCMPLX_1: 6

              .= 1 by SQUARE_1: 18;

              hence contradiction by A9, A10, EUCLID: 52, JGRAPH_2: 3;

            end;

            hence thesis by A1, A2, A3, A6, A7, Th84;

          end;

            case

             A11: ((q `2 ) / |.q.|) < sn & (q `1 ) >= 0 ;

            set q9 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

            

             A12: (q9 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

            now

              assume

               A13: q9 = ( 0. ( TOP-REAL 2));

              

               A14: |.q.| <> 0 by A6, TOPRNS_1: 24;

              

              then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) = ( sqrt (1 - ( 0 ^2 ))) by A12, A13, JGRAPH_2: 3, XCMPLX_1: 6

              .= 1 by SQUARE_1: 18;

              hence contradiction by A13, A14, EUCLID: 52, JGRAPH_2: 3;

            end;

            hence thesis by A1, A2, A3, A6, A11, Th84;

          end;

            case (q `1 ) < 0 ;

            then (f . p) = p by Th82;

            hence thesis by A6, Th82, JGRAPH_2: 3;

          end;

        end;

        hence thesis;

      end;

      

       A15: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

        let V be Subset of ( TOP-REAL 2);

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm11;

        assume that

         A16: (f . ( 0. ( TOP-REAL 2))) in V and

         A17: V is open;

        VV is open by A17, Lm11, PRE_TOPC: 30;

        then

        consider r be Real such that

         A18: r > 0 and

         A19: ( Ball (u0,r)) c= V by A3, A16, TOPMETR: 15;

        reconsider r as Real;

         the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

        then

        reconsider W1 = ( Ball (u0,r)) as Subset of ( TOP-REAL 2);

        

         A20: W1 is open by GOBOARD6: 3;

        

         A21: (f .: W1) c= W1

        proof

          let z be object;

          assume z in (f .: W1);

          then

          consider y be object such that

           A22: y in ( dom f) and

           A23: y in W1 and

           A24: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A22, A24, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider q = y as Point of ( TOP-REAL 2) by A22;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          ( dist (u0,qy)) < r by A23, METRIC_1: 11;

          then

           A25: |.(( 0. ( TOP-REAL 2)) - q).| < r by JGRAPH_1: 28;

          now

            per cases by JGRAPH_2: 3;

              case (q `1 ) <= 0 ;

              hence thesis by A23, A24, Th82;

            end;

              case

               A26: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) / |.q.|) >= sn & (q `1 ) >= 0 ;

              then

               A27: (((q `2 ) / |.q.|) - sn) >= 0 by XREAL_1: 48;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then

               A28: (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              

               A29: (1 - sn) > 0 by A2, XREAL_1: 149;

               |.q.| <> 0 by A26, TOPRNS_1: 24;

              then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A28, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

              then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

              then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

              then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A29, XREAL_1: 72;

              then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A29, XCMPLX_1: 197;

              then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A29, A27, SQUARE_1: 49;

              then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A30: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

              

               A31: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A26, Th84;

              then

               A32: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by A24, EUCLID: 52;

              (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by A24, A31, EUCLID: 52;

              

              then

               A33: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

              .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A30, SQUARE_1:def 2;

              ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.q.| ^2 ) by A32, A33;

              then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

              then

               A34: |.qz.| = |.q.| by SQUARE_1: 22;

               |.( - q).| < r by A25, RLVECT_1: 4;

              then |.q.| < r by TOPRNS_1: 26;

              then |.( - qz).| < r by A34, TOPRNS_1: 26;

              then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

              then ( dist (u0,pz)) < r by JGRAPH_1: 28;

              hence thesis by METRIC_1: 11;

            end;

              case

               A35: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) / |.q.|) < sn & (q `1 ) >= 0 ;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then

               A36: (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              

               A37: (1 + sn) > 0 by A1, XREAL_1: 148;

               |.q.| <> 0 by A35, TOPRNS_1: 24;

              then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A36, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

              then ( - ( - 1)) >= ( - ((q `2 ) / |.q.|)) by XREAL_1: 24;

              then (1 + sn) >= (( - ((q `2 ) / |.q.|)) + sn) by XREAL_1: 7;

              then

               A38: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A37, XREAL_1: 185;

              (sn - ((q `2 ) / |.q.|)) >= 0 by A35, XREAL_1: 48;

              then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A37;

              then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A38, SQUARE_1: 49;

              then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

              then

               A39: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

              

               A40: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A35, Th84;

              then

               A41: (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A24, EUCLID: 52;

              (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by A24, A40, EUCLID: 52;

              

              then

               A42: ((qz `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

              .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A39, SQUARE_1:def 2;

              ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

              .= ( |.q.| ^2 ) by A41, A42;

              then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

              then

               A43: |.qz.| = |.q.| by SQUARE_1: 22;

               |.( - q).| < r by A25, RLVECT_1: 4;

              then |.q.| < r by TOPRNS_1: 26;

              then |.( - qz).| < r by A43, TOPRNS_1: 26;

              then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

              then ( dist (u0,pz)) < r by JGRAPH_1: 28;

              hence thesis by METRIC_1: 11;

            end;

          end;

          hence thesis;

        end;

        u0 in W1 by A18, GOBOARD6: 1;

        hence thesis by A19, A20, A21, XBOOLE_1: 1;

      end;

      

       A44: (D ` ) = {( 0. ( TOP-REAL 2))} by JGRAPH_3: 20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((sn -FanMorphE ) | D) & h is continuous by A1, A2, Th100;

      hence thesis by A3, A44, A4, A15, JGRAPH_3: 3;

    end;

    theorem :: JGRAPH_4:102

    

     Th102: for sn be Real st ( - 1) < sn & sn < 1 holds (sn -FanMorphE ) is one-to-one

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      for x1,x2 be object st x1 in ( dom (sn -FanMorphE )) & x2 in ( dom (sn -FanMorphE )) & ((sn -FanMorphE ) . x1) = ((sn -FanMorphE ) . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A3: x1 in ( dom (sn -FanMorphE )) and

         A4: x2 in ( dom (sn -FanMorphE )) and

         A5: ((sn -FanMorphE ) . x1) = ((sn -FanMorphE ) . x2);

        reconsider p2 = x2 as Point of ( TOP-REAL 2) by A4;

        reconsider p1 = x1 as Point of ( TOP-REAL 2) by A3;

        set q = p1, p = p2;

        

         A6: (1 - sn) > 0 by A2, XREAL_1: 149;

        now

          per cases by JGRAPH_2: 3;

            case

             A7: (q `1 ) <= 0 ;

            then

             A8: ((sn -FanMorphE ) . q) = q by Th82;

            now

              per cases by JGRAPH_2: 3;

                case (p `1 ) <= 0 ;

                hence thesis by A5, A8, Th82;

              end;

                case

                 A9: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 ;

                then

                 A10: (((p `2 ) / |.p.|) - sn) >= 0 by XREAL_1: 48;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

                

                 A11: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A12: |.p.| <> 0 by A9, TOPRNS_1: 24;

                then

                 A13: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A11, XREAL_1: 72;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A13, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `2 ) / |.p.|) by SQUARE_1: 51;

                then (1 - sn) >= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A14: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                

                 A15: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A2, A9, Th84;

                (((p `2 ) / |.p.|) - sn) >= 0 by A9, XREAL_1: 48;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A14, SQUARE_1: 49;

                then

                 A16: (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) & (q `1 ) = 0 by A5, A7, A8, A15, EUCLID: 52;

                then

                 A17: ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) = 0 by A5, A8, A15, A12, XCMPLX_1: 6;

                (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 - sn))) ^2 )) >= 0 by A16, XCMPLX_1: 187;

                then (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )) = 0 by A17, SQUARE_1: 24;

                then 1 = ((((p `2 ) / |.p.|) - sn) / (1 - sn)) by A6, A10, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - sn)) = (((p `2 ) / |.p.|) - sn) by A6, XCMPLX_1: 87;

                then (1 * |.p.|) = (p `2 ) by A9, TOPRNS_1: 24, XCMPLX_1: 87;

                then (p `1 ) = 0 by A11, XCMPLX_1: 6;

                hence thesis by A5, A8, Th82;

              end;

                case

                 A18: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) >= 0 ;

                then

                 A19: |.p.| <> 0 by TOPRNS_1: 24;

                then

                 A20: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

                

                 A21: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A22: (1 + sn) > 0 by A1, XREAL_1: 148;

                

                 A23: (((p `2 ) / |.p.|) - sn) <= 0 by A18, XREAL_1: 47;

                then

                 A24: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by A22;

                

                 A25: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A2, A18, Th84;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A21, XREAL_1: 72;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A20, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then (( - ((p `2 ) / |.p.|)) ^2 ) <= 1;

                then 1 >= ( - ((p `2 ) / |.p.|)) by SQUARE_1: 51;

                then (1 + sn) >= (( - ((p `2 ) / |.p.|)) + sn) by XREAL_1: 7;

                then (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) <= 1 by A22, XREAL_1: 185;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A24, SQUARE_1: 49;

                then

                 A26: (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((p `2 ) / |.p.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p4 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) & (q `1 ) = 0 by A5, A7, A8, A25, EUCLID: 52;

                then

                 A27: ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) = 0 by A5, A8, A25, A19, XCMPLX_1: 6;

                (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) >= 0 by A26, XCMPLX_1: 187;

                then (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )) = 0 by A27, SQUARE_1: 24;

                then 1 = ( sqrt (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) by SQUARE_1: 18;

                then 1 = ( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by A22, A23, SQUARE_1: 22;

                then 1 = (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by XCMPLX_1: 187;

                then (1 * (1 + sn)) = ( - (((p `2 ) / |.p.|) - sn)) by A22, XCMPLX_1: 87;

                then ((1 + sn) - sn) = ( - ((p `2 ) / |.p.|));

                then 1 = (( - (p `2 )) / |.p.|) by XCMPLX_1: 187;

                then (1 * |.p.|) = ( - (p `2 )) by A18, TOPRNS_1: 24, XCMPLX_1: 87;

                then (((p `2 ) ^2 ) - ((p `2 ) ^2 )) = ((p `1 ) ^2 ) by A21, XCMPLX_1: 26;

                then (p `1 ) = 0 by XCMPLX_1: 6;

                hence thesis by A5, A8, Th82;

              end;

            end;

            hence thesis;

          end;

            case

             A28: ((q `2 ) / |.q.|) >= sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

            then |.q.| <> 0 by TOPRNS_1: 24;

            then

             A29: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;

            

             A30: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

            

             A31: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th84;

            

             A32: (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

            now

              per cases by JGRAPH_2: 3;

                case

                 A33: (p `1 ) <= 0 ;

                then

                 A34: ((sn -FanMorphE ) . p) = p by Th82;

                

                 A35: |.q.| <> 0 by A28, TOPRNS_1: 24;

                then

                 A36: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

                

                 A37: (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                

                 A38: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A39: (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                

                 A40: (1 - sn) > 0 by A2, XREAL_1: 149;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A38, XREAL_1: 72;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A36, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

                then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A40, XREAL_1: 72;

                then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A40, XCMPLX_1: 197;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A40, A37, SQUARE_1: 49;

                then

                 A41: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) >= 0 by SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p `1 ) = 0 by A5, A31, A33, A34, EUCLID: 52;

                then

                 A42: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) = 0 by A5, A31, A32, A34, A35, XCMPLX_1: 6;

                (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by A41, XCMPLX_1: 187;

                then (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) = 0 by A42, SQUARE_1: 24;

                then 1 = ((((q `2 ) / |.q.|) - sn) / (1 - sn)) by A40, A39, SQUARE_1: 18, SQUARE_1: 22;

                then (1 * (1 - sn)) = (((q `2 ) / |.q.|) - sn) by A40, XCMPLX_1: 87;

                then (1 * |.q.|) = (q `2 ) by A28, TOPRNS_1: 24, XCMPLX_1: 87;

                then (q `1 ) = 0 by A38, XCMPLX_1: 6;

                hence thesis by A5, A34, Th82;

              end;

                case

                 A43: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 ;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A29, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((q `2 ) / |.q.|) by SQUARE_1: 51;

                then (1 - sn) >= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A44: ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                (((q `2 ) / |.q.|) - sn) >= 0 by A28, XREAL_1: 48;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A44, SQUARE_1: 49;

                then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A45: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

                

                then

                 A46: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

                .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) by A45, SQUARE_1:def 2;

                

                 A47: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) by EUCLID: 52;

                ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.q.| ^2 ) by A47, A46;

                then

                 A48: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

                then

                 A49: |.q4.| = |.q.| by SQUARE_1: 22;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then

                 A50: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

                 |.p.| <> 0 by A43, TOPRNS_1: 24;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A50, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then 1 >= ((p `2 ) / |.p.|) by SQUARE_1: 51;

                then (1 - sn) >= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

                then ( - (1 - sn)) <= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

                then (( - (1 - sn)) / (1 - sn)) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XREAL_1: 72;

                then

                 A51: ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) by A6, XCMPLX_1: 197;

                (((p `2 ) / |.p.|) - sn) >= 0 by A43, XREAL_1: 48;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 ) <= (1 ^2 ) by A6, A51, SQUARE_1: 49;

                then (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 - sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A52: (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 - sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

                

                 A53: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) by EUCLID: 52;

                (p4 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

                

                then

                 A54: ((p4 `1 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) ^2 ))

                .= (( |.p.| ^2 ) * (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 ))) by A52, SQUARE_1:def 2;

                ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.p.| ^2 ) by A53, A54;

                then

                 A55: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

                then

                 A56: |.p4.| = |.p.| by SQUARE_1: 22;

                

                 A57: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| by A1, A2, A43, Th84;

                then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) = (( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn))) / |.p.|) by A5, A31, A30, A43, A53, TOPRNS_1: 24, XCMPLX_1: 89;

                then ((((p `2 ) / |.p.|) - sn) / (1 - sn)) = ((((q `2 ) / |.q.|) - sn) / (1 - sn)) by A5, A31, A43, A57, A48, A55, TOPRNS_1: 24, XCMPLX_1: 89;

                then (((((p `2 ) / |.p.|) - sn) / (1 - sn)) * (1 - sn)) = (((q `2 ) / |.q.|) - sn) by A6, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) - sn) = (((q `2 ) / |.q.|) - sn) by A6, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) * |.p.|) = (q `2 ) by A5, A31, A43, A57, A49, A56, TOPRNS_1: 24, XCMPLX_1: 87;

                then

                 A58: (p `2 ) = (q `2 ) by A43, TOPRNS_1: 24, XCMPLX_1: 87;

                

                 A59: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

                ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                then (p `1 ) = ( sqrt ((q `1 ) ^2 )) by A5, A31, A43, A57, A49, A56, A58, SQUARE_1: 22;

                then (p `1 ) = (q `1 ) by A28, SQUARE_1: 22;

                hence thesis by A58, A59, EUCLID: 53;

              end;

                case

                 A60: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) >= 0 ;

                then (((p `2 ) / |.p.|) - sn) < 0 by XREAL_1: 49;

                then

                 A61: ((((p `2 ) / |.p.|) - sn) / (1 + sn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

                

                 A62: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) & (((q `2 ) / |.q.|) - sn) >= 0 by A28, EUCLID: 52, XREAL_1: 48;

                

                 A63: (1 - sn) > 0 by A2, XREAL_1: 149;

                ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| & |.p.| <> 0 by A1, A2, A60, Th84, TOPRNS_1: 24;

                hence thesis by A5, A31, A30, A61, A62, A63, XREAL_1: 132;

              end;

            end;

            hence thesis;

          end;

            case

             A64: ((q `2 ) / |.q.|) < sn & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

            then

             A65: |.q.| <> 0 by TOPRNS_1: 24;

            then

             A66: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            set q4 = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;

            

             A67: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

            

             A68: ((sn -FanMorphE ) . q) = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A64, Th84;

            

             A69: (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

            now

              per cases by JGRAPH_2: 3;

                case

                 A70: (p `1 ) <= 0 ;

                

                 A71: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A72: (1 + sn) > 0 by A1, XREAL_1: 148;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A71, XREAL_1: 72;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A66, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then (( - ((q `2 ) / |.q.|)) ^2 ) <= 1;

                then 1 >= ( - ((q `2 ) / |.q.|)) by SQUARE_1: 51;

                then (1 + sn) >= (( - ((q `2 ) / |.q.|)) + sn) by XREAL_1: 7;

                then

                 A73: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A72, XREAL_1: 185;

                

                 A74: (((q `2 ) / |.q.|) - sn) <= 0 by A64, XREAL_1: 47;

                then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A72;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A73, SQUARE_1: 49;

                then

                 A75: (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A76: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                

                 A77: ((sn -FanMorphE ) . p) = p by A70, Th82;

                ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) >= 0 by A75, SQUARE_1:def 2;

                then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) >= 0 by XCMPLX_1: 76;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) >= 0 ;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) >= 0 by XCMPLX_1: 76;

                then (p `1 ) = 0 by A5, A68, A70, A77, EUCLID: 52;

                then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) = 0 by A5, A68, A69, A65, A77, XCMPLX_1: 6;

                then (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) = 0 by A76, SQUARE_1: 24;

                then 1 = ( sqrt (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) by SQUARE_1: 18;

                then 1 = ( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by A72, A74, SQUARE_1: 22;

                then 1 = (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by XCMPLX_1: 187;

                then (1 * (1 + sn)) = ( - (((q `2 ) / |.q.|) - sn)) by A72, XCMPLX_1: 87;

                then ((1 + sn) - sn) = ( - ((q `2 ) / |.q.|));

                then 1 = (( - (q `2 )) / |.q.|) by XCMPLX_1: 187;

                then (1 * |.q.|) = ( - (q `2 )) by A64, TOPRNS_1: 24, XCMPLX_1: 87;

                then (((q `2 ) ^2 ) - ((q `2 ) ^2 )) = ((q `1 ) ^2 ) by A71, XCMPLX_1: 26;

                then (q `1 ) = 0 by XCMPLX_1: 6;

                hence thesis by A5, A77, Th82;

              end;

                case

                 A78: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) >= sn & (p `1 ) >= 0 ;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]|;

                

                 A79: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn))) & |.q.| <> 0 by A64, EUCLID: 52, TOPRNS_1: 24;

                (((q `2 ) / |.q.|) - sn) < 0 by A64, XREAL_1: 49;

                then

                 A80: ((((q `2 ) / |.q.|) - sn) / (1 + sn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

                

                 A81: (1 - sn) > 0 by A2, XREAL_1: 149;

                ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 - sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 - sn)))]| & (((p `2 ) / |.p.|) - sn) >= 0 by A1, A2, A78, Th84, XREAL_1: 48;

                hence thesis by A5, A68, A67, A80, A79, A81, XREAL_1: 132;

              end;

                case

                 A82: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) / |.p.|) < sn & (p `1 ) >= 0 ;

                 0 <= ((p `1 ) ^2 ) by XREAL_1: 63;

                then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `2 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then

                 A83: (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

                

                 A84: (1 + sn) > 0 by A1, XREAL_1: 148;

                 0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

                then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

                then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A66, XCMPLX_1: 60;

                then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then ( - 1) <= ((q `2 ) / |.q.|) by SQUARE_1: 51;

                then (( - 1) - sn) <= (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

                then ( - (( - 1) - sn)) >= ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

                then

                 A85: (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) <= 1 by A84, XREAL_1: 185;

                (((q `2 ) / |.q.|) - sn) <= 0 by A64, XREAL_1: 47;

                then ( - 1) <= (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A84;

                then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A85, SQUARE_1: 49;

                then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A86: (1 - (( - ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (q4 `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

