jgraph_5.miz



    begin

    theorem :: JGRAPH_5:1

    

     Th1: for p be Point of ( TOP-REAL 2) st |.p.| <= 1 holds ( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1

    proof

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

      set a = |.p.|;

      

       A1: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

      then ((a ^2 ) - ((p `1 ) ^2 )) >= 0 by XREAL_1: 63;

      then (((a ^2 ) - ((p `1 ) ^2 )) + ((p `1 ) ^2 )) >= ( 0 + ((p `1 ) ^2 )) by XREAL_1: 7;

      then

       A2: ( - a) <= (p `1 ) & (p `1 ) <= a by SQUARE_1: 47;

      ((a ^2 ) - ((p `2 ) ^2 )) >= 0 by A1, XREAL_1: 63;

      then (((a ^2 ) - ((p `2 ) ^2 )) + ((p `2 ) ^2 )) >= ( 0 + ((p `2 ) ^2 )) by XREAL_1: 7;

      then

       A3: ( - a) <= (p `2 ) & (p `2 ) <= a by SQUARE_1: 47;

      assume

       A4: |.p.| <= 1;

      then ( - a) >= ( - 1) by XREAL_1: 24;

      hence thesis by A4, A2, A3, XXREAL_0: 2;

    end;

    theorem :: JGRAPH_5:2

    

     Th2: for p be Point of ( TOP-REAL 2) st |.p.| <= 1 & (p `1 ) <> 0 & (p `2 ) <> 0 holds ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1

    proof

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

      assume that

       A1: |.p.| <= 1 and

       A2: (p `1 ) <> 0 and

       A3: (p `2 ) <> 0 ;

      set a = |.p.|;

      

       A4: ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_3: 1;

      then (((a ^2 ) - ((p `1 ) ^2 )) + ((p `1 ) ^2 )) > ( 0 + ((p `1 ) ^2 )) by A3, SQUARE_1: 12, XREAL_1: 8;

      then

       A5: ( - a) < (p `1 ) & (p `1 ) < a by SQUARE_1: 48;

      (((a ^2 ) - ((p `2 ) ^2 )) + ((p `2 ) ^2 )) > ( 0 + ((p `2 ) ^2 )) by A2, A4, SQUARE_1: 12, XREAL_1: 8;

      then

       A6: ( - a) < (p `2 ) & (p `2 ) < a by SQUARE_1: 48;

      ( - a) >= ( - 1) by A1, XREAL_1: 24;

      hence thesis by A1, A5, A6, XXREAL_0: 2;

    end;

    theorem :: JGRAPH_5:3

    for a,b,d,e,r3 be Real, PM,PM2 be non empty MetrStruct, x be Element of PM, x2 be Element of PM2 st d <= a & a <= b & b <= e & PM = ( Closed-Interval-MSpace (a,b)) & PM2 = ( Closed-Interval-MSpace (d,e)) & x = x2 holds ( Ball (x,r3)) c= ( Ball (x2,r3))

    proof

      let a,b,d,e,