                

                then

                 A87: ((q4 `1 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

                .= (( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) by A86, SQUARE_1:def 2;

                

                 A88: (q4 `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

                set p4 = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]|;

                

                 A89: (p4 `2 ) = ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn))) by EUCLID: 52;

                 |.p.| <> 0 by A82, TOPRNS_1: 24;

                then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

                then (((p `2 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A83, XCMPLX_1: 60;

                then (((p `2 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

                then ( - 1) <= ((p `2 ) / |.p.|) by SQUARE_1: 51;

                then (( - 1) - sn) <= (((p `2 ) / |.p.|) - sn) by XREAL_1: 9;

                then ( - (( - 1) - sn)) >= ( - (((p `2 ) / |.p.|) - sn)) by XREAL_1: 24;

                then

                 A90: (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) <= 1 by A84, XREAL_1: 185;

                (((p `2 ) / |.p.|) - sn) <= 0 by A82, XREAL_1: 47;

                then ( - 1) <= (( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) by A84;

                then ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 ) <= (1 ^2 ) by A90, SQUARE_1: 49;

                then (1 - ((( - (((p `2 ) / |.p.|) - sn)) / (1 + sn)) ^2 )) >= 0 by XREAL_1: 48;

                then

                 A91: (1 - (( - ((((p `2 ) / |.p.|) - sn) / (1 + sn))) ^2 )) >= 0 by XCMPLX_1: 187;

                (p4 `1 ) = ( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))) by EUCLID: 52;

                

                then

                 A92: ((p4 `1 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) ^2 ))

                .= (( |.p.| ^2 ) * (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 ))) by A91, SQUARE_1:def 2;

                ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.p.| ^2 ) by A89, A92;

                then

                 A93: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

                then

                 A94: |.p4.| = |.p.| by SQUARE_1: 22;

                ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

                .= ( |.q.| ^2 ) by A88, A87;

                then

                 A95: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

                then

                 A96: |.q4.| = |.q.| by SQUARE_1: 22;

                

                 A97: ((sn -FanMorphE ) . p) = |[( |.p.| * ( sqrt (1 - (((((p `2 ) / |.p.|) - sn) / (1 + sn)) ^2 )))), ( |.p.| * ((((p `2 ) / |.p.|) - sn) / (1 + sn)))]| by A1, A2, A82, Th84;

                then ((((p `2 ) / |.p.|) - sn) / (1 + sn)) = (( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) / |.p.|) by A5, A68, A67, A82, A89, TOPRNS_1: 24, XCMPLX_1: 89;

                then ((((p `2 ) / |.p.|) - sn) / (1 + sn)) = ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by A5, A68, A82, A97, A95, A93, TOPRNS_1: 24, XCMPLX_1: 89;

                then (((((p `2 ) / |.p.|) - sn) / (1 + sn)) * (1 + sn)) = (((q `2 ) / |.q.|) - sn) by A84, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) - sn) = (((q `2 ) / |.q.|) - sn) by A84, XCMPLX_1: 87;

                then (((p `2 ) / |.p.|) * |.p.|) = (q `2 ) by A5, A68, A82, A97, A96, A94, TOPRNS_1: 24, XCMPLX_1: 87;

                then

                 A98: (p `2 ) = (q `2 ) by A82, TOPRNS_1: 24, XCMPLX_1: 87;

                

                 A99: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

                ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

                then (p `1 ) = ( sqrt ((q `1 ) ^2 )) by A5, A68, A82, A97, A96, A94, A98, SQUARE_1: 22;

                then (p `1 ) = (q `1 ) by A64, SQUARE_1: 22;

                hence thesis by A98, A99, EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      hence thesis by FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_4:103

    

     Th103: for sn be Real st ( - 1) < sn & sn < 1 holds (sn -FanMorphE ) is Function of ( TOP-REAL 2), ( TOP-REAL 2) & ( rng (sn -FanMorphE )) = the carrier of ( TOP-REAL 2)

    proof

      let sn be Real;

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1;

      thus (sn -FanMorphE ) is Function of ( TOP-REAL 2), ( TOP-REAL 2);

      for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (sn -FanMorphE ) holds ( rng (sn -FanMorphE )) = the carrier of ( TOP-REAL 2)

      proof

        let f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

        assume

         A3: f = (sn -FanMorphE );

        

         A4: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        the carrier of ( TOP-REAL 2) c= ( rng f)

        proof

          let y be object;

          assume y in the carrier of ( TOP-REAL 2);

          then

          reconsider p2 = y as Point of ( TOP-REAL 2);

          set q = p2;

          now

            per cases by JGRAPH_2: 3;

              suppose (q `1 ) <= 0 ;

              then y = ((sn -FanMorphE ) . q) by Th82;

              hence ex x be set st x in ( dom (sn -FanMorphE )) & y = ((sn -FanMorphE ) . x) by A3, A4;

            end;

              suppose

               A5: ((q `2 ) / |.q.|) >= 0 & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

              ( - ( - (1 + sn))) > 0 by A1, XREAL_1: 148;

              then

               A6: ( - (( - 1) - sn)) > 0 ;

              

               A7: (1 - sn) >= 0 by A2, XREAL_1: 149;

              then (((q `2 ) / |.q.|) * (1 - sn)) >= 0 by A5;

              then (( - 1) - sn) <= (((q `2 ) / |.q.|) * (1 - sn)) by A6;

              then

               A8: ((( - 1) - sn) + sn) <= ((((q `2 ) / |.q.|) * (1 - sn)) + sn) by XREAL_1: 7;

              set px = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn))]|;

              

               A9: (px `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn)) by EUCLID: 52;

               |.q.| <> 0 by A5, TOPRNS_1: 24;

              then

               A10: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              

               A11: ( dom (sn -FanMorphE )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A12: (1 - sn) > 0 by A2, XREAL_1: 149;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A10, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `2 ) / |.q.|) <= 1 by SQUARE_1: 51;

              then (((q `2 ) / |.q.|) * (1 - sn)) <= (1 * (1 - sn)) by A12, XREAL_1: 64;

              then (((((q `2 ) / |.q.|) * (1 - sn)) + sn) - sn) <= (1 - sn);

              then ((((q `2 ) / |.q.|) * (1 - sn)) + sn) <= 1 by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ) by A8, SQUARE_1: 49;

              then

               A13: (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A14: ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A15: (px `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )))) ^2 ) + (( |.q.| * ((((q `2 ) / |.q.|) * (1 - sn)) + sn)) ^2 )) by A9, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 )));

              

              then

               A16: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 - sn)) + sn) ^2 ))) by A13, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A17: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A18: px <> ( 0. ( TOP-REAL 2)) by A5, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `2 ) / |.q.|) * (1 - sn)) + sn) >= ( 0 + sn) by A5, A7, XREAL_1: 7;

              then ((px `2 ) / |.px.|) >= sn by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A19: ((sn -FanMorphE ) . px) = |[( |.px.| * ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 - sn)) ^2 )))), ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 - sn)))]| by A1, A2, A15, A14, A18, Th84;

              

               A20: ( |.px.| * ( sqrt (((q `1 ) / |.q.|) ^2 ))) = ( |.q.| * ((q `1 ) / |.q.|)) by A5, A17, SQUARE_1: 22

              .= (q `1 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

               A21: ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 - sn))) = ( |.q.| * ((((((q `2 ) / |.q.|) * (1 - sn)) + sn) - sn) / (1 - sn))) by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `2 ) / |.q.|)) by A12, XCMPLX_1: 89

              .= (q `2 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 - sn)) ^2 )))) = ( |.px.| * ( sqrt (1 - (((q `2 ) / |.px.|) ^2 )))) by A5, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( sqrt (1 - (((q `2 ) ^2 ) / ( |.px.| ^2 ))))) by XCMPLX_1: 76

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `2 ) ^2 ) / ( |.px.| ^2 ))))) by A10, A16, XCMPLX_1: 60

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) - ((q `2 ) ^2 )) / ( |.px.| ^2 )))) by XCMPLX_1: 120

              .= ( |.px.| * ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `2 ) ^2 )) / ( |.px.| ^2 )))) by A16, JGRAPH_3: 1

              .= ( |.px.| * ( sqrt (((q `1 ) / |.q.|) ^2 ))) by A17, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (sn -FanMorphE )) & y = ((sn -FanMorphE ) . x) by A19, A21, A20, A11, EUCLID: 53;

            end;

              suppose

               A22: ((q `2 ) / |.q.|) < 0 & (q `1 ) >= 0 & q <> ( 0. ( TOP-REAL 2));

              

               A23: (1 + sn) >= 0 by A1, XREAL_1: 148;

              (1 - sn) > 0 by A2, XREAL_1: 149;

              then

               A24: ((1 - sn) + sn) >= ((((q `2 ) / |.q.|) * (1 + sn)) + sn) by A22, A23, XREAL_1: 7;

              

               A25: (1 + sn) > 0 by A1, XREAL_1: 148;

               |.q.| <> 0 by A22, TOPRNS_1: 24;

              then

               A26: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

               0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `2 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A26, XCMPLX_1: 60;

              then (((q `2 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `2 ) / |.q.|) >= ( - 1) by SQUARE_1: 51;

              then (((q `2 ) / |.q.|) * (1 + sn)) >= (( - 1) * (1 + sn)) by A25, XREAL_1: 64;

              then (((((q `2 ) / |.q.|) * (1 + sn)) + sn) - sn) >= (( - 1) - sn);

              then ((((q `2 ) / |.q.|) * (1 + sn)) + sn) >= ( - 1) by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ) by A24, SQUARE_1: 49;

              then

               A27: (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A28: ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A29: ( dom (sn -FanMorphE )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              set px = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn))]|;

              

               A30: (px `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn)) by EUCLID: 52;

              

               A31: (px `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )))) ^2 ) + (( |.q.| * ((((q `2 ) / |.q.|) * (1 + sn)) + sn)) ^2 )) by A30, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 )));

              

              then

               A32: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) + (( |.q.| ^2 ) * (((((q `2 ) / |.q.|) * (1 + sn)) + sn) ^2 ))) by A27, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A33: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A34: px <> ( 0. ( TOP-REAL 2)) by A22, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `2 ) / |.q.|) * (1 + sn)) + sn) <= ( 0 + sn) by A22, A23, XREAL_1: 7;

              then ((px `2 ) / |.px.|) <= sn by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A35: ((sn -FanMorphE ) . px) = |[( |.px.| * ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 + sn)) ^2 )))), ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 + sn)))]| by A1, A2, A31, A28, A34, Th84;

              

               A36: ( |.px.| * ( sqrt (((q `1 ) / |.q.|) ^2 ))) = ( |.q.| * ((q `1 ) / |.q.|)) by A22, A33, SQUARE_1: 22

              .= (q `1 ) by A22, TOPRNS_1: 24, XCMPLX_1: 87;

              

               A37: ( |.px.| * ((((px `2 ) / |.px.|) - sn) / (1 + sn))) = ( |.q.| * ((((((q `2 ) / |.q.|) * (1 + sn)) + sn) - sn) / (1 + sn))) by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `2 ) / |.q.|)) by A25, XCMPLX_1: 89

              .= (q `2 ) by A22, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( sqrt (1 - (((((px `2 ) / |.px.|) - sn) / (1 + sn)) ^2 )))) = ( |.px.| * ( sqrt (1 - (((q `2 ) / |.px.|) ^2 )))) by A22, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( sqrt (1 - (((q `2 ) ^2 ) / ( |.px.| ^2 ))))) by XCMPLX_1: 76

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `2 ) ^2 ) / ( |.px.| ^2 ))))) by A26, A32, XCMPLX_1: 60

              .= ( |.px.| * ( sqrt ((( |.px.| ^2 ) - ((q `2 ) ^2 )) / ( |.px.| ^2 )))) by XCMPLX_1: 120

              .= ( |.px.| * ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `2 ) ^2 )) / ( |.px.| ^2 )))) by A32, JGRAPH_3: 1

              .= ( |.px.| * ( sqrt (((q `1 ) / |.q.|) ^2 ))) by A33, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (sn -FanMorphE )) & y = ((sn -FanMorphE ) . x) by A35, A37, A36, A29, EUCLID: 53;

            end;

          end;

          hence thesis by A3, FUNCT_1:def 3;

        end;

        hence thesis by A3, XBOOLE_0:def 10;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_4:104

    

     Th104: for sn be Real, p2 be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = ((sn -FanMorphE ) .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & ((sn -FanMorphE ) . p2) in V2

    proof

      reconsider O = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

      let sn be Real, p2 be Point of ( TOP-REAL 2);

      

       A1: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

       the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

      then

      reconsider V0 = ( Ball (O,( |.p2.| + 1))) as Subset of ( TOP-REAL 2);

      O in V0 & V0 c= ( cl_Ball (O,( |.p2.| + 1))) by GOBOARD6: 1, METRIC_1: 14;

      then

      reconsider K0 = ( cl_Ball (O,( |.p2.| + 1))) as non empty compact Subset of ( TOP-REAL 2) by A1, Th15;

      set q3 = ((sn -FanMorphE ) . p2);

      reconsider VV0 = V0 as Subset of ( TopSpaceMetr ( Euclid 2));

      reconsider u2 = p2 as Point of ( Euclid 2) by EUCLID: 67;

      reconsider u3 = q3 as Point of ( Euclid 2) by EUCLID: 67;

      

       A2: ((sn -FanMorphE ) .: K0) c= K0

      proof

        let y be object;

        assume y in ((sn -FanMorphE ) .: K0);

        then

        consider x be object such that

         A3: x in ( dom (sn -FanMorphE )) and

         A4: x in K0 and

         A5: y = ((sn -FanMorphE ) . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A3;

        reconsider uq = q as Point of ( Euclid 2) by EUCLID: 67;

        ( dist (O,uq)) <= ( |.p2.| + 1) by A4, METRIC_1: 12;

        then |.(( 0. ( TOP-REAL 2)) - q).| <= ( |.p2.| + 1) by JGRAPH_1: 28;

        then |.( - q).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then

         A6: |.q.| <= ( |.p2.| + 1) by TOPRNS_1: 26;

        

         A7: y in ( rng (sn -FanMorphE )) by A3, A5, FUNCT_1:def 3;

        then

        reconsider u = y as Point of ( Euclid 2) by EUCLID: 67;

        reconsider q4 = y as Point of ( TOP-REAL 2) by A7;

         |.q4.| = |.q.| by A5, Th97;

        then |.( - q4).| <= ( |.p2.| + 1) by A6, TOPRNS_1: 26;

        then |.(( 0. ( TOP-REAL 2)) - q4).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then ( dist (O,u)) <= ( |.p2.| + 1) by JGRAPH_1: 28;

        hence thesis by METRIC_1: 12;

      end;

      VV0 is open by TOPMETR: 14;

      then

       A8: V0 is open by Lm11, PRE_TOPC: 30;

      

       A9: |.p2.| < ( |.p2.| + 1) by XREAL_1: 29;

      then |.( - p2).| < ( |.p2.| + 1) by TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - p2).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u2)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A10: p2 in V0 by METRIC_1: 11;

       |.q3.| = |.p2.| by Th97;

      then |.( - q3).| < ( |.p2.| + 1) by A9, TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - q3).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u3)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A11: ((sn -FanMorphE ) . p2) in V0 by METRIC_1: 11;

      assume

       A12: ( - 1) < sn & sn < 1;

      K0 c= ((sn -FanMorphE ) .: K0)

      proof

        let y be object;

        assume

         A13: y in K0;

        then

        reconsider q4 = y as Point of ( TOP-REAL 2);

        reconsider y as Point of ( Euclid 2) by A13;

        the carrier of ( TOP-REAL 2) c= ( rng (sn -FanMorphE )) by A12, Th103;

        then q4 in ( rng (sn -FanMorphE ));

        then

        consider x be object such that

         A14: x in ( dom (sn -FanMorphE )) and

         A15: y = ((sn -FanMorphE ) . x) by FUNCT_1:def 3;

        reconsider x as Point of ( Euclid 2) by A14, Lm11;

        reconsider q = x as Point of ( TOP-REAL 2) by A14;

         |.q4.| = |.q.| by A15, Th97;

        then q in K0 by A13, Lm12;

        hence thesis by A14, A15, FUNCT_1:def 6;

      end;

      then K0 = ((sn -FanMorphE ) .: K0) by A2, XBOOLE_0:def 10;

      hence thesis by A10, A8, A11, METRIC_1: 14;

    end;

    theorem :: JGRAPH_4:105

    for sn be Real st ( - 1) < sn & sn < 1 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (sn -FanMorphE ) & f is being_homeomorphism

    proof

      let sn be Real;

      reconsider f = (sn -FanMorphE ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume

       A1: ( - 1) < sn & sn < 1;

      then

       A2: for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2 by Th104;

      ( rng (sn -FanMorphE )) = the carrier of ( TOP-REAL 2) & ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = (sn -FanMorphE ) & h is continuous by A1, Th101, Th103;

      then f is being_homeomorphism by A1, A2, Th3, Th102;

      hence thesis;

    end;

    theorem :: JGRAPH_4:106

    

     Th106: for sn be Real, q be Point of ( TOP-REAL 2) st sn < 1 & (q `1 ) > 0 & ((q `2 ) / |.q.|) >= sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphE ) . q) holds (p `1 ) > 0 & (p `2 ) >= 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: sn < 1 and

       A2: (q `1 ) > 0 and

       A3: ((q `2 ) / |.q.|) >= sn;

      

       A4: (((q `2 ) / |.q.|) - sn) >= 0 by A3, XREAL_1: 48;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      

       A5: (1 - sn) > 0 by A1, XREAL_1: 149;

      

       A6: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, XREAL_1: 74;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

      then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then 1 > ((q `2 ) / |.q.|) by SQUARE_1: 52;

      then (1 - sn) > (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

      then ( - (1 - sn)) < ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

      then (( - (1 - sn)) / (1 - sn)) < (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A5, XREAL_1: 74;

      then ( - 1) < (( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) by A5, XCMPLX_1: 197;

      then ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) < (1 ^2 ) by A5, A4, SQUARE_1: 50;

      then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 )))) > 0 ;

      then

       A8: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) > 0 by XCMPLX_1: 76;

      assume p = ((sn -FanMorphE ) . q);

      then

       A9: p = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A2, A3, Th82;

      then (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

      hence thesis by A9, A6, A5, A4, A8, EUCLID: 52, XREAL_1: 129;

    end;

    theorem :: JGRAPH_4:107

    

     Th107: for sn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < sn & (q `1 ) > 0 & ((q `2 ) / |.q.|) < sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphE ) . q) holds (p `1 ) > 0 & (p `2 ) < 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: (q `1 ) > 0 and

       A3: ((q `2 ) / |.q.|) < sn;

      

       A4: (1 + sn) > 0 by A1, XREAL_1: 148;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((sn -FanMorphE ) . q);

      then p = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th83;

      then

       A5: (qz `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) & (qz `2 ) = ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn))) by EUCLID: 52;

      

       A6: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      

       A8: (((q `2 ) / |.q.|) - sn) < 0 by A3, XREAL_1: 49;

      then ( - (((q `2 ) / |.q.|) - sn)) > 0 by XREAL_1: 58;

      then (( - (1 + sn)) / (1 + sn)) < (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A4, XREAL_1: 74;

      then

       A9: ( - 1) < (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) by A4, XCMPLX_1: 197;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `2 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, XREAL_1: 74;

      then (((q `2 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

      then (((q `2 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then ( - 1) < ((q `2 ) / |.q.|) by SQUARE_1: 52;

      then (( - 1) - sn) < (((q `2 ) / |.q.|) - sn) by XREAL_1: 9;

      then ( - ( - (1 + sn))) > ( - (((q `2 ) / |.q.|) - sn)) by XREAL_1: 24;

      then (( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) < 1 by A4, XREAL_1: 191;

      then ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) < (1 ^2 ) by A9, SQUARE_1: 50;

      then (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 )))) > 0 ;

      then

       A10: ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) > 0 by XCMPLX_1: 76;

      ((((q `2 ) / |.q.|) - sn) / (1 + sn)) < 0 by A1, A8, XREAL_1: 141, XREAL_1: 148;

      hence thesis by A6, A5, A10, XREAL_1: 129, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:108

    

     Th108: for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st sn < 1 & (q1 `1 ) > 0 & ((q1 `2 ) / |.q1.|) >= sn & (q2 `1 ) > 0 & ((q2 `2 ) / |.q2.|) >= sn & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphE ) . q1) & p2 = ((sn -FanMorphE ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: sn < 1 and

       A2: (q1 `1 ) > 0 and

       A3: ((q1 `2 ) / |.q1.|) >= sn and

       A4: (q2 `1 ) > 0 and

       A5: ((q2 `2 ) / |.q2.|) >= sn and

       A6: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      

       A7: (((q1 `2 ) / |.q1.|) - sn) < (((q2 `2 ) / |.q2.|) - sn) & (1 - sn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 149;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((sn -FanMorphE ) . q1) and

       A9: p2 = ((sn -FanMorphE ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th97;

      p2 = |[( |.q2.| * ( sqrt (1 - (((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)) ^2 )))), ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)))]| by A4, A5, A9, Th82;

      then

       A11: (p2 `2 ) = ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `2 ) / |.p2.|) = ((((q2 `2 ) / |.q2.|) - sn) / (1 - sn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ( sqrt (1 - (((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)) ^2 )))), ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)))]| by A2, A3, A8, Th82;

      then

       A13: (p1 `2 ) = ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th97;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `2 ) / |.p1.|) = ((((q1 `2 ) / |.q1.|) - sn) / (1 - sn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:109

    

     Th109: for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < sn & (q1 `1 ) > 0 & ((q1 `2 ) / |.q1.|) < sn & (q2 `1 ) > 0 & ((q2 `2 ) / |.q2.|) < sn & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphE ) . q1) & p2 = ((sn -FanMorphE ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: (q1 `1 ) > 0 and

       A3: ((q1 `2 ) / |.q1.|) < sn and

       A4: (q2 `1 ) > 0 and

       A5: ((q2 `2 ) / |.q2.|) < sn and

       A6: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      

       A7: (((q1 `2 ) / |.q1.|) - sn) < (((q2 `2 ) / |.q2.|) - sn) & (1 + sn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 148;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((sn -FanMorphE ) . q1) and

       A9: p2 = ((sn -FanMorphE ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th97;

      p2 = |[( |.q2.| * ( sqrt (1 - (((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)) ^2 )))), ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)))]| by A4, A5, A9, Th83;

      then

       A11: (p2 `2 ) = ( |.q2.| * ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `2 ) / |.p2.|) = ((((q2 `2 ) / |.q2.|) - sn) / (1 + sn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ( sqrt (1 - (((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)) ^2 )))), ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)))]| by A2, A3, A8, Th83;

      then

       A13: (p1 `2 ) = ( |.q1.| * ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th97;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `2 ) / |.p1.|) = ((((q1 `2 ) / |.q1.|) - sn) / (1 + sn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:110

    for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (q1 `1 ) > 0 & (q2 `1 ) > 0 & ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((sn -FanMorphE ) . q1) & p2 = ((sn -FanMorphE ) . q2) holds ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|)

    proof

      let sn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1 and

       A3: (q1 `1 ) > 0 and

       A4: (q2 `1 ) > 0 and

       A5: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A6: p1 = ((sn -FanMorphE ) . q1) and

       A7: p2 = ((sn -FanMorphE ) . q2);

      per cases ;

        suppose ((q1 `2 ) / |.q1.|) >= sn & ((q2 `2 ) / |.q2.|) >= sn;

        hence thesis by A2, A3, A4, A5, A6, A7, Th108;

      end;

        suppose ((q1 `2 ) / |.q1.|) >= sn & ((q2 `2 ) / |.q2.|) < sn;

        hence thesis by A5, XXREAL_0: 2;

      end;

        suppose

         A8: ((q1 `2 ) / |.q1.|) < sn & ((q2 `2 ) / |.q2.|) >= sn;

        then (p2 `2 ) >= 0 by A2, A4, A7, Th106;

        then

         A9: ((p2 `2 ) / |.p2.|) >= 0 ;

        (p1 `2 ) < 0 by A1, A3, A6, A8, Th107;

        hence thesis by A9, Lm1, JGRAPH_2: 3, XREAL_1: 141;

      end;

        suppose ((q1 `2 ) / |.q1.|) < sn & ((q2 `2 ) / |.q2.|) < sn;

        hence thesis by A1, A3, A4, A5, A6, A7, Th109;

      end;

    end;

    theorem :: JGRAPH_4:111

    for sn be Real, q be Point of ( TOP-REAL 2) st (q `1 ) > 0 & ((q `2 ) / |.q.|) = sn holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphE ) . q) holds (p `1 ) > 0 & (p `2 ) = 0

    proof

      let sn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: (q `1 ) > 0 and

       A2: ((q `2 ) / |.q.|) = sn;

      

       A3: |.q.| <> 0 & ( sqrt (1 - ((( - (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ))) > 0 by A1, A2, JGRAPH_2: 3, SQUARE_1: 25, TOPRNS_1: 24;

      let p be Point of ( TOP-REAL 2);

      assume p = ((sn -FanMorphE ) . q);

      then

       A4: p = |[( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))), ( |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, Th82;

      then (p `1 ) = ( |.q.| * ( sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID: 52;

      hence thesis by A2, A4, A3, EUCLID: 52, XREAL_1: 129;

    end;

    theorem :: JGRAPH_4:112

    for sn be Real holds ( 0. ( TOP-REAL 2)) = ((sn -FanMorphE ) . ( 0. ( TOP-REAL 2))) by Th82, JGRAPH_2: 3;

    begin

    definition

      let s be Real, q be Point of ( TOP-REAL 2);

      :: JGRAPH_4:def8

      func FanS (s,q) -> Point of ( TOP-REAL 2) equals

      : Def8: ( |.q.| * |[((((q `1 ) / |.q.|) - s) / (1 - s)), ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - s) / (1 - s)) ^2 ))))]|) if ((q `1 ) / |.q.|) >= s & (q `2 ) < 0 ,

( |.q.| * |[((((q `1 ) / |.q.|) - s) / (1 + s)), ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - s) / (1 + s)) ^2 ))))]|) if ((q `1 ) / |.q.|) < s & (q `2 ) < 0

      otherwise q;

      correctness ;

    end

    definition

      let c be Real;

      :: JGRAPH_4:def9

      func c -FanMorphS -> Function of ( TOP-REAL 2), ( TOP-REAL 2) means

      : Def9: for q be Point of ( TOP-REAL 2) holds (it . q) = ( FanS (c,q));

      existence

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanS (c,$1));

        thus ex IT be Function of ( TOP-REAL 2), ( TOP-REAL 2) st for q be Point of ( TOP-REAL 2) holds (IT . q) = F(q) from FUNCT_2:sch 4;

      end;

      uniqueness

      proof

        deffunc F( Point of ( TOP-REAL 2)) = ( FanS (c,$1));

        thus for a,b be Function of ( TOP-REAL 2), ( TOP-REAL 2) st (for q be Point of ( TOP-REAL 2) holds (a . q) = F(q)) & (for q be Point of ( TOP-REAL 2) holds (b . q) = F(q)) holds a = b from BINOP_2:sch 1;

      end;

    end

    theorem :: JGRAPH_4:113

    

     Th113: for cn be Real holds (((q `1 ) / |.q.|) >= cn & (q `2 ) < 0 implies ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]|) & ((q `2 ) >= 0 implies ((cn -FanMorphS ) . q) = q)

    proof

      let cn be Real;

      hereby

        assume ((q `1 ) / |.q.|) >= cn & (q `2 ) < 0 ;

        

        then ( FanS (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 - cn)), ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|) by Def8

        .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by EUCLID: 58;

        hence ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by Def9;

      end;

      assume

       A1: (q `2 ) >= 0 ;

      ((cn -FanMorphS ) . q) = ( FanS (cn,q)) by Def9;

      hence thesis by A1, Def8;

    end;

    theorem :: JGRAPH_4:114

    

     Th114: for cn be Real holds (((q `1 ) / |.q.|) <= cn & (q `2 ) < 0 implies ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]|)

    proof

      let cn be Real;

      assume that

       A1: ((q `1 ) / |.q.|) <= cn and

       A2: (q `2 ) < 0 ;

      per cases by A1, XXREAL_0: 1;

        suppose ((q `1 ) / |.q.|) < cn;

        

        then ( FanS (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 + cn)), ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|) by A2, Def8

        .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by EUCLID: 58;

        hence thesis by Def9;

      end;

        suppose

         A3: ((q `1 ) / |.q.|) = cn;

        then ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = 0 ;

        hence thesis by A2, A3, Th113;

      end;

    end;

    theorem :: JGRAPH_4:115

    

     Th115: for cn be Real st ( - 1) < cn & cn < 1 holds (((q `1 ) / |.q.|) >= cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]|) & (((q `1 ) / |.q.|) <= cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) implies ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]|)

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      per cases ;

        suppose

         A3: ((q `1 ) / |.q.|) >= cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose

           A4: (q `2 ) < 0 ;

          

          then ( FanS (cn,q)) = ( |.q.| * |[((((q `1 ) / |.q.|) - cn) / (1 - cn)), ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|) by A3, Def8

          .= |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by EUCLID: 58;

          hence thesis by A4, Def9, Th114;

        end;

          suppose

           A5: (q `2 ) >= 0 ;

          then

           A6: ((cn -FanMorphS ) . q) = q by Th113;

          

           A7: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

          

           A8: (1 - cn) > 0 by A2, XREAL_1: 149;

          

           A9: (q `2 ) = 0 by A3, A5;

           |.q.| <> 0 by A3, TOPRNS_1: 24;

          then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          then (((q `1 ) ^2 ) / ( |.q.| ^2 )) = (1 ^2 ) by A7, A9, XCMPLX_1: 60;

          then (((q `1 ) / |.q.|) ^2 ) = (1 ^2 ) by XCMPLX_1: 76;

          then

           A10: ( sqrt (((q `1 ) / |.q.|) ^2 )) = 1 by SQUARE_1: 22;

           A11:

          now

            assume (q `1 ) < 0 ;

            then ( - ((q `1 ) / |.q.|)) = 1 by A10, SQUARE_1: 23;

            hence contradiction by A1, A3;

          end;

          ( sqrt ( |.q.| ^2 )) = |.q.| by SQUARE_1: 22;

          then

           A12: |.q.| = (q `1 ) by A7, A9, A11, SQUARE_1: 22;

          then 1 = ((q `1 ) / |.q.|) by A3, TOPRNS_1: 24, XCMPLX_1: 60;

          then ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = 1 by A8, XCMPLX_1: 60;

          hence thesis by A2, A6, A9, A12, EUCLID: 53, SQUARE_1: 17, TOPRNS_1: 24, XCMPLX_1: 60;

        end;

      end;

        suppose

         A13: ((q `1 ) / |.q.|) <= cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

        per cases ;

          suppose (q `2 ) < 0 ;

          hence thesis by Th113, Th114;

        end;

          suppose

           A14: (q `2 ) >= 0 ;

          then

           A15: (q `2 ) = 0 by A13;

          

           A16: (1 + cn) > 0 by A1, XREAL_1: 148;

          

           A17: |.q.| <> 0 by A13, TOPRNS_1: 24;

          1 > ((q `1 ) / |.q.|) by A2, A13, XXREAL_0: 2;

          then (1 * |.q.|) > (((q `1 ) / |.q.|) * |.q.|) by A17, XREAL_1: 68;

          then

           A18: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & |.q.| > (q `1 ) by A13, JGRAPH_3: 1, TOPRNS_1: 24, XCMPLX_1: 87;

          then

           A19: |.q.| = ( - (q `1 )) by A15, SQUARE_1: 40;

          

           A20: (q `1 ) = ( - |.q.|) by A15, A18, SQUARE_1: 40;

          then ( - 1) = ((q `1 ) / |.q.|) by A13, TOPRNS_1: 24, XCMPLX_1: 197;

          

          then ((((q `1 ) / |.q.|) - cn) / (1 + cn)) = (( - (1 + cn)) / (1 + cn))

          .= ( - 1) by A16, XCMPLX_1: 197;

          then |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| = q by A15, A19, EUCLID: 53, SQUARE_1: 17;

          hence thesis by A1, A14, A17, A20, Th113, XCMPLX_1: 197;

        end;

      end;

        suppose (q `2 ) > 0 or q = ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

    end;

    theorem :: JGRAPH_4:116

    

     Th116: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st cn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 - cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: cn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in K1 by A7, A8, A9, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        

         A11: (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        (f . r) = ( |.r.| * ((((r `1 ) / |.r.|) - cn) / (1 - cn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:117

    

     Th117: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < cn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 + cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: ( - 1) < cn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      (1 + cn) > 0 by A1, XREAL_1: 148;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * (((r1 / r2) - a) / b)) and

       A6: g3 is continuous by A4, Th5;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A8: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

        reconsider s = x as Point of (( TOP-REAL 2) | K1);

        x in ( dom g3) by A7, A9;

        then x in K1 by A7, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A10: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        

         A11: (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        (f . r) = ( |.r.| * ((((r `1 ) / |.r.|) - cn) / (1 + cn))) by A2, A9;

        hence thesis by A5, A11, A10;

      end;

      ( dom f) = ( dom g3) by A7, FUNCT_2:def 1;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:118

    

     Th118: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st cn < 1 & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 - cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: cn < 1 and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      b > 0 by A1, XREAL_1: 149;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))) and

       A6: g3 is continuous by A4, Th9;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        

         A9: (1 - cn) > 0 by A1, XREAL_1: 149;

        assume

         A10: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        

         A11: |.r.| <> 0 by A3, A10, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `1 ) - |.r.|) * ((r `1 ) + |.r.|)) = ( - ((r `2 ) ^2 ));

        ((r `2 ) ^2 ) >= 0 by XREAL_1: 63;

        then (r `1 ) <= |.r.| by A12, XREAL_1: 93;

        then ((r `1 ) / |.r.|) <= ( |.r.| / |.r.|) by XREAL_1: 72;

        then ((r `1 ) / |.r.|) <= 1 by A11, XCMPLX_1: 60;

        then

         A13: (((r `1 ) / |.r.|) - cn) <= (1 - cn) by XREAL_1: 9;

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A10;

         A14:

        now

          assume ((1 - cn) ^2 ) = 0 ;

          then ((1 - cn) + cn) = ( 0 + cn) by XCMPLX_1: 6;

          hence contradiction by A1;

        end;

        (cn - ((r `1 ) / |.r.|)) <= 0 by A3, A10, XREAL_1: 47;

        then ( - (cn - ((r `1 ) / |.r.|))) >= ( - (1 - cn)) by A9, XREAL_1: 24;

        then ((1 - cn) ^2 ) >= 0 & ((((r `1 ) / |.r.|) - cn) ^2 ) <= ((1 - cn) ^2 ) by A13, SQUARE_1: 49, XREAL_1: 63;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 - cn) ^2 )) <= (((1 - cn) ^2 ) / ((1 - cn) ^2 )) by XREAL_1: 72;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 - cn) ^2 )) <= 1 by A14, XCMPLX_1: 60;

        then (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )).| = (1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )) by ABSVALUE:def 1;

        then

         A15: (f . r) = ( |.r.| * ( - ( sqrt |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 - cn)) ^2 )).|))) by A2, A10;

        

         A16: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        hence thesis by A5, A15, A16;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:119

    

     Th119: for cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st ( - 1) < cn & (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2))) holds f is continuous

    proof

      let cn be Real, K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 ;

      reconsider g1 = ((2 NormF ) | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      set a = cn, b = (1 + cn);

      reconsider g2 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm2;

      assume that

       A1: ( - 1) < cn and

       A2: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) and

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2));

      

       A4: (1 + cn) > 0 by A1, XREAL_1: 148;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds q <> ( 0. ( TOP-REAL 2)) by A3;

      then for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0 by Lm6;

      then

      consider g3 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A5: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( - ( sqrt |.(1 - ((((r1 / r2) - a) / b) ^2 )).|))) and

       A6: g3 is continuous by A4, Th9;

      

       A7: ( dom g3) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      then

       A8: ( dom f) = ( dom g3) by FUNCT_2:def 1;

      for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then x in K1 by A7, A8, PRE_TOPC: 8;

        then

        reconsider r = x as Point of ( TOP-REAL 2);

        reconsider s = x as Point of (( TOP-REAL 2) | K1) by A9;

        

         A10: ((1 + cn) ^2 ) > 0 by A4, SQUARE_1: 12;

        

         A11: |.r.| <> 0 by A3, A9, TOPRNS_1: 24;

        ( |.r.| ^2 ) = (((r `1 ) ^2 ) + ((r `2 ) ^2 )) by JGRAPH_3: 1;

        then

         A12: (((r `1 ) - |.r.|) * ((r `1 ) + |.r.|)) = ( - ((r `2 ) ^2 ));

        ((r `2 ) ^2 ) >= 0 by XREAL_1: 63;

        then ( - |.r.|) <= (r `1 ) by A12, XREAL_1: 93;

        then ((r `1 ) / |.r.|) >= (( - |.r.|) / |.r.|) by XREAL_1: 72;

        then ((r `1 ) / |.r.|) >= ( - 1) by A11, XCMPLX_1: 197;

        then (((r `1 ) / |.r.|) - cn) >= (( - 1) - cn) by XREAL_1: 9;

        then

         A13: (((r `1 ) / |.r.|) - cn) >= ( - (1 + cn));

        (cn - ((r `1 ) / |.r.|)) >= 0 by A3, A9, XREAL_1: 48;

        then ( - (cn - ((r `1 ) / |.r.|))) <= ( - 0 );

        then ((((r `1 ) / |.r.|) - cn) ^2 ) <= ((1 + cn) ^2 ) by A4, A13, SQUARE_1: 49;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 + cn) ^2 )) <= (((1 + cn) ^2 ) / ((1 + cn) ^2 )) by A4, XREAL_1: 72;

        then (((((r `1 ) / |.r.|) - cn) ^2 ) / ((1 + cn) ^2 )) <= 1 by A10, XCMPLX_1: 60;

        then (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 ) <= 1 by XCMPLX_1: 76;

        then (1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

        then |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )).| = (1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )) by ABSVALUE:def 1;

        then

         A14: (f . r) = ( |.r.| * ( - ( sqrt |.(1 - (((((r `1 ) / |.r.|) - cn) / (1 + cn)) ^2 )).|))) by A2, A9;

        

         A15: ( proj1 . r) = (r `1 ) & ((2 NormF ) . r) = |.r.| by Def1, PSCOMP_1:def 5;

        (g2 . s) = ( proj1 . s) & (g1 . s) = ((2 NormF ) . s) by Lm2, Lm5;

        hence thesis by A5, A14, A15;

      end;

      hence thesis by A6, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_4:120

    

     Th120: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( - ( sqrt (1 - (cn ^2 ))));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - sn) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - sn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      (( - sn) ^2 ) = (1 - (cn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `1 ) / |.p0.|) = cn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A7: p0 in K0 by A3, A1, A6, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A8: ( rng ( proj2 * ((cn -FanMorphS ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A9: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) / |.p8.|) >= cn & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A10: ( dom ((cn -FanMorphS ) | K1)) c= ( dom ( proj1 * ((cn -FanMorphS ) | K1)))

      proof

        let x be object;

        assume

         A11: x in ( dom ((cn -FanMorphS ) | K1));

        then x in (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphS )) by XBOOLE_0:def 4;

        then

         A12: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphS ) . x) in ( rng (cn -FanMorphS )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphS ) | K1) . x) = ((cn -FanMorphS ) . x) by A11, FUNCT_1: 47;

        hence thesis by A11, A12, FUNCT_1: 11;

      end;

      

       A13: ( rng ( proj1 * ((cn -FanMorphS ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj1 * ((cn -FanMorphS ) | K1))) c= ( dom ((cn -FanMorphS ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((cn -FanMorphS ) | K1))) = ( dom ((cn -FanMorphS ) | K1)) by A10, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj1 * ((cn -FanMorphS ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A13, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A14: ( dom ((cn -FanMorphS ) | K1)) = (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A15: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A16: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A15;

        then

         A17: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| by A3, Th115;

        (((cn -FanMorphS ) | K1) . p) = ((cn -FanMorphS ) . p) by A16, A15, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]|) by A16, A14, A15, A17, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A18: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)));

      

       A19: ( dom ((cn -FanMorphS ) | K1)) c= ( dom ( proj2 * ((cn -FanMorphS ) | K1)))

      proof

        let x be object;

        assume

         A20: x in ( dom ((cn -FanMorphS ) | K1));

        then x in (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphS )) by XBOOLE_0:def 4;

        then

         A21: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphS ) . x) in ( rng (cn -FanMorphS )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphS ) | K1) . x) = ((cn -FanMorphS ) . x) by A20, FUNCT_1: 47;

        hence thesis by A20, A21, FUNCT_1: 11;

      end;

      ( dom ( proj2 * ((cn -FanMorphS ) | K1))) c= ( dom ((cn -FanMorphS ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((cn -FanMorphS ) | K1))) = ( dom ((cn -FanMorphS ) | K1)) by A19, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj2 * ((cn -FanMorphS ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A8, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A22: ( dom ((cn -FanMorphS ) | K1)) = (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A23: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A24: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A23;

        then

         A25: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| by A3, Th115;

        (((cn -FanMorphS ) | K1) . p) = ((cn -FanMorphS ) . p) by A24, A23, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]|) by A24, A22, A23, A25, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A26: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) >= cn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A27: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A27;

        hence thesis;

      end;

      then

       A28: f1 is continuous by A3, A26, Th118;

      

       A29: for x,y,s,r be Real st |[x, y]| in K1 & s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|) holds (f . |[x, y]|) = |[s, r]|

      proof

        let x,y,s,r be Real;

        assume that

         A30: |[x, y]| in K1 and

         A31: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A32: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A30;

        

         A33: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A34: (f1 . p99) = ( |.p99.| * ( - ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 - cn)) ^2 ))))) by A26, A30;

        (((cn -FanMorphS ) | K0) . |[x, y]|) = ((cn -FanMorphS ) . |[x, y]|) by A30, FUNCT_1: 49

        .= |[( |.p99.| * ((((p99 `1 ) / |.p99.|) - cn) / (1 - cn))), ( |.p99.| * ( - ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 - cn)) ^2 )))))]| by A3, A32, Th115

        .= |[s, r]| by A18, A30, A31, A33, A34;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A35: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) >= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A35;

        hence thesis;

      end;

      then f2 is continuous by A3, A18, Th116;

      hence thesis by A7, A9, A28, A29, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:121

    

     Th121: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( - ( sqrt (1 - (cn ^2 ))));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = { q where q be Point of ( TOP-REAL 2) : (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2)) } & K0 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - sn) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - sn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      (( - sn) ^2 ) = (1 - (cn ^2 )) by A4, SQUARE_1:def 2;

      then ((p0 `1 ) / |.p0.|) = cn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A7: p0 in K0 by A3, A1, A6, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      

       A8: ( rng ( proj2 * ((cn -FanMorphS ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A9: K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) / |.p8.|) <= cn & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A3;

        hence thesis by A3;

      end;

      

       A10: ( dom ((cn -FanMorphS ) | K1)) c= ( dom ( proj1 * ((cn -FanMorphS ) | K1)))

      proof

        let x be object;

        assume

         A11: x in ( dom ((cn -FanMorphS ) | K1));

        then x in (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphS )) by XBOOLE_0:def 4;

        then

         A12: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphS ) . x) in ( rng (cn -FanMorphS )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphS ) | K1) . x) = ((cn -FanMorphS ) . x) by A11, FUNCT_1: 47;

        hence thesis by A11, A12, FUNCT_1: 11;

      end;

      

       A13: ( rng ( proj1 * ((cn -FanMorphS ) | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj1 * ((cn -FanMorphS ) | K1))) c= ( dom ((cn -FanMorphS ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ((cn -FanMorphS ) | K1))) = ( dom ((cn -FanMorphS ) | K1)) by A10, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g2 = ( proj1 * ((cn -FanMorphS ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A13, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A14: ( dom ((cn -FanMorphS ) | K1)) = (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A15: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A16: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A15;

        then

         A17: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| by A3, Th115;

        (((cn -FanMorphS ) | K1) . p) = ((cn -FanMorphS ) . p) by A16, A15, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]|) by A16, A14, A15, A17, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| `1 ) by PSCOMP_1:def 5

        .= ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f2 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A18: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn)));

      

       A19: ( dom ((cn -FanMorphS ) | K1)) c= ( dom ( proj2 * ((cn -FanMorphS ) | K1)))

      proof

        let x be object;

        assume

         A20: x in ( dom ((cn -FanMorphS ) | K1));

        then x in (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61;

        then x in ( dom (cn -FanMorphS )) by XBOOLE_0:def 4;

        then

         A21: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ((cn -FanMorphS ) . x) in ( rng (cn -FanMorphS )) by FUNCT_1: 3, FUNCT_2:def 1;

        (((cn -FanMorphS ) | K1) . x) = ((cn -FanMorphS ) . x) by A20, FUNCT_1: 47;

        hence thesis by A20, A21, FUNCT_1: 11;

      end;

      ( dom ( proj2 * ((cn -FanMorphS ) | K1))) c= ( dom ((cn -FanMorphS ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ((cn -FanMorphS ) | K1))) = ( dom ((cn -FanMorphS ) | K1)) by A19, XBOOLE_0:def 10

      .= (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

      .= the carrier of (( TOP-REAL 2) | K1) by PRE_TOPC: 8;

      then

      reconsider g1 = ( proj2 * ((cn -FanMorphS ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A8, FUNCT_2: 2;

      for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (g1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))

      proof

        let p be Point of ( TOP-REAL 2);

        

         A22: ( dom ((cn -FanMorphS ) | K1)) = (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

        

         A23: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A24: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A23;

        then

         A25: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| by A3, Th115;

        (((cn -FanMorphS ) | K1) . p) = ((cn -FanMorphS ) . p) by A24, A23, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]|) by A24, A22, A23, A25, FUNCT_1: 13

        .= ( |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| `2 ) by PSCOMP_1:def 6

        .= ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

        hence thesis;

      end;

      then

      consider f1 be Function of (( TOP-REAL 2) | K1), R^1 such that

       A26: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))));

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & ((q `1 ) / |.q.|) <= cn & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A27: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A27;

        hence thesis;

      end;

      then

       A28: f1 is continuous by A3, A26, Th119;

      

       A29: for x,y,s,r be Real st |[x, y]| in K1 & s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|) holds (f . |[x, y]|) = |[s, r]|

      proof

        let x,y,s,r be Real;

        assume that

         A30: |[x, y]| in K1 and

         A31: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A32: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A30;

        

         A33: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A34: (f1 . p99) = ( |.p99.| * ( - ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 + cn)) ^2 ))))) by A26, A30;

        (((cn -FanMorphS ) | K0) . |[x, y]|) = ((cn -FanMorphS ) . |[x, y]|) by A30, FUNCT_1: 49

        .= |[( |.p99.| * ((((p99 `1 ) / |.p99.|) - cn) / (1 + cn))), ( |.p99.| * ( - ( sqrt (1 - (((((p99 `1 ) / |.p99.|) - cn) / (1 + cn)) ^2 )))))]| by A3, A32, Th115

        .= |[s, r]| by A18, A30, A31, A33, A34;

        hence thesis by A3;

      end;

      for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2))

      proof

        let q be Point of ( TOP-REAL 2);

        

         A35: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume q in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) / |.p3.|) <= cn & (p3 `2 ) <= 0 & p3 <> ( 0. ( TOP-REAL 2)) by A3, A35;

        hence thesis;

      end;

      then f2 is continuous by A3, A18, Th117;

      hence thesis by A7, A9, A28, A29, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_4:122

    

     Th122: for cn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `1 ) >= (cn * |.p.|) & (p `2 ) <= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `1 ) >= (sn * |.p.|) & (p `2 ) <= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) >= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm8, JORDAN6: 8;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:123

    

     Th123: for cn be Real, K03 be Subset of ( TOP-REAL 2) st K03 = { p : (p `1 ) <= (cn * |.p.|) & (p `2 ) <= 0 } holds K03 is closed

    proof

      defpred Q[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= 0 ;

      let sn be Real, K003 be Subset of ( TOP-REAL 2);

      assume

       A1: K003 = { p : (p `1 ) <= (sn * |.p.|) & (p `2 ) <= 0 };

      reconsider KX = { p where p be Point of ( TOP-REAL 2) : Q[p] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) <= (sn * |.$1.|));

      reconsider K1 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A2: { p : P[p] & Q[p] } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ { p1 where p1 be Point of ( TOP-REAL 2) : Q[p1] }) from DOMAIN_1:sch 10;

      K1 is closed & KX is closed by Lm10, JORDAN6: 8;

      hence thesis by A1, A2, TOPS_1: 8;

    end;

    theorem :: JGRAPH_4:124

    

     Th124: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( - ( sqrt (1 - (cn ^2 ))));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      (p0 `1 ) = cn by EUCLID: 52;

      then

       A2: |.p0.| = ( sqrt ((sn ^2 ) + (cn ^2 ))) by A1, JGRAPH_3: 1;

      assume

       A3: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A4: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then

       A5: ( - sn) > 0 by SQUARE_1: 25;

       A6:

      now

        assume p0 = ( 0. ( TOP-REAL 2));

        then ( - ( - sn)) = ( - 0 ) by EUCLID: 52, JGRAPH_2: 3;

        hence contradiction by A4, SQUARE_1: 25;

      end;

      then p0 in K0 by A3, A1, A5;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      (( - sn) ^2 ) = (1 - (cn ^2 )) by A4, SQUARE_1:def 2;

      then

       A7: ((p0 `1 ) / |.p0.|) = cn by A2, EUCLID: 52, SQUARE_1: 18;

      then

       A8: p0 in { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5;

       not p0 in {( 0. ( TOP-REAL 2))} by A6, TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A3, XBOOLE_0:def 5;

      K1 c= D

      proof

        let x be object;

        assume

         A9: x in K1;

        then ex p6 be Point of ( TOP-REAL 2) st p6 = x & (p6 `2 ) <= 0 & p6 <> ( 0. ( TOP-REAL 2)) by A3;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A3, A9, XBOOLE_0:def 5;

      end;

      then D = (K1 \/ D) by XBOOLE_1: 12;

      then

       A10: (( TOP-REAL 2) | K1) is SubSpace of (( TOP-REAL 2) | D) by TOPMETR: 4;

      

       A11: { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      

       A12: { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } c= K1

      proof

        let x be object;

        assume x in { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

        then ex p st p = x & ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2));

        hence thesis by A3;

      end;

      then

      reconsider K00 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A8, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A13: ( rng (f | K00)) c= D;

      p0 in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5, A7;

      then

      reconsider K11 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | K1) by A11, PRE_TOPC: 8;

      the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      then

       A14: ( rng (f | K11)) c= D;

      the carrier of (( TOP-REAL 2) | B0) = the carrier of (( TOP-REAL 2) | D);

      

      then

       A15: ( dom f) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1

      .= K1 by PRE_TOPC: 8;

      

      then ( dom (f | K00)) = K00 by A12, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K00) by PRE_TOPC: 8;

      then

      reconsider f1 = (f | K00) as Function of ((( TOP-REAL 2) | K1) | K00), (( TOP-REAL 2) | D) by A13, FUNCT_2: 2;

      ( dom (f | K11)) = K11 by A11, A15, RELAT_1: 62

      .= the carrier of ((( TOP-REAL 2) | K1) | K11) by PRE_TOPC: 8;

      then

      reconsider f2 = (f | K11) as Function of ((( TOP-REAL 2) | K1) | K11), (( TOP-REAL 2) | D) by A14, FUNCT_2: 2;

      

       A16: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) / |.$1.|) >= cn & ($1 `2 ) <= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A17: ( dom f2) = the carrier of ((( TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1

      .= K11 by PRE_TOPC: 8;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K001 = { p : ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A8;

      

       A18: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) >= (cn * |.$1.|) & ($1 `2 ) <= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K003 = { p : (p `1 ) >= (cn * |.p.|) & (p `2 ) <= 0 } as Subset of ( TOP-REAL 2);

      defpred P[ Point of ( TOP-REAL 2)] means (($1 `1 ) / |.$1.|) <= cn & ($1 `2 ) <= 0 & $1 <> ( 0. ( TOP-REAL 2));

      

       A19: { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      

       A20: ( rng ((cn -FanMorphS ) | K001)) c= K1

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphS ) | K001));

        then

        consider x be object such that

         A21: x in ( dom ((cn -FanMorphS ) | K001)) and

         A22: y = (((cn -FanMorphS ) | K001) . x) by FUNCT_1:def 3;

        x in ( dom (cn -FanMorphS )) by A21, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A23: y = ((cn -FanMorphS ) . q) by A21, A22, FUNCT_1: 47;

        ( dom ((cn -FanMorphS ) | K001)) = (( dom (cn -FanMorphS )) /\ K001) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K001) by FUNCT_2:def 1

        .= K001 by XBOOLE_1: 28;

        then

         A24: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `1 ) / |.p2.|) >= cn & (p2 `2 ) <= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A21;

        then

         A25: (((q `1 ) / |.q.|) - cn) >= 0 by XREAL_1: 48;

         |.q.| <> 0 by A24, TOPRNS_1: 24;

        then

         A26: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]|;

        

         A27: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

        

         A28: (1 - cn) > 0 by A3, XREAL_1: 149;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then ((q `1 ) ^2 ) <= ( |.q.| ^2 ) by JGRAPH_3: 1;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A26, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

        then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A28, XREAL_1: 72;

        then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A28, XCMPLX_1: 197;

        then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A28, A25, SQUARE_1: 49;

        then

         A29: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

        then

         A30: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

        ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by A29, SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

        then

         A31: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A32: (q4 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

        

        then

         A33: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A30, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A27, A33;

        then

         A34: q4 <> ( 0. ( TOP-REAL 2)) by A26, TOPRNS_1: 23;

        ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by A3, A24, Th115;

        hence thesis by A3, A23, A32, A31, A34;

      end;

      

       A35: ( dom (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then ( dom ((cn -FanMorphS ) | K001)) = K001 by RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K001) by PRE_TOPC: 8;

      then

      reconsider f3 = ((cn -FanMorphS ) | K001) as Function of (( TOP-REAL 2) | K001), (( TOP-REAL 2) | K1) by A18, A20, FUNCT_2: 2;

      

       A36: K003 is closed by Th122;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) <= (cn * |.$1.|) & ($1 `2 ) <= 0 ;

      { p : P[p] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider K004 = { p : (p `1 ) <= (cn * |.p.|) & (p `2 ) <= 0 } as Subset of ( TOP-REAL 2);

      

       A37: (K004 /\ K1) c= K11

      proof

        let x be object;

        assume

         A38: x in (K004 /\ K1);

        then x in K004 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A39: q1 = x and

         A40: (q1 `1 ) <= (cn * |.q1.|) and (q1 `2 ) <= 0 ;

        x in K1 by A38, XBOOLE_0:def 4;

        then

         A41: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `2 ) <= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `1 ) / |.q1.|) <= ((cn * |.q1.|) / |.q1.|) by A40, XREAL_1: 72;

        then ((q1 `1 ) / |.q1.|) <= cn by A39, A41, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A39, A41;

      end;

      

       A42: K004 is closed by Th123;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K00) = (( TOP-REAL 2) | K001) & f1 = f3 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A43: f1 is continuous by A3, A10, Th120, PRE_TOPC: 26;

      

       A44: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

      p0 in { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } by A1, A6, A5, A7;

      then

      reconsider K111 = { p : ((p `1 ) / |.p.|) <= cn & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of ( TOP-REAL 2) by A19;

      

       A45: ( rng ((cn -FanMorphS ) | K111)) c= K1

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphS ) | K111));

        then

        consider x be object such that

         A46: x in ( dom ((cn -FanMorphS ) | K111)) and

         A47: y = (((cn -FanMorphS ) | K111) . x) by FUNCT_1:def 3;

        x in ( dom (cn -FanMorphS )) by A46, RELAT_1: 57;

        then

        reconsider q = x as Point of ( TOP-REAL 2);

        

         A48: y = ((cn -FanMorphS ) . q) by A46, A47, FUNCT_1: 47;

        ( dom ((cn -FanMorphS ) | K111)) = (( dom (cn -FanMorphS )) /\ K111) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K111) by FUNCT_2:def 1

        .= K111 by XBOOLE_1: 28;

        then

         A49: ex p2 be Point of ( TOP-REAL 2) st p2 = q & ((p2 `1 ) / |.p2.|) <= cn & (p2 `2 ) <= 0 & p2 <> ( 0. ( TOP-REAL 2)) by A46;

        then

         A50: (((q `1 ) / |.q.|) - cn) <= 0 by XREAL_1: 47;

         |.q.| <> 0 by A49, TOPRNS_1: 24;

        then

         A51: ( |.q.| ^2 ) > ( 0 ^2 ) by SQUARE_1: 12;

        set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]|;

        

         A52: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

        

         A53: (1 + cn) > 0 by A3, XREAL_1: 148;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A51, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        then (( - (1 + cn)) / (1 + cn)) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A53, XREAL_1: 72;

        then ( - 1) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A53, XCMPLX_1: 197;

        then

         A54: (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) <= (1 ^2 ) by A53, A50, SQUARE_1: 49;

        then

         A55: (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

        (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by A54, XREAL_1: 48;

        then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XCMPLX_1: 187;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

        then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

        then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

        then

         A56: ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

        

         A57: (q4 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

        

        then

         A58: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

        .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A55, SQUARE_1:def 2;

        ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

        .= ( |.q.| ^2 ) by A52, A58;

        then

         A59: q4 <> ( 0. ( TOP-REAL 2)) by A51, TOPRNS_1: 23;

        ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by A3, A49, Th115;

        hence thesis by A3, A48, A57, A56, A59;

      end;

      ( dom ((cn -FanMorphS ) | K111)) = K111 by A35, RELAT_1: 62

      .= the carrier of (( TOP-REAL 2) | K111) by PRE_TOPC: 8;

      then

      reconsider f4 = ((cn -FanMorphS ) | K111) as Function of (( TOP-REAL 2) | K111), (( TOP-REAL 2) | K1) by A16, A45, FUNCT_2: 2;

      the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      then ((( TOP-REAL 2) | K1) | K11) = (( TOP-REAL 2) | K111) & f2 = f4 by A3, FUNCT_1: 51, GOBOARD9: 2;

      then

       A60: f2 is continuous by A3, A10, Th121, PRE_TOPC: 26;

      set T1 = ((( TOP-REAL 2) | K1) | K00), T2 = ((( TOP-REAL 2) | K1) | K11);

      

       A61: ( [#] ((( TOP-REAL 2) | K1) | K11)) = K11 by PRE_TOPC:def 5;

      K11 c= (K004 /\ K1)

      proof

        let x be object;

        assume x in K11;

        then

        consider p such that

         A62: p = x and

         A63: ((p `1 ) / |.p.|) <= cn and

         A64: (p `2 ) <= 0 and

         A65: p <> ( 0. ( TOP-REAL 2));

        (((p `1 ) / |.p.|) * |.p.|) <= (cn * |.p.|) by A63, XREAL_1: 64;

        then (p `1 ) <= (cn * |.p.|) by A65, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A66: x in K004 by A62, A64;

        x in K1 by A3, A62, A64, A65;

        hence thesis by A66, XBOOLE_0:def 4;

      end;

      then K11 = (K004 /\ ( [#] (( TOP-REAL 2) | K1))) by A44, A37, XBOOLE_0:def 10;

      then

       A67: K11 is closed by A42, PRE_TOPC: 13;

      

       A68: (K003 /\ K1) c= K00

      proof

        let x be object;

        assume

         A69: x in (K003 /\ K1);

        then x in K003 by XBOOLE_0:def 4;

        then

        consider q1 be Point of ( TOP-REAL 2) such that

         A70: q1 = x and

         A71: (q1 `1 ) >= (cn * |.q1.|) and (q1 `2 ) <= 0 ;

        x in K1 by A69, XBOOLE_0:def 4;

        then

         A72: ex q2 be Point of ( TOP-REAL 2) st q2 = x & (q2 `2 ) <= 0 & q2 <> ( 0. ( TOP-REAL 2)) by A3;

        ((q1 `1 ) / |.q1.|) >= ((cn * |.q1.|) / |.q1.|) by A71, XREAL_1: 72;

        then ((q1 `1 ) / |.q1.|) >= cn by A70, A72, TOPRNS_1: 24, XCMPLX_1: 89;

        hence thesis by A70, A72;

      end;

      

       A73: the carrier of (( TOP-REAL 2) | K1) = K0 by PRE_TOPC: 8;

      

       A74: D <> {} ;

      

       A75: ( [#] ((( TOP-REAL 2) | K1) | K00)) = K00 by PRE_TOPC:def 5;

      

       A76: for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f1 . p) = (f2 . p)

      proof

        let p be object;

        assume

         A77: p in (( [#] T1) /\ ( [#] T2));

        then p in K00 by A75, XBOOLE_0:def 4;

        

        hence (f1 . p) = (f . p) by FUNCT_1: 49

        .= (f2 . p) by A61, A77, FUNCT_1: 49;

      end;

      K00 c= (K003 /\ K1)

      proof

        let x be object;

        assume x in K00;

        then

        consider p such that

         A78: p = x and

         A79: ((p `1 ) / |.p.|) >= cn and

         A80: (p `2 ) <= 0 and

         A81: p <> ( 0. ( TOP-REAL 2));

        (((p `1 ) / |.p.|) * |.p.|) >= (cn * |.p.|) by A79, XREAL_1: 64;

        then (p `1 ) >= (cn * |.p.|) by A81, TOPRNS_1: 24, XCMPLX_1: 87;

        then

         A82: x in K003 by A78, A80;

        x in K1 by A3, A78, A80, A81;

        hence thesis by A82, XBOOLE_0:def 4;

      end;

      then K00 = (K003 /\ ( [#] (( TOP-REAL 2) | K1))) by A44, A68, XBOOLE_0:def 10;

      then

       A83: K00 is closed by A36, PRE_TOPC: 13;

      

       A84: K1 c= (K00 \/ K11)

      proof

        let x be object;

        assume x in K1;

        then

        consider p such that

         A85: p = x & (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) by A3;

        per cases ;

          suppose ((p `1 ) / |.p.|) >= cn;

          then x in K00 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

          suppose ((p `1 ) / |.p.|) < cn;

          then x in K11 by A85;

          hence thesis by XBOOLE_0:def 3;

        end;

      end;

      then (( [#] ((( TOP-REAL 2) | K1) | K00)) \/ ( [#] ((( TOP-REAL 2) | K1) | K11))) = ( [#] (( TOP-REAL 2) | K1)) by A75, A61, A44, XBOOLE_0:def 10;

      then

      consider h be Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D) such that

       A86: h = (f1 +* f2) and

       A87: h is continuous by A75, A61, A83, A67, A43, A60, A76, JGRAPH_2: 1;

      

       A88: ( dom h) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1;

      

       A89: ( dom f1) = the carrier of ((( TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1

      .= K00 by PRE_TOPC: 8;

      

       A90: for y be object st y in ( dom h) holds (h . y) = (f . y)

      proof

        let y be object;

        assume

         A91: y in ( dom h);

        per cases by A84, A88, A73, A91, XBOOLE_0:def 3;

          suppose

           A92: y in K00 & not y in K11;

          then y in (( dom f1) \/ ( dom f2)) by A89, XBOOLE_0:def 3;

          

          hence (h . y) = (f1 . y) by A17, A86, A92, FUNCT_4:def 1

          .= (f . y) by A92, FUNCT_1: 49;

        end;

          suppose

           A93: y in K11;

          then y in (( dom f1) \/ ( dom f2)) by A17, XBOOLE_0:def 3;

          

          hence (h . y) = (f2 . y) by A17, A86, A93, FUNCT_4:def 1

          .= (f . y) by A93, FUNCT_1: 49;

        end;

      end;

      K0 = the carrier of (( TOP-REAL 2) | K0) by PRE_TOPC: 8

      .= ( dom f) by A74, FUNCT_2:def 1;

      hence thesis by A87, A88, A90, FUNCT_1: 2, PRE_TOPC: 8;

    end;

    theorem :: JGRAPH_4:125

    

     Th125: for cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0);

      set sn = ( sqrt (1 - (cn ^2 )));

      set p0 = |[cn, sn]|;

      

       A1: (p0 `2 ) = sn by EUCLID: 52;

      assume

       A2: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then (cn ^2 ) < (1 ^2 ) by SQUARE_1: 50;

      then

       A3: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

      then sn > 0 by SQUARE_1: 25;

      then p0 in K0 by A2, A1, JGRAPH_2: 3;

      then

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2);

      (p0 `2 ) > 0 by A1, A3, SQUARE_1: 25;

      then not p0 in {( 0. ( TOP-REAL 2))} by JGRAPH_2: 3, TARSKI:def 1;

      then

      reconsider D = B0 as non empty Subset of ( TOP-REAL 2) by A2, XBOOLE_0:def 5;

      

       A4: K1 c= D

      proof

        let x be object;

        assume x in K1;

        then

        consider p2 be Point of ( TOP-REAL 2) such that

         A5: p2 = x and (p2 `2 ) >= 0 and

         A6: p2 <> ( 0. ( TOP-REAL 2)) by A2;

         not p2 in {( 0. ( TOP-REAL 2))} by A6, TARSKI:def 1;

        hence thesis by A2, A5, XBOOLE_0:def 5;

      end;

      for p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D) st (f . p) in V & V is open holds ex W be Subset of (( TOP-REAL 2) | K1) st p in W & W is open & (f .: W) c= V

      proof

        let p be Point of (( TOP-REAL 2) | K1), V be Subset of (( TOP-REAL 2) | D);

        assume that

         A7: (f . p) in V and

         A8: V is open;

        consider V2 be Subset of ( TOP-REAL 2) such that

         A9: V2 is open and

         A10: (V2 /\ ( [#] (( TOP-REAL 2) | D))) = V by A8, TOPS_2: 24;

        reconsider W2 = (V2 /\ ( [#] (( TOP-REAL 2) | K1))) as Subset of (( TOP-REAL 2) | K1);

        

         A11: ( [#] (( TOP-REAL 2) | K1)) = K1 by PRE_TOPC:def 5;

        then

         A12: (f . p) = ((cn -FanMorphS ) . p) by A2, FUNCT_1: 49;

        

         A13: (f .: W2) c= V

        proof

          let y be object;

          assume y in (f .: W2);

          then

          consider x be object such that

           A14: x in ( dom f) and

           A15: x in W2 and

           A16: y = (f . x) by FUNCT_1:def 6;

          f is Function of (( TOP-REAL 2) | K1), (( TOP-REAL 2) | D);

          then ( dom f) = K1 by A11, FUNCT_2:def 1;

          then

          consider p4 be Point of ( TOP-REAL 2) such that

           A17: x = p4 and

           A18: (p4 `2 ) >= 0 and p4 <> ( 0. ( TOP-REAL 2)) by A2, A14;

          

           A19: p4 in V2 by A15, A17, XBOOLE_0:def 4;

          p4 in ( [#] (( TOP-REAL 2) | K1)) by A14, A17;

          then p4 in D by A4, A11;

          then

           A20: p4 in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

          (f . p4) = ((cn -FanMorphS ) . p4) by A2, A11, A14, A17, FUNCT_1: 49

          .= p4 by A18, Th113;

          hence thesis by A10, A16, A17, A19, A20, XBOOLE_0:def 4;

        end;

        p in the carrier of (( TOP-REAL 2) | K1);

        then

        consider q be Point of ( TOP-REAL 2) such that

         A21: q = p and

         A22: (q `2 ) >= 0 and q <> ( 0. ( TOP-REAL 2)) by A2, A11;

        ((cn -FanMorphS ) . q) = q by A22, Th113;

        then p in V2 by A7, A10, A12, A21, XBOOLE_0:def 4;

        then

         A23: p in W2 by XBOOLE_0:def 4;

        W2 is open by A9, TOPS_2: 24;

        hence thesis by A23, A13;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JGRAPH_4:126

    

     Th126: for cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th124;

    end;

    theorem :: JGRAPH_4:127

    

     Th127: for cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

      let cn be Real, B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0);

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: ( - 1) < cn & cn < 1 & f = ((cn -FanMorphS ) | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

      hence thesis by A1, Th125;

    end;

    theorem :: JGRAPH_4:128

    

     Th128: for cn be Real, p be Point of ( TOP-REAL 2) holds |.((cn -FanMorphS ) . p).| = |.p.|

    proof

      let cn be Real, p be Point of ( TOP-REAL 2);

      set f = (cn -FanMorphS );

      set z = (f . p);

      set q = p;

      reconsider qz = z as Point of ( TOP-REAL 2);

      per cases ;

        suppose

         A1: ((q `1 ) / |.q.|) >= cn & (q `2 ) < 0 ;

        then

         A2: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by Th113;

        then

         A3: (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

        

         A4: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by A2, EUCLID: 52;

        

         A5: (((q `1 ) / |.q.|) - cn) >= 0 by A1, XREAL_1: 48;

        

         A6: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         |.q.| <> 0 by A1, JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A7: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A6, XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A7, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then

         A8: (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        per cases ;

          suppose

           A9: (1 - cn) = 0 ;

          

           A10: ((((q `1 ) / |.q.|) - cn) / (1 - cn)) = ((((q `1 ) / |.q.|) - cn) * ((1 - cn) " )) by XCMPLX_0:def 9

          .= ((((q `1 ) / |.q.|) - cn) * 0 ) by A9

          .= 0 ;

          then (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) = 1;

          

          then ((cn -FanMorphS ) . q) = |[( |.q.| * 0 ), ( |.q.| * ( - 1))]| by A1, A10, Th113, SQUARE_1: 18

          .= |[ 0 , ( - |.q.|)]|;

          then (((cn -FanMorphS ) . q) `2 ) = ( - |.q.|) & (((cn -FanMorphS ) . q) `1 ) = 0 by EUCLID: 52;

          

          then |.((cn -FanMorphS ) . p).| = ( sqrt ((( - |.q.|) ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ( |.q.| ^2 ))

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A11: (1 - cn) <> 0 ;

          per cases by A11;

            suppose

             A12: (1 - cn) > 0 ;

            ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by A8, XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A12, XREAL_1: 72;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A12, XCMPLX_1: 197;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A5, A12, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A13: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A14: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 )) by A3

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A13, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A4, A14;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A15: (1 - cn) < 0 ;

            ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A1, SQUARE_1: 12, XREAL_1: 8;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A7, A6, XREAL_1: 74;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A7, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then

             A16: 1 > ((q `1 ) / |.p.|) by SQUARE_1: 52;

            (((q `1 ) / |.q.|) - cn) >= 0 by A1, XREAL_1: 48;

            hence thesis by A15, A16, XREAL_1: 9;

          end;

        end;

      end;

        suppose

         A17: ((q `1 ) / |.q.|) < cn & (q `2 ) < 0 ;

        then |.q.| <> 0 by JGRAPH_2: 3, TOPRNS_1: 24;

        then

         A18: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

        

         A19: (((q `1 ) / |.q.|) - cn) < 0 by A17, XREAL_1: 49;

        

         A20: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A20, XREAL_1: 72;

        then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A18, XCMPLX_1: 60;

        then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

        then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

        then

         A21: (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

        

         A22: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by A17, Th114;

        then

         A23: (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

        

         A24: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A22, EUCLID: 52;

        per cases ;

          suppose

           A25: (1 + cn) = 0 ;

          ((((q `1 ) / |.q.|) - cn) / (1 + cn)) = ((((q `1 ) / |.q.|) - cn) * ((1 + cn) " )) by XCMPLX_0:def 9

          .= ((((q `1 ) / |.q.|) - cn) * 0 ) by A25

          .= 0 ;

          then (((cn -FanMorphS ) . q) `2 ) = ( - |.q.|) & (((cn -FanMorphS ) . q) `1 ) = 0 by A22, EUCLID: 52, SQUARE_1: 18;

          

          then |.((cn -FanMorphS ) . p).| = ( sqrt ((( - |.q.|) ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ( |.q.| ^2 ))

          .= |.q.| by SQUARE_1: 22;

          hence thesis;

        end;

          suppose

           A26: (1 + cn) <> 0 ;

          per cases by A26;

            suppose

             A27: (1 + cn) > 0 ;

            then (( - (1 + cn)) / (1 + cn)) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A21, XREAL_1: 72;

            then ( - 1) <= ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A27, XCMPLX_1: 197;

            then (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) <= (1 ^2 ) by A19, A27, SQUARE_1: 49;

            then

             A28: (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            

             A29: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 )) by A23

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A28, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A24, A29;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            hence thesis by SQUARE_1: 22;

          end;

            suppose

             A30: (1 + cn) < 0 ;

            ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A17, SQUARE_1: 12, XREAL_1: 8;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A18, A20, XREAL_1: 74;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A18, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

            then ( - 1) < ((q `1 ) / |.p.|) by SQUARE_1: 52;

            then

             A31: (((q `1 ) / |.q.|) - cn) > (( - 1) - cn) by XREAL_1: 9;

            ( - (1 + cn)) > ( - 0 ) by A30, XREAL_1: 24;

            hence thesis by A17, A31, XREAL_1: 49;

          end;

        end;

      end;

        suppose (q `2 ) >= 0 ;

        hence thesis by Th113;

      end;

    end;

    theorem :: JGRAPH_4:129

    

     Th129: for cn be Real, x,K0 be set st ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((cn -FanMorphS ) . x) in K0

    proof

      let cn be Real, x,K0 be set;

      assume

       A1: ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then

      consider p such that

       A2: p = x and

       A3: (p `2 ) <= 0 and

       A4: p <> ( 0. ( TOP-REAL 2));

       A5:

      now

        assume |.p.| <= 0 ;

        then |.p.| = 0 ;

        hence contradiction by A4, TOPRNS_1: 24;

      end;

      then

       A6: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

      per cases ;

        suppose

         A7: ((p `1 ) / |.p.|) <= cn;

        reconsider p9 = ((cn -FanMorphS ) . p) as Point of ( TOP-REAL 2);

        ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A3, A4, A7, Th115;

        then

         A8: (p9 `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

        

         A9: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A10: (1 + cn) > 0 by A1, XREAL_1: 148;

        per cases ;

          suppose (p `2 ) = 0 ;

          hence thesis by A1, A2, Th113;

        end;

          suppose (p `2 ) <> 0 ;

          then ( 0 + ((p `1 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A9, XREAL_1: 74;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `1 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ( - 1) < ((p `1 ) / |.p.|) by SQUARE_1: 52;

          then (( - 1) - cn) < (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

          then ((( - 1) * (1 + cn)) / (1 + cn)) < ((((p `1 ) / |.p.|) - cn) / (1 + cn)) by A10, XREAL_1: 74;

          then

           A11: ( - 1) < ((((p `1 ) / |.p.|) - cn) / (1 + cn)) by A10, XCMPLX_1: 89;

          (((p `1 ) / |.p.|) - cn) <= 0 by A7, XREAL_1: 47;

          then (1 ^2 ) > (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ) by A10, A11, SQUARE_1: 50;

          then (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )) > 0 by XREAL_1: 50;

          then ( - ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) > 0 by SQUARE_1: 25;

          then ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) < 0 ;

          then ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) < 0 by A5, XREAL_1: 132;

          hence thesis by A1, A2, A8, JGRAPH_2: 3;

        end;

      end;

        suppose

         A12: ((p `1 ) / |.p.|) > cn;

        reconsider p9 = ((cn -FanMorphS ) . p) as Point of ( TOP-REAL 2);

        ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A3, A4, A12, Th115;

        then

         A13: (p9 `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

        

         A14: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

        

         A15: (1 - cn) > 0 by A1, XREAL_1: 149;

        per cases ;

          suppose (p `2 ) = 0 ;

          hence thesis by A1, A2, Th113;

        end;

          suppose (p `2 ) <> 0 ;

          then ( 0 + ((p `1 ) ^2 )) < (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by SQUARE_1: 12, XREAL_1: 8;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < (( |.p.| ^2 ) / ( |.p.| ^2 )) by A6, A14, XREAL_1: 74;

          then (((p `1 ) ^2 ) / ( |.p.| ^2 )) < 1 by A6, XCMPLX_1: 60;

          then (((p `1 ) / |.p.|) ^2 ) < 1 by XCMPLX_1: 76;

          then ((p `1 ) / |.p.|) < 1 by SQUARE_1: 52;

          then (((p `1 ) / |.p.|) - cn) < (1 - cn) by XREAL_1: 9;

          then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) < ((1 - cn) / (1 - cn)) by A15, XREAL_1: 74;

          then

           A16: ((((p `1 ) / |.p.|) - cn) / (1 - cn)) < 1 by A15, XCMPLX_1: 60;

          ( - (1 - cn)) < ( - 0 ) & (((p `1 ) / |.p.|) - cn) >= (cn - cn) by A12, A15, XREAL_1: 9, XREAL_1: 24;

          then ((( - 1) * (1 - cn)) / (1 - cn)) < ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A15, XREAL_1: 74;

          then ( - 1) < ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A15, XCMPLX_1: 89;

          then (1 ^2 ) > (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ) by A16, SQUARE_1: 50;

          then (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )) > 0 by XREAL_1: 50;

          then ( - ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) > 0 by SQUARE_1: 25;

          then ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) < 0 ;

          then (p9 `2 ) < 0 by A5, A13, XREAL_1: 132;

          hence thesis by A1, A2, JGRAPH_2: 3;

        end;

      end;

    end;

    theorem :: JGRAPH_4:130

    

     Th130: for cn be Real, x,K0 be set st ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } holds ((cn -FanMorphS ) . x) in K0

    proof

      let cn be Real, x,K0 be set;

      assume

       A1: ( - 1) < cn & cn < 1 & x in K0 & K0 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) };

      then ex p st p = x & (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2));

      hence thesis by A1, Th113;

    end;

    theorem :: JGRAPH_4:131

    

     Th131: for cn be Real, D be non empty Subset of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (D ` ) = {( 0. ( TOP-REAL 2))} holds ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((cn -FanMorphS ) | D) & h is continuous

    proof

      set Y1 = |[ 0 , 1]|;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) <= 0 ;

      reconsider B0 = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

      let cn be Real, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn & cn < 1 and

       A2: (D ` ) = {( 0. ( TOP-REAL 2))};

      

       A3: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      ( dom (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then

       A4: ( dom ((cn -FanMorphS ) | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A3, XBOOLE_1: 28;

      ( |[ 0 , ( - 1)]| `2 ) = ( - 1) & |[ 0 , ( - 1)]| <> ( 0. ( TOP-REAL 2)) by EUCLID: 52, JGRAPH_2: 3;

      then

       A5: |[ 0 , ( - 1)]| in { p where p be Point of ( TOP-REAL 2) : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) };

      (Y1 `2 ) = 1 by EUCLID: 52;

      then

       A6: Y1 in { p where p be Point of ( TOP-REAL 2) : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } by JGRAPH_2: 3;

      

       A7: D = (B0 ` ) by A2

      .= ( NonZero ( TOP-REAL 2)) by SUBSET_1:def 4;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A7);

      then

      reconsider K0 = { p : (p `2 ) <= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A5;

      

       A8: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      defpred P[ Point of ( TOP-REAL 2)] means ($1 `2 ) >= 0 ;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D) from InclSub( A7);

      then

      reconsider K1 = { p : (p `2 ) >= 0 & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A6;

      

       A9: K0 is closed & K1 is closed by A7, Th62, Th63;

      

       A10: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A11: ( rng ((cn -FanMorphS ) | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphS ) | K0));

        then

        consider x be object such that

         A12: x in ( dom ((cn -FanMorphS ) | K0)) and

         A13: y = (((cn -FanMorphS ) | K0) . x) by FUNCT_1:def 3;

        x in (( dom (cn -FanMorphS )) /\ K0) by A12, RELAT_1: 61;

        then

         A14: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A10, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A14;

        ((cn -FanMorphS ) . p) = y by A13, A14, FUNCT_1: 49;

        then y in K0 by A1, A14, Th129;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A15: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `2 ) <= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ((cn -FanMorphS ) | K0)) = (( dom (cn -FanMorphS )) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by FUNCT_2:def 1

      .= K0 by A15, XBOOLE_1: 28;

      then

      reconsider f = ((cn -FanMorphS ) | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A8, A11, FUNCT_2: 2, XBOOLE_1: 1;

      

       A16: K1 = the carrier of ((( TOP-REAL 2) | D) | K1) by PRE_TOPC: 8;

      

       A17: ( rng ((cn -FanMorphS ) | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        let y be object;

        assume y in ( rng ((cn -FanMorphS ) | K1));

        then

        consider x be object such that

         A18: x in ( dom ((cn -FanMorphS ) | K1)) and

         A19: y = (((cn -FanMorphS ) | K1) . x) by FUNCT_1:def 3;

        x in (( dom (cn -FanMorphS )) /\ K1) by A18, RELAT_1: 61;

        then

         A20: x in K1 by XBOOLE_0:def 4;

        K1 c= the carrier of ( TOP-REAL 2) by A10, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A20;

        ((cn -FanMorphS ) . p) = y by A19, A20, FUNCT_1: 49;

        then y in K1 by A1, A20, Th130;

        hence thesis by PRE_TOPC: 8;

      end;

      

       A21: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & (p8 `2 ) >= 0 & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ((cn -FanMorphS ) | K1)) = (( dom (cn -FanMorphS )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by FUNCT_2:def 1

      .= K1 by A21, XBOOLE_1: 28;

      then

      reconsider g = ((cn -FanMorphS ) | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A16, A17, FUNCT_2: 2, XBOOLE_1: 1;

      

       A22: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A23: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A24: x in D;

        then

        reconsider px = x as Point of ( TOP-REAL 2);

         not x in {( 0. ( TOP-REAL 2))} by A7, A24, XBOOLE_0:def 5;

        then (px `2 ) >= 0 & px <> ( 0. ( TOP-REAL 2)) or (px `2 ) <= 0 & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        then x in K1 or x in K0;

        hence thesis by XBOOLE_0:def 3;

      end;

      

       A25: ( dom f) = K0 by A8, FUNCT_2:def 1;

      

       A26: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A27: for x be object st x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1))) holds (f . x) = (g . x)

      proof

        let x be object;

        assume

         A28: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A26, XBOOLE_0:def 4;

        then (f . x) = ((cn -FanMorphS ) . x) by FUNCT_1: 49;

        hence thesis by A22, A28, FUNCT_1: 49;

      end;

      D = ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

      then

       A29: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A26, A22, A23, XBOOLE_0:def 10;

      

       A30: f is continuous & g is continuous by A1, A7, Th126, Th127;

      then

      consider h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) such that

       A31: h = (f +* g) and h is continuous by A26, A22, A29, A9, A27, JGRAPH_2: 1;

      

       A32: ( dom g) = K1 by A16, FUNCT_2:def 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A33: f tolerates g by A27, A25, A32, PARTFUN1:def 4;

      

       A34: the carrier of (( TOP-REAL 2) | D) = ( NonZero ( TOP-REAL 2)) by A7, PRE_TOPC: 8;

      

       A35: for x be object st x in ( dom h) holds (h . x) = (((cn -FanMorphS ) | D) . x)

      proof

        let x be object;

        assume

         A36: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A34, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A7, A3, A36, XBOOLE_0:def 5;

        then

         A37: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

           A38: x in K0;

          

           A39: (((cn -FanMorphS ) | D) . p) = ((cn -FanMorphS ) . p) by A3, A36, FUNCT_1: 49

          .= (f . p) by A38, FUNCT_1: 49;

          (h . p) = ((g +* f) . p) by A31, A33, FUNCT_4: 34

          .= (f . p) by A25, A38, FUNCT_4: 13;

          hence thesis by A39;

        end;

          suppose not x in K0;

          then not (p `2 ) <= 0 by A37;

          then

           A40: x in K1 by A37;

          (((cn -FanMorphS ) | D) . p) = ((cn -FanMorphS ) . p) by A3, A36, FUNCT_1: 49

          .= (g . p) by A40, FUNCT_1: 49;

          hence thesis by A31, A32, A40, FUNCT_4: 13;

        end;

      end;

      ( dom h) = the carrier of (( TOP-REAL 2) | D) by FUNCT_2:def 1;

      then (f +* g) = ((cn -FanMorphS ) | D) by A31, A4, A35, FUNCT_1: 2;

      hence thesis by A26, A22, A29, A30, A9, A27, JGRAPH_2: 1;

    end;

    theorem :: JGRAPH_4:132

    

     Th132: for cn be Real st ( - 1) < cn & cn < 1 holds (cn -FanMorphS ) is continuous

    proof

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      reconsider f = (cn -FanMorphS ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A3: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Th113, JGRAPH_2: 3;

      

       A4: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

        let p be Point of (( TOP-REAL 2) | D);

        

         A5: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A5, XBOOLE_0:def 5;

        then

         A6: p <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

           A7: ((q `1 ) / |.q.|) >= cn & (q `2 ) <= 0 ;

          set q9 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]|;

          

           A8: (q9 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

          

           A9: (q9 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

          now

            assume

             A10: q9 = ( 0. ( TOP-REAL 2));

            

             A11: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

            

            then ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) = ( - ( sqrt (1 - 0 ))) by A8, A10, JGRAPH_2: 3, XCMPLX_1: 6

            .= ( - 1) by SQUARE_1: 18;

            hence contradiction by A9, A10, A11, JGRAPH_2: 3, XCMPLX_1: 6;

          end;

          hence thesis by A1, A2, A3, A6, A7, Th115;

        end;

          suppose

           A12: ((q `1 ) / |.q.|) < cn & (q `2 ) <= 0 ;

          set q9 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]|;

          

           A13: (q9 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

          

           A14: (q9 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

          now

            assume

             A15: q9 = ( 0. ( TOP-REAL 2));

            

             A16: |.q.| <> ( 0 ^2 ) by A6, TOPRNS_1: 24;

            

            then ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) = ( - ( sqrt (1 - 0 ))) by A13, A15, JGRAPH_2: 3, XCMPLX_1: 6

            .= ( - 1) by SQUARE_1: 18;

            hence contradiction by A14, A15, A16, JGRAPH_2: 3, XCMPLX_1: 6;

          end;

          hence thesis by A1, A2, A3, A6, A12, Th115;

        end;

          suppose (q `2 ) > 0 ;

          then (f . p) = p by Th113;

          hence thesis by A6, Th113, JGRAPH_2: 3;

        end;

      end;

      

       A17: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

        let V be Subset of ( TOP-REAL 2);

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm11;

        assume that

         A18: (f . ( 0. ( TOP-REAL 2))) in V and

         A19: V is open;

        VV is open by A19, Lm11, PRE_TOPC: 30;

        then

        consider r be Real such that

         A20: r > 0 and

         A21: ( Ball (u0,r)) c= V by A3, A18, TOPMETR: 15;

        reconsider r as Real;

         the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

        then

        reconsider W1 = ( Ball (u0,r)) as Subset of ( TOP-REAL 2);

        

         A22: W1 is open by GOBOARD6: 3;

        

         A23: (f .: W1) c= W1

        proof

          let z be object;

          assume z in (f .: W1);

          then

          consider y be object such that

           A24: y in ( dom f) and

           A25: y in W1 and

           A26: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A24, A26, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          reconsider q = y as Point of ( TOP-REAL 2) by A24;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          ( dist (u0,qy)) < r by A25, METRIC_1: 11;

          then

           A27: |.(( 0. ( TOP-REAL 2)) - q).| < r by JGRAPH_1: 28;

          per cases by JGRAPH_2: 3;

            suppose (q `2 ) >= 0 ;

            hence thesis by A25, A26, Th113;

          end;

            suppose

             A28: q <> ( 0. ( TOP-REAL 2)) & ((q `1 ) / |.q.|) >= cn & (q `2 ) <= 0 ;

            then

             A29: (((q `1 ) / |.q.|) - cn) >= 0 by XREAL_1: 48;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A30: (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            

             A31: (1 - cn) > 0 by A2, XREAL_1: 149;

             |.q.| <> 0 by A28, TOPRNS_1: 24;

            then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A30, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

            then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

            then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A31, XREAL_1: 72;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A31, XCMPLX_1: 197;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A31, A29, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A32: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A33: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, A28, Th115;

            then

             A34: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by A26, EUCLID: 52;

            (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by A26, A33, EUCLID: 52;

            

            then

             A35: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A32, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A34, A35;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A36: |.qz.| = |.q.| by SQUARE_1: 22;

             |.( - q).| < r by A27, RLVECT_1: 4;

            then |.q.| < r by TOPRNS_1: 26;

            then |.( - qz).| < r by A36, TOPRNS_1: 26;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

            suppose

             A37: q <> ( 0. ( TOP-REAL 2)) & ((q `1 ) / |.q.|) < cn & (q `2 ) <= 0 ;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A38: (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            

             A39: (1 + cn) > 0 by A1, XREAL_1: 148;

             |.q.| <> 0 by A37, TOPRNS_1: 24;

            then ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A38, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

            then ( - ( - 1)) >= ( - ((q `1 ) / |.q.|)) by XREAL_1: 24;

            then (1 + cn) >= (( - ((q `1 ) / |.q.|)) + cn) by XREAL_1: 7;

            then

             A40: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A39, XREAL_1: 185;

            (cn - ((q `1 ) / |.q.|)) >= 0 by A37, XREAL_1: 48;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A39;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A40, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A41: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            

             A42: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A2, A37, Th115;

            then

             A43: (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A26, EUCLID: 52;

            (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by A26, A42, EUCLID: 52;

            

            then

             A44: ((qz `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A41, SQUARE_1:def 2;

            ( |.qz.| ^2 ) = (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A43, A44;

            then ( sqrt ( |.qz.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A45: |.qz.| = |.q.| by SQUARE_1: 22;

             |.( - q).| < r by A27, RLVECT_1: 4;

            then |.q.| < r by TOPRNS_1: 26;

            then |.( - qz).| < r by A45, TOPRNS_1: 26;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by RLVECT_1: 4;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

        end;

        u0 in W1 by A20, GOBOARD6: 1;

        hence thesis by A21, A22, A23, XBOOLE_1: 1;

      end;

      

       A46: (D ` ) = {( 0. ( TOP-REAL 2))} by JGRAPH_3: 20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ((cn -FanMorphS ) | D) & h is continuous by A1, A2, Th131;

      hence thesis by A3, A46, A4, A17, JGRAPH_3: 3;

    end;

    theorem :: JGRAPH_4:133

    

     Th133: for cn be Real st ( - 1) < cn & cn < 1 holds (cn -FanMorphS ) is one-to-one

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      for x1,x2 be object st x1 in ( dom (cn -FanMorphS )) & x2 in ( dom (cn -FanMorphS )) & ((cn -FanMorphS ) . x1) = ((cn -FanMorphS ) . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A3: x1 in ( dom (cn -FanMorphS )) and

         A4: x2 in ( dom (cn -FanMorphS )) and

         A5: ((cn -FanMorphS ) . x1) = ((cn -FanMorphS ) . x2);

        reconsider p2 = x2 as Point of ( TOP-REAL 2) by A4;

        reconsider p1 = x1 as Point of ( TOP-REAL 2) by A3;

        set q = p1, p = p2;

        

         A6: (1 - cn) > 0 by A2, XREAL_1: 149;

        per cases by JGRAPH_2: 3;

          suppose

           A7: (q `2 ) >= 0 ;

          then

           A8: ((cn -FanMorphS ) . q) = q by Th113;

          per cases by JGRAPH_2: 3;

            suppose (p `2 ) >= 0 ;

            hence thesis by A5, A8, Th113;

          end;

            suppose

             A9: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 ;

            then

             A10: |.p.| <> 0 by TOPRNS_1: 24;

            then

             A11: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

            

             A12: (((p `1 ) / |.p.|) - cn) >= 0 by A9, XREAL_1: 48;

            

             A13: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

             0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

            then ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A13, XREAL_1: 72;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A11, XCMPLX_1: 60;

            then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((p `1 ) / |.p.|) by SQUARE_1: 51;

            then (1 - cn) >= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

            then ( - (1 - cn)) <= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

            then

             A14: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

            

             A15: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, A9, Th115;

            then

             A16: (q `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) by A5, A8, EUCLID: 52;

            (((p `1 ) / |.p.|) - cn) >= 0 by A9, XREAL_1: 48;

            then ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A14, SQUARE_1: 49;

            then

             A17: (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

            then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

            then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

            then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

            then (q `2 ) = 0 by A5, A7, A8, A15, EUCLID: 52;

            then

             A18: ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))) = ( - 0 ) by A16, A10, XCMPLX_1: 6;

            (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 - cn))) ^2 )) >= 0 by A17, XCMPLX_1: 187;

            then (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )) = 0 by A18, SQUARE_1: 24;

            then 1 = ((((p `1 ) / |.p.|) - cn) / (1 - cn)) by A6, A12, SQUARE_1: 18, SQUARE_1: 22;

            then (1 * (1 - cn)) = (((p `1 ) / |.p.|) - cn) by A6, XCMPLX_1: 87;

            then (1 * |.p.|) = (p `1 ) by A9, TOPRNS_1: 24, XCMPLX_1: 87;

            then (p `2 ) = 0 by A13, XCMPLX_1: 6;

            hence thesis by A5, A8, Th113;

          end;

            suppose

             A19: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) <= 0 ;

            then

             A20: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A2, Th115;

            then

             A21: (q `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) by A5, A8, EUCLID: 52;

            

             A22: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

            

             A23: |.p.| <> 0 by A19, TOPRNS_1: 24;

            then

             A24: ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

            

             A25: (1 + cn) > 0 by A1, XREAL_1: 148;

            

             A26: (((p `1 ) / |.p.|) - cn) <= 0 by A19, XREAL_1: 47;

            then

             A27: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by A25;

             0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

            then ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 7;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by A22, XREAL_1: 72;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A24, XCMPLX_1: 60;

            then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then (( - ((p `1 ) / |.p.|)) ^2 ) <= 1;

            then 1 >= ( - ((p `1 ) / |.p.|)) by SQUARE_1: 51;

            then (1 + cn) >= (( - ((p `1 ) / |.p.|)) + cn) by XREAL_1: 7;

            then (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) <= 1 by A25, XREAL_1: 185;

            then ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A27, SQUARE_1: 49;

            then

             A28: (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

            then ( sqrt (1 - ((( - (((p `1 ) / |.p.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

            then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

            then ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

            then (q `2 ) = 0 by A5, A7, A8, A20, EUCLID: 52;

            then

             A29: ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))) = ( - 0 ) by A21, A23, XCMPLX_1: 6;

            (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 )) >= 0 by A28, XCMPLX_1: 187;

            then (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )) = 0 by A29, SQUARE_1: 24;

            then 1 = (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 );

            then 1 = ( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by A25, A26, SQUARE_1: 18, SQUARE_1: 22;

            then 1 = (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by XCMPLX_1: 187;

            then (1 * (1 + cn)) = ( - (((p `1 ) / |.p.|) - cn)) by A25, XCMPLX_1: 87;

            then ((1 + cn) - cn) = ( - ((p `1 ) / |.p.|));

            then 1 = (( - (p `1 )) / |.p.|) by XCMPLX_1: 187;

            then (1 * |.p.|) = ( - (p `1 )) by A19, TOPRNS_1: 24, XCMPLX_1: 87;

            then (((p `1 ) ^2 ) - ((p `1 ) ^2 )) = ((p `2 ) ^2 ) by A22, XCMPLX_1: 26;

            then (p `2 ) = 0 by XCMPLX_1: 6;

            hence thesis by A5, A8, Th113;

          end;

        end;

          suppose

           A30: ((q `1 ) / |.q.|) >= cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

          then |.q.| <> 0 by TOPRNS_1: 24;

          then

           A31: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]|;

          

           A32: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

          

           A33: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, A30, Th115;

          per cases by JGRAPH_2: 3;

            suppose

             A34: (p `2 ) >= 0 ;

            then

             A35: ((cn -FanMorphS ) . p) = p by Th113;

            then

             A36: (p `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by A5, A33, EUCLID: 52;

            

             A37: (((q `1 ) / |.q.|) - cn) >= 0 by A30, XREAL_1: 48;

            

             A38: (1 - cn) > 0 by A2, XREAL_1: 149;

            

             A39: |.q.| <> 0 by A30, TOPRNS_1: 24;

            then

             A40: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

            

             A41: (((q `1 ) / |.q.|) - cn) >= 0 by A30, XREAL_1: 48;

            

             A42: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A42, XREAL_1: 72;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A40, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

            then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

            then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A38, XREAL_1: 72;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A38, XCMPLX_1: 197;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A38, A41, SQUARE_1: 49;

            then

             A43: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ))) >= 0 by SQUARE_1:def 2;

            then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 )))) >= 0 by XCMPLX_1: 76;

            then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 )))) >= 0 ;

            then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

            then (p `2 ) = 0 by A5, A33, A34, A35, EUCLID: 52;

            then

             A44: ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))) = ( - 0 ) by A36, A39, XCMPLX_1: 6;

            (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by A43, XCMPLX_1: 187;

            then (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) = 0 by A44, SQUARE_1: 24;

            then 1 = ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by A38, A37, SQUARE_1: 18, SQUARE_1: 22;

            then (1 * (1 - cn)) = (((q `1 ) / |.q.|) - cn) by A38, XCMPLX_1: 87;

            then (1 * |.q.|) = (q `1 ) by A30, TOPRNS_1: 24, XCMPLX_1: 87;

            then (q `2 ) = 0 by A42, XCMPLX_1: 6;

            hence thesis by A5, A35, Th113;

          end;

            suppose

             A45: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 ;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A31, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((q `1 ) / |.q.|) by SQUARE_1: 51;

            then (1 - cn) >= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

            then ( - (1 - cn)) <= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

            then

             A46: ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

            (((q `1 ) / |.q.|) - cn) >= 0 by A30, XREAL_1: 48;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A46, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A47: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            (q4 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

            

            then

             A48: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by A47, SQUARE_1:def 2;

            

             A49: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) by EUCLID: 52;

            ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A49, A48;

            then

             A50: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A51: |.q4.| = |.q.| by SQUARE_1: 22;

             0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

            then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A52: (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

             |.p.| <> 0 by A45, TOPRNS_1: 24;

            then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A52, XCMPLX_1: 60;

            then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then 1 >= ((p `1 ) / |.p.|) by SQUARE_1: 51;

            then (1 - cn) >= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

            then ( - (1 - cn)) <= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

            then (( - (1 - cn)) / (1 - cn)) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XREAL_1: 72;

            then

             A53: ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) by A6, XCMPLX_1: 197;

            (((p `1 ) / |.p.|) - cn) >= 0 by A45, XREAL_1: 48;

            then ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 ) <= (1 ^2 ) by A6, A53, SQUARE_1: 49;

            then (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 - cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A54: (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 - cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]|;

            

             A55: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) by EUCLID: 52;

            (p4 `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

            

            then

             A56: ((p4 `2 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) ^2 ))

            .= (( |.p.| ^2 ) * (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) by A54, SQUARE_1:def 2;

            ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.p.| ^2 ) by A55, A56;

            then

             A57: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

            then

             A58: |.p4.| = |.p.| by SQUARE_1: 22;

            

             A59: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, A45, Th115;

            then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) = (( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))) / |.p.|) by A5, A33, A32, A45, A55, TOPRNS_1: 24, XCMPLX_1: 89;

            then ((((p `1 ) / |.p.|) - cn) / (1 - cn)) = ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by A5, A33, A45, A59, A50, A57, TOPRNS_1: 24, XCMPLX_1: 89;

            then (((((p `1 ) / |.p.|) - cn) / (1 - cn)) * (1 - cn)) = (((q `1 ) / |.q.|) - cn) by A6, XCMPLX_1: 87;

            then (((p `1 ) / |.p.|) - cn) = (((q `1 ) / |.q.|) - cn) by A6, XCMPLX_1: 87;

            then (((p `1 ) / |.p.|) * |.p.|) = (q `1 ) by A5, A33, A45, A59, A51, A58, TOPRNS_1: 24, XCMPLX_1: 87;

            then

             A60: (p `1 ) = (q `1 ) by A45, TOPRNS_1: 24, XCMPLX_1: 87;

            ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

            then (( - (p `2 )) ^2 ) = ((q `2 ) ^2 ) by A5, A33, A59, A51, A58, A60;

            then ( - (p `2 )) = ( sqrt (( - (q `2 )) ^2 )) by A45, SQUARE_1: 22;

            then

             A61: ( - ( - (p `2 ))) = ( - ( - (q `2 ))) by A30, SQUARE_1: 22;

            p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

            hence thesis by A60, A61, EUCLID: 53;

          end;

            suppose

             A62: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) <= 0 ;

            then (((p `1 ) / |.p.|) - cn) < 0 by XREAL_1: 49;

            then

             A63: ((((p `1 ) / |.p.|) - cn) / (1 + cn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

            set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]|;

            

             A64: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) & (((q `1 ) / |.q.|) - cn) >= 0 by A30, EUCLID: 52, XREAL_1: 48;

            

             A65: (1 - cn) > 0 by A2, XREAL_1: 149;

            ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| & |.p.| <> 0 by A1, A2, A62, Th115, TOPRNS_1: 24;

            hence thesis by A5, A33, A32, A63, A64, A65, XREAL_1: 132;

          end;

        end;

          suppose

           A66: ((q `1 ) / |.q.|) < cn & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

          then

           A67: |.q.| <> 0 by TOPRNS_1: 24;

          then

           A68: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

          set q4 = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]|;

          

           A69: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

          

           A70: ((cn -FanMorphS ) . q) = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A2, A66, Th115;

          per cases by JGRAPH_2: 3;

            suppose

             A71: (p `2 ) >= 0 ;

            then

             A72: ((cn -FanMorphS ) . p) = p by Th113;

            then

             A73: (p `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by A5, A70, EUCLID: 52;

            

             A74: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

            

             A75: (1 + cn) > 0 by A1, XREAL_1: 148;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by XREAL_1: 7;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by A74, XREAL_1: 72;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A68, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then (( - ((q `1 ) / |.q.|)) ^2 ) <= 1;

            then 1 >= ( - ((q `1 ) / |.q.|)) by SQUARE_1: 51;

            then (1 + cn) >= (( - ((q `1 ) / |.q.|)) + cn) by XREAL_1: 7;

            then

             A76: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A75, XREAL_1: 185;

            

             A77: (((q `1 ) / |.q.|) - cn) <= 0 by A66, XREAL_1: 47;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A75;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A76, SQUARE_1: 49;

            then

             A78: (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A79: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) >= 0 by A78, SQUARE_1:def 2;

            then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) >= 0 by XCMPLX_1: 76;

            then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) >= 0 ;

            then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) >= 0 by XCMPLX_1: 76;

            then (p `2 ) = 0 by A5, A70, A71, A72, EUCLID: 52;

            then ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) = ( - 0 ) by A67, A73, XCMPLX_1: 6;

            then (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) = 0 by A79, SQUARE_1: 24;

            then 1 = (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 );

            then 1 = ( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by A75, A77, SQUARE_1: 18, SQUARE_1: 22;

            then 1 = (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by XCMPLX_1: 187;

            then (1 * (1 + cn)) = ( - (((q `1 ) / |.q.|) - cn)) by A75, XCMPLX_1: 87;

            then ((1 + cn) - cn) = ( - ((q `1 ) / |.q.|));

            then 1 = (( - (q `1 )) / |.q.|) by XCMPLX_1: 187;

            then (1 * |.q.|) = ( - (q `1 )) by A66, TOPRNS_1: 24, XCMPLX_1: 87;

            then (((q `1 ) ^2 ) - ((q `1 ) ^2 )) = ((q `2 ) ^2 ) by A74, XCMPLX_1: 26;

            then (q `2 ) = 0 by XCMPLX_1: 6;

            hence thesis by A5, A72, Th113;

          end;

            suppose

             A80: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) >= cn & (p `2 ) <= 0 ;

            set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]|;

            

             A81: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))) & |.q.| <> 0 by A66, EUCLID: 52, TOPRNS_1: 24;

            (((q `1 ) / |.q.|) - cn) < 0 by A66, XREAL_1: 49;

            then

             A82: ((((q `1 ) / |.q.|) - cn) / (1 + cn)) < 0 by A1, XREAL_1: 141, XREAL_1: 148;

            

             A83: (1 - cn) > 0 by A2, XREAL_1: 149;

            ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )))))]| & (((p `1 ) / |.p.|) - cn) >= 0 by A1, A2, A80, Th115, XREAL_1: 48;

            hence thesis by A5, A70, A69, A82, A81, A83, XREAL_1: 132;

          end;

            suppose

             A84: p <> ( 0. ( TOP-REAL 2)) & ((p `1 ) / |.p.|) < cn & (p `2 ) <= 0 ;

             0 <= ((p `2 ) ^2 ) by XREAL_1: 63;

            then ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( 0 + ((p `1 ) ^2 )) <= (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then

             A85: (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= (( |.p.| ^2 ) / ( |.p.| ^2 )) by XREAL_1: 72;

            

             A86: (1 + cn) > 0 by A1, XREAL_1: 148;

             0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

            then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

            then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A68, XCMPLX_1: 60;

            then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then ( - 1) <= ((q `1 ) / |.q.|) by SQUARE_1: 51;

            then (( - 1) - cn) <= (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

            then ( - (( - 1) - cn)) >= ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

            then

             A87: (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) <= 1 by A86, XREAL_1: 185;

            (((q `1 ) / |.q.|) - cn) <= 0 by A66, XREAL_1: 47;

            then ( - 1) <= (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A86;

            then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A87, SQUARE_1: 49;

            then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A88: (1 - (( - ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            (q4 `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

            

            then

             A89: ((q4 `2 ) ^2 ) = (( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

            .= (( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by A88, SQUARE_1:def 2;

            

             A90: (q4 `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

            set p4 = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]|;

            

             A91: (p4 `1 ) = ( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))) by EUCLID: 52;

             |.p.| <> 0 by A84, TOPRNS_1: 24;

            then ( |.p.| ^2 ) > 0 by SQUARE_1: 12;

            then (((p `1 ) ^2 ) / ( |.p.| ^2 )) <= 1 by A85, XCMPLX_1: 60;

            then (((p `1 ) / |.p.|) ^2 ) <= 1 by XCMPLX_1: 76;

            then ( - 1) <= ((p `1 ) / |.p.|) by SQUARE_1: 51;

            then (( - 1) - cn) <= (((p `1 ) / |.p.|) - cn) by XREAL_1: 9;

            then ( - (( - 1) - cn)) >= ( - (((p `1 ) / |.p.|) - cn)) by XREAL_1: 24;

            then

             A92: (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) <= 1 by A86, XREAL_1: 185;

            (((p `1 ) / |.p.|) - cn) <= 0 by A84, XREAL_1: 47;

            then ( - 1) <= (( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) by A86;

            then ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 ) <= (1 ^2 ) by A92, SQUARE_1: 49;

            then (1 - ((( - (((p `1 ) / |.p.|) - cn)) / (1 + cn)) ^2 )) >= 0 by XREAL_1: 48;

            then

             A93: (1 - (( - ((((p `1 ) / |.p.|) - cn) / (1 + cn))) ^2 )) >= 0 by XCMPLX_1: 187;

            (p4 `2 ) = ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))))) by EUCLID: 52;

            

            then

             A94: ((p4 `2 ) ^2 ) = (( |.p.| ^2 ) * (( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) ^2 ))

            .= (( |.p.| ^2 ) * (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) by A93, SQUARE_1:def 2;

            ( |.p4.| ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.p.| ^2 ) by A91, A94;

            then

             A95: ( sqrt ( |.p4.| ^2 )) = |.p.| by SQUARE_1: 22;

            then

             A96: |.p4.| = |.p.| by SQUARE_1: 22;

            ( |.q4.| ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by JGRAPH_3: 1

            .= ( |.q.| ^2 ) by A90, A89;

            then

             A97: ( sqrt ( |.q4.| ^2 )) = |.q.| by SQUARE_1: 22;

            then

             A98: |.q4.| = |.q.| by SQUARE_1: 22;

            

             A99: ((cn -FanMorphS ) . p) = |[( |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 + cn))), ( |.p.| * ( - ( sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A2, A84, Th115;

            then ((((p `1 ) / |.p.|) - cn) / (1 + cn)) = (( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) / |.p.|) by A5, A70, A69, A84, A91, TOPRNS_1: 24, XCMPLX_1: 89;

            then ((((p `1 ) / |.p.|) - cn) / (1 + cn)) = ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A5, A70, A84, A99, A97, A95, TOPRNS_1: 24, XCMPLX_1: 89;

            then (((((p `1 ) / |.p.|) - cn) / (1 + cn)) * (1 + cn)) = (((q `1 ) / |.q.|) - cn) by A86, XCMPLX_1: 87;

            then (((p `1 ) / |.p.|) - cn) = (((q `1 ) / |.q.|) - cn) by A86, XCMPLX_1: 87;

            then (((p `1 ) / |.p.|) * |.p.|) = (q `1 ) by A5, A70, A84, A99, A98, A96, TOPRNS_1: 24, XCMPLX_1: 87;

            then

             A100: (p `1 ) = (q `1 ) by A84, TOPRNS_1: 24, XCMPLX_1: 87;

            ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;

            then (( - (p `2 )) ^2 ) = ((q `2 ) ^2 ) by A5, A70, A99, A98, A96, A100;

            then ( - (p `2 )) = ( sqrt (( - (q `2 )) ^2 )) by A84, SQUARE_1: 22;

            then

             A101: ( - ( - (p `2 ))) = ( - ( - (q `2 ))) by A66, SQUARE_1: 22;

            p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

            hence thesis by A100, A101, EUCLID: 53;

          end;

        end;

      end;

      hence thesis by FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_4:134

    

     Th134: for cn be Real st ( - 1) < cn & cn < 1 holds (cn -FanMorphS ) is Function of ( TOP-REAL 2), ( TOP-REAL 2) & ( rng (cn -FanMorphS )) = the carrier of ( TOP-REAL 2)

    proof

      let cn be Real;

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1;

      thus (cn -FanMorphS ) is Function of ( TOP-REAL 2), ( TOP-REAL 2);

      for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (cn -FanMorphS ) holds ( rng (cn -FanMorphS )) = the carrier of ( TOP-REAL 2)

      proof

        let f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

        assume

         A3: f = (cn -FanMorphS );

        

         A4: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        the carrier of ( TOP-REAL 2) c= ( rng f)

        proof

          let y be object;

          assume y in the carrier of ( TOP-REAL 2);

          then

          reconsider p2 = y as Point of ( TOP-REAL 2);

          set q = p2;

          now

            per cases by JGRAPH_2: 3;

              case (q `2 ) >= 0 ;

              then y = ((cn -FanMorphS ) . q) by Th113;

              hence ex x be set st x in ( dom (cn -FanMorphS )) & y = ((cn -FanMorphS ) . x) by A3, A4;

            end;

              case

               A5: ((q `1 ) / |.q.|) >= 0 & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

              ( - ( - (1 + cn))) > 0 by A1, XREAL_1: 148;

              then

               A6: ( - (( - 1) - cn)) > 0 ;

              

               A7: (1 - cn) >= 0 by A2, XREAL_1: 149;

              then (((q `1 ) / |.q.|) * (1 - cn)) >= 0 by A5;

              then (( - 1) - cn) <= (((q `1 ) / |.q.|) * (1 - cn)) by A6;

              then

               A8: ((( - 1) - cn) + cn) <= ((((q `1 ) / |.q.|) * (1 - cn)) + cn) by XREAL_1: 7;

              set px = |[( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)), ( - ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )))))]|;

              

               A9: (px `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)) by EUCLID: 52;

               |.q.| <> 0 by A5, TOPRNS_1: 24;

              then

               A10: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

              

               A11: ( dom (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A12: (1 - cn) > 0 by A2, XREAL_1: 149;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A10, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `1 ) / |.q.|) <= 1 by SQUARE_1: 51;

              then (((q `1 ) / |.q.|) * (1 - cn)) <= (1 * (1 - cn)) by A12, XREAL_1: 64;

              then (((((q `1 ) / |.q.|) * (1 - cn)) + cn) - cn) <= (1 - cn);

              then ((((q `1 ) / |.q.|) * (1 - cn)) + cn) <= 1 by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ) by A8, SQUARE_1: 49;

              then

               A13: (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A14: ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A15: (px `2 ) = ( - ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = (((( - |.q.|) * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )))) ^2 ) + (( |.q.| * ((((q `1 ) / |.q.|) * (1 - cn)) + cn)) ^2 )) by A9, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 )));

              

              then

               A16: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 - cn)) + cn) ^2 ))) by A13, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A17: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A18: px <> ( 0. ( TOP-REAL 2)) by A5, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `1 ) / |.q.|) * (1 - cn)) + cn) >= ( 0 + cn) by A5, A7, XREAL_1: 7;

              then ((px `1 ) / |.px.|) >= cn by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A19: ((cn -FanMorphS ) . px) = |[( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 - cn))), ( |.px.| * ( - ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, A15, A14, A18, Th115;

              ( |.px.| * ( sqrt (( - ((q `2 ) / |.q.|)) ^2 ))) = ( |.q.| * ( - ((q `2 ) / |.q.|))) by A5, A17, SQUARE_1: 22

              .= ((( - (q `2 )) / |.q.|) * |.q.|) by XCMPLX_1: 187

              .= ( - (q `2 )) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              then

               A20: ( |.px.| * ( - ( sqrt (( - ((q `2 ) / |.q.|)) ^2 )))) = (q `2 );

              

               A21: ( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 - cn))) = ( |.q.| * ((((((q `1 ) / |.q.|) * (1 - cn)) + cn) - cn) / (1 - cn))) by A5, A9, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `1 ) / |.q.|)) by A12, XCMPLX_1: 89

              .= (q `1 ) by A5, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( - ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 - cn)) ^2 ))))) = ( |.px.| * ( - ( sqrt (1 - (((q `1 ) / |.px.|) ^2 ))))) by A5, A17, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( - ( sqrt (1 - (((q `1 ) ^2 ) / ( |.px.| ^2 )))))) by XCMPLX_1: 76

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `1 ) ^2 ) / ( |.px.| ^2 )))))) by A10, A16, XCMPLX_1: 60

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) - ((q `1 ) ^2 )) / ( |.px.| ^2 ))))) by XCMPLX_1: 120

              .= ( |.px.| * ( - ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `1 ) ^2 )) / ( |.px.| ^2 ))))) by A16, JGRAPH_3: 1

              .= ( |.px.| * ( - ( sqrt (((q `2 ) / |.q.|) ^2 )))) by A17, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (cn -FanMorphS )) & y = ((cn -FanMorphS ) . x) by A19, A21, A20, A11, EUCLID: 53;

            end;

              case

               A22: ((q `1 ) / |.q.|) < 0 & (q `2 ) <= 0 & q <> ( 0. ( TOP-REAL 2));

              

               A23: (1 + cn) >= 0 by A1, XREAL_1: 148;

              (1 - cn) > 0 by A2, XREAL_1: 149;

              then

               A24: ((1 - cn) + cn) >= ((((q `1 ) / |.q.|) * (1 + cn)) + cn) by A22, A23, XREAL_1: 7;

              

               A25: (1 + cn) > 0 by A1, XREAL_1: 148;

               |.q.| <> 0 by A22, TOPRNS_1: 24;

              then

               A26: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

               0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

              then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1, XREAL_1: 7;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= (( |.q.| ^2 ) / ( |.q.| ^2 )) by XREAL_1: 72;

              then (((q `1 ) ^2 ) / ( |.q.| ^2 )) <= 1 by A26, XCMPLX_1: 60;

              then (((q `1 ) / |.q.|) ^2 ) <= 1 by XCMPLX_1: 76;

              then ((q `1 ) / |.q.|) >= ( - 1) by SQUARE_1: 51;

              then (((q `1 ) / |.q.|) * (1 + cn)) >= (( - 1) * (1 + cn)) by A25, XREAL_1: 64;

              then (((((q `1 ) / |.q.|) * (1 + cn)) + cn) - cn) >= (( - 1) - cn);

              then ((((q `1 ) / |.q.|) * (1 + cn)) + cn) >= ( - 1) by XREAL_1: 9;

              then (1 ^2 ) >= (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ) by A24, SQUARE_1: 49;

              then

               A27: (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )) >= 0 by XREAL_1: 48;

              then

               A28: ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) >= 0 by SQUARE_1:def 2;

              

               A29: ( dom (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              set px = |[( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)), ( - ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )))))]|;

              

               A30: (px `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)) by EUCLID: 52;

              

               A31: (px `2 ) = ( - ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))))) by EUCLID: 52;

              

              then ( |.px.| ^2 ) = ((( - ( |.q.| * ( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))))) ^2 ) + (( |.q.| * ((((q `1 ) / |.q.|) * (1 + cn)) + cn)) ^2 )) by A30, JGRAPH_3: 1

              .= ((( |.q.| ^2 ) * (( sqrt (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) ^2 )) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 )));

              

              then

               A32: ( |.px.| ^2 ) = ((( |.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) + (( |.q.| ^2 ) * (((((q `1 ) / |.q.|) * (1 + cn)) + cn) ^2 ))) by A27, SQUARE_1:def 2

              .= ( |.q.| ^2 );

              

              then

               A33: |.px.| = ( sqrt ( |.q.| ^2 )) by SQUARE_1: 22

              .= |.q.| by SQUARE_1: 22;

              then

               A34: px <> ( 0. ( TOP-REAL 2)) by A22, TOPRNS_1: 23, TOPRNS_1: 24;

              ((((q `1 ) / |.q.|) * (1 + cn)) + cn) <= ( 0 + cn) by A22, A23, XREAL_1: 7;

              then ((px `1 ) / |.px.|) <= cn by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89;

              then

               A35: ((cn -FanMorphS ) . px) = |[( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 + cn))), ( |.px.| * ( - ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 + cn)) ^2 )))))]| by A1, A2, A31, A28, A34, Th115;

              

               A36: ( |.px.| * ( - ( sqrt (((q `2 ) / |.q.|) ^2 )))) = ( |.px.| * ( - ( sqrt (( - ((q `2 ) / |.q.|)) ^2 ))))

              .= ( |.px.| * ( - ( - ((q `2 ) / |.q.|)))) by A22, SQUARE_1: 22

              .= (q `2 ) by A22, A33, TOPRNS_1: 24, XCMPLX_1: 87;

              

               A37: ( |.px.| * ((((px `1 ) / |.px.|) - cn) / (1 + cn))) = ( |.q.| * ((((((q `1 ) / |.q.|) * (1 + cn)) + cn) - cn) / (1 + cn))) by A22, A30, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.q.| * ((q `1 ) / |.q.|)) by A25, XCMPLX_1: 89

              .= (q `1 ) by A22, TOPRNS_1: 24, XCMPLX_1: 87;

              

              then ( |.px.| * ( - ( sqrt (1 - (((((px `1 ) / |.px.|) - cn) / (1 + cn)) ^2 ))))) = ( |.px.| * ( - ( sqrt (1 - (((q `1 ) / |.px.|) ^2 ))))) by A22, A33, TOPRNS_1: 24, XCMPLX_1: 89

              .= ( |.px.| * ( - ( sqrt (1 - (((q `1 ) ^2 ) / ( |.px.| ^2 )))))) by XCMPLX_1: 76

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) / ( |.px.| ^2 )) - (((q `1 ) ^2 ) / ( |.px.| ^2 )))))) by A26, A32, XCMPLX_1: 60

              .= ( |.px.| * ( - ( sqrt ((( |.px.| ^2 ) - ((q `1 ) ^2 )) / ( |.px.| ^2 ))))) by XCMPLX_1: 120

              .= ( |.px.| * ( - ( sqrt (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) - ((q `1 ) ^2 )) / ( |.px.| ^2 ))))) by A32, JGRAPH_3: 1

              .= ( |.px.| * ( - ( sqrt (((q `2 ) / |.q.|) ^2 )))) by A33, XCMPLX_1: 76;

              hence ex x be set st x in ( dom (cn -FanMorphS )) & y = ((cn -FanMorphS ) . x) by A35, A37, A36, A29, EUCLID: 53;

            end;

          end;

          hence thesis by A3, FUNCT_1:def 3;

        end;

        hence thesis by A3, XBOOLE_0:def 10;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_4:135

    

     Th135: for cn be Real, p2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = ((cn -FanMorphS ) .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & ((cn -FanMorphS ) . p2) in V2

    proof

      reconsider O = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

      let cn be Real, p2 be Point of ( TOP-REAL 2);

      

       A1: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

       the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

      then

      reconsider V0 = ( Ball (O,( |.p2.| + 1))) as Subset of ( TOP-REAL 2);

      O in V0 & V0 c= ( cl_Ball (O,( |.p2.| + 1))) by GOBOARD6: 1, METRIC_1: 14;

      then

      reconsider K0 = ( cl_Ball (O,( |.p2.| + 1))) as non empty compact Subset of ( TOP-REAL 2) by A1, Th15;

      set q3 = ((cn -FanMorphS ) . p2);

      reconsider VV0 = V0 as Subset of ( TopSpaceMetr ( Euclid 2));

      reconsider u2 = p2 as Point of ( Euclid 2) by EUCLID: 67;

      reconsider u3 = q3 as Point of ( Euclid 2) by EUCLID: 67;

      

       A2: ((cn -FanMorphS ) .: K0) c= K0

      proof

        let y be object;

        assume y in ((cn -FanMorphS ) .: K0);

        then

        consider x be object such that

         A3: x in ( dom (cn -FanMorphS )) and

         A4: x in K0 and

         A5: y = ((cn -FanMorphS ) . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A3;

        reconsider uq = q as Point of ( Euclid 2) by EUCLID: 67;

        ( dist (O,uq)) <= ( |.p2.| + 1) by A4, METRIC_1: 12;

        then |.(( 0. ( TOP-REAL 2)) - q).| <= ( |.p2.| + 1) by JGRAPH_1: 28;

        then |.( - q).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then

         A6: |.q.| <= ( |.p2.| + 1) by TOPRNS_1: 26;

        

         A7: y in ( rng (cn -FanMorphS )) by A3, A5, FUNCT_1:def 3;

        then

        reconsider u = y as Point of ( Euclid 2) by EUCLID: 67;

        reconsider q4 = y as Point of ( TOP-REAL 2) by A7;

         |.q4.| = |.q.| by A5, Th128;

        then |.( - q4).| <= ( |.p2.| + 1) by A6, TOPRNS_1: 26;

        then |.(( 0. ( TOP-REAL 2)) - q4).| <= ( |.p2.| + 1) by RLVECT_1: 4;

        then ( dist (O,u)) <= ( |.p2.| + 1) by JGRAPH_1: 28;

        hence thesis by METRIC_1: 12;

      end;

      VV0 is open by TOPMETR: 14;

      then

       A8: V0 is open by Lm11, PRE_TOPC: 30;

      

       A9: |.p2.| < ( |.p2.| + 1) by XREAL_1: 29;

      then |.( - p2).| < ( |.p2.| + 1) by TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - p2).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u2)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A10: p2 in V0 by METRIC_1: 11;

       |.q3.| = |.p2.| by Th128;

      then |.( - q3).| < ( |.p2.| + 1) by A9, TOPRNS_1: 26;

      then |.(( 0. ( TOP-REAL 2)) - q3).| < ( |.p2.| + 1) by RLVECT_1: 4;

      then ( dist (O,u3)) < ( |.p2.| + 1) by JGRAPH_1: 28;

      then

       A11: ((cn -FanMorphS ) . p2) in V0 by METRIC_1: 11;

      assume

       A12: ( - 1) < cn & cn < 1;

      K0 c= ((cn -FanMorphS ) .: K0)

      proof

        let y be object;

        assume

         A13: y in K0;

        then

        reconsider q4 = y as Point of ( TOP-REAL 2);

        reconsider y as Point of ( Euclid 2) by A13;

        the carrier of ( TOP-REAL 2) c= ( rng (cn -FanMorphS )) by A12, Th134;

        then q4 in ( rng (cn -FanMorphS ));

        then

        consider x be object such that

         A14: x in ( dom (cn -FanMorphS )) and

         A15: y = ((cn -FanMorphS ) . x) by FUNCT_1:def 3;

        reconsider x as Point of ( Euclid 2) by A14, Lm11;

        reconsider q = x as Point of ( TOP-REAL 2) by A14;

         |.q4.| = |.q.| by A15, Th128;

        then q in K0 by A13, Lm12;

        hence thesis by A14, A15, FUNCT_1:def 6;

      end;

      then K0 = ((cn -FanMorphS ) .: K0) by A2, XBOOLE_0:def 10;

      hence thesis by A10, A8, A11, METRIC_1: 14;

    end;

    theorem :: JGRAPH_4:136

    for cn be Real st ( - 1) < cn & cn < 1 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = (cn -FanMorphS ) & f is being_homeomorphism

    proof

      let cn be Real;

      set f = (cn -FanMorphS );

      assume

       A1: ( - 1) < cn & cn < 1;

      then

       A2: for p2 be Point of ( TOP-REAL 2) holds ex K be non empty compact Subset of ( TOP-REAL 2) st K = (f .: K) & ex V2 be Subset of ( TOP-REAL 2) st p2 in V2 & V2 is open & V2 c= K & (f . p2) in V2 by Th135;

      ( rng (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) & (cn -FanMorphS ) is continuous by A1, Th132, Th134;

      then f is being_homeomorphism by A1, A2, Th3, Th133;

      hence thesis;

    end;

    theorem :: JGRAPH_4:137

    

     Th137: for cn be Real, q be Point of ( TOP-REAL 2) st cn < 1 & (q `2 ) < 0 & ((q `1 ) / |.q.|) >= cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphS ) . q) holds (p `2 ) < 0 & (p `1 ) >= 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: cn < 1 and

       A2: (q `2 ) < 0 and

       A3: ((q `1 ) / |.q.|) >= cn;

      

       A4: (1 - cn) > 0 by A1, XREAL_1: 149;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((cn -FanMorphS ) . q);

      then

       A5: p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by A2, A3, Th113;

      then

       A6: (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

      

       A7: (((q `1 ) / |.q.|) - cn) >= 0 by A3, XREAL_1: 48;

      

       A8: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A9: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A9, XREAL_1: 74;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A9, XCMPLX_1: 60;

      then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then 1 > ((q `1 ) / |.q.|) by SQUARE_1: 52;

      then (1 - cn) > (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

      then ( - (1 - cn)) < ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

      then (( - (1 - cn)) / (1 - cn)) < (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A4, XREAL_1: 74;

      then ( - 1) < (( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) by A4, XCMPLX_1: 197;

      then ((( - (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) < (1 ^2 ) by A4, A7, SQUARE_1: 50;

      hence thesis by A5, A8, A4, A6, A7, Lm13, EUCLID: 52, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:138

    

     Th138: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & (q `2 ) < 0 & ((q `1 ) / |.q.|) < cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphS ) . q) holds (p `2 ) < 0 & (p `1 ) < 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: (q `2 ) < 0 and

       A3: ((q `1 ) / |.q.|) < cn;

      

       A4: (1 + cn) > 0 by A1, XREAL_1: 148;

      

       A5: (((q `1 ) / |.q.|) - cn) < 0 by A3, XREAL_1: 49;

      then ( - (((q `1 ) / |.q.|) - cn)) > 0 by XREAL_1: 58;

      then (( - (1 + cn)) / (1 + cn)) < (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A4, XREAL_1: 74;

      then

       A6: ( - 1) < (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) by A4, XCMPLX_1: 197;

      

       A7: |.q.| <> 0 by A2, JGRAPH_2: 3, TOPRNS_1: 24;

      then

       A8: ( |.q.| ^2 ) > 0 by SQUARE_1: 12;

      ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) & ( 0 + ((q `1 ) ^2 )) < (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A2, JGRAPH_3: 1, SQUARE_1: 12, XREAL_1: 8;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < (( |.q.| ^2 ) / ( |.q.| ^2 )) by A8, XREAL_1: 74;

      then (((q `1 ) ^2 ) / ( |.q.| ^2 )) < 1 by A8, XCMPLX_1: 60;

      then (((q `1 ) / |.q.|) ^2 ) < 1 by XCMPLX_1: 76;

      then ( - 1) < ((q `1 ) / |.q.|) by SQUARE_1: 52;

      then (( - 1) - cn) < (((q `1 ) / |.q.|) - cn) by XREAL_1: 9;

      then ( - ( - (1 + cn))) > ( - (((q `1 ) / |.q.|) - cn)) by XREAL_1: 24;

      then (( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) < 1 by A4, XREAL_1: 191;

      then ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) < (1 ^2 ) by A6, SQUARE_1: 50;

      then (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) > 0 by XREAL_1: 50;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ))) > 0 by SQUARE_1: 25;

      then ( sqrt (1 - ((( - (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 )))) > 0 by XCMPLX_1: 76;

      then ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 )))) > 0 ;

      then ( - ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) > 0 by XCMPLX_1: 76;

      then

       A9: ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))) < 0 ;

      let p be Point of ( TOP-REAL 2);

      set qz = p;

      assume p = ((cn -FanMorphS ) . q);

      then p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )))))]| by A2, A3, Th114;

      then

       A10: (qz `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))) & (qz `1 ) = ( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))) by EUCLID: 52;

      ((((q `1 ) / |.q.|) - cn) / (1 + cn)) < 0 by A1, A5, XREAL_1: 141, XREAL_1: 148;

      hence thesis by A7, A10, A9, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:139

    

     Th139: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st cn < 1 & (q1 `2 ) < 0 & ((q1 `1 ) / |.q1.|) >= cn & (q2 `2 ) < 0 & ((q2 `1 ) / |.q2.|) >= cn & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphS ) . q1) & p2 = ((cn -FanMorphS ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: cn < 1 and

       A2: (q1 `2 ) < 0 and

       A3: ((q1 `1 ) / |.q1.|) >= cn and

       A4: (q2 `2 ) < 0 and

       A5: ((q2 `1 ) / |.q2.|) >= cn and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      

       A7: (((q1 `1 ) / |.q1.|) - cn) < (((q2 `1 ) / |.q2.|) - cn) & (1 - cn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 149;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((cn -FanMorphS ) . q1) and

       A9: p2 = ((cn -FanMorphS ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th128;

      p2 = |[( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn))), ( |.q2.| * ( - ( sqrt (1 - (((((q2 `1 ) / |.q2.|) - cn) / (1 - cn)) ^2 )))))]| by A4, A5, A9, Th113;

      then

       A11: (p2 `1 ) = ( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `1 ) / |.p2.|) = ((((q2 `1 ) / |.q2.|) - cn) / (1 - cn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn))), ( |.q1.| * ( - ( sqrt (1 - (((((q1 `1 ) / |.q1.|) - cn) / (1 - cn)) ^2 )))))]| by A2, A3, A8, Th113;

      then

       A13: (p1 `1 ) = ( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th128;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `1 ) / |.p1.|) = ((((q1 `1 ) / |.q1.|) - cn) / (1 - cn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:140

    

     Th140: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & (q1 `2 ) < 0 & ((q1 `1 ) / |.q1.|) < cn & (q2 `2 ) < 0 & ((q2 `1 ) / |.q2.|) < cn & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphS ) . q1) & p2 = ((cn -FanMorphS ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: (q1 `2 ) < 0 and

       A3: ((q1 `1 ) / |.q1.|) < cn and

       A4: (q2 `2 ) < 0 and

       A5: ((q2 `1 ) / |.q2.|) < cn and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      

       A7: (((q1 `1 ) / |.q1.|) - cn) < (((q2 `1 ) / |.q2.|) - cn) & (1 + cn) > 0 by A1, A6, XREAL_1: 9, XREAL_1: 148;

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A8: p1 = ((cn -FanMorphS ) . q1) and

       A9: p2 = ((cn -FanMorphS ) . q2);

      

       A10: |.p2.| = |.q2.| by A9, Th128;

      p2 = |[( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn))), ( |.q2.| * ( - ( sqrt (1 - (((((q2 `1 ) / |.q2.|) - cn) / (1 + cn)) ^2 )))))]| by A4, A5, A9, Th114;

      then

       A11: (p2 `1 ) = ( |.q2.| * ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn))) by EUCLID: 52;

       |.q2.| > 0 by A4, Lm1, JGRAPH_2: 3;

      then

       A12: ((p2 `1 ) / |.p2.|) = ((((q2 `1 ) / |.q2.|) - cn) / (1 + cn)) by A11, A10, XCMPLX_1: 89;

      p1 = |[( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn))), ( |.q1.| * ( - ( sqrt (1 - (((((q1 `1 ) / |.q1.|) - cn) / (1 + cn)) ^2 )))))]| by A2, A3, A8, Th114;

      then

       A13: (p1 `1 ) = ( |.q1.| * ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn))) by EUCLID: 52;

      

       A14: |.p1.| = |.q1.| by A8, Th128;

       |.q1.| > 0 by A2, Lm1, JGRAPH_2: 3;

      then ((p1 `1 ) / |.p1.|) = ((((q1 `1 ) / |.q1.|) - cn) / (1 + cn)) by A13, A14, XCMPLX_1: 89;

      hence thesis by A12, A7, XREAL_1: 74;

    end;

    theorem :: JGRAPH_4:141

    for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q1 `2 ) < 0 & (q2 `2 ) < 0 & ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|) holds for p1,p2 be Point of ( TOP-REAL 2) st p1 = ((cn -FanMorphS ) . q1) & p2 = ((cn -FanMorphS ) . q2) holds ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|)

    proof

      let cn be Real, q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q1 `2 ) < 0 and

       A4: (q2 `2 ) < 0 and

       A5: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      let p1,p2 be Point of ( TOP-REAL 2);

      assume that

       A6: p1 = ((cn -FanMorphS ) . q1) and

       A7: p2 = ((cn -FanMorphS ) . q2);

      per cases ;

        suppose ((q1 `1 ) / |.q1.|) >= cn & ((q2 `1 ) / |.q2.|) >= cn;

        hence thesis by A2, A3, A4, A5, A6, A7, Th139;

      end;

        suppose ((q1 `1 ) / |.q1.|) >= cn & ((q2 `1 ) / |.q2.|) < cn;

        hence thesis by A5, XXREAL_0: 2;

      end;

        suppose

         A8: ((q1 `1 ) / |.q1.|) < cn & ((q2 `1 ) / |.q2.|) >= cn;

        then (p2 `1 ) >= 0 by A2, A4, A7, Th137;

        then

         A9: ((p2 `1 ) / |.p2.|) >= 0 ;

        (p1 `1 ) < 0 by A1, A3, A6, A8, Th138;

        hence thesis by A9, Lm1, JGRAPH_2: 3, XREAL_1: 141;

      end;

        suppose ((q1 `1 ) / |.q1.|) < cn & ((q2 `1 ) / |.q2.|) < cn;

        hence thesis by A1, A3, A4, A5, A6, A7, Th140;

      end;

    end;

    theorem :: JGRAPH_4:142

    for cn be Real, q be Point of ( TOP-REAL 2) st (q `2 ) < 0 & ((q `1 ) / |.q.|) = cn holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphS ) . q) holds (p `2 ) < 0 & (p `1 ) = 0

    proof

      let cn be Real, q be Point of ( TOP-REAL 2);

      assume that

       A1: (q `2 ) < 0 and

       A2: ((q `1 ) / |.q.|) = cn;

      let p be Point of ( TOP-REAL 2);

      

       A3: |.q.| <> 0 by A1, JGRAPH_2: 3, TOPRNS_1: 24;

      assume p = ((cn -FanMorphS ) . q);

      then

       A4: p = |[( |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))), ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )))))]| by A1, A2, Th113;

      then (p `2 ) = ( |.q.| * ( - ( sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))) by EUCLID: 52;

      hence thesis by A2, A4, A3, Lm13, EUCLID: 52, XREAL_1: 132;

    end;

    theorem :: JGRAPH_4:143

    ( 0. ( TOP-REAL 2)) = ((a -FanMorphS ) . ( 0. ( TOP-REAL 2))) by Th113, JGRAPH_2: 3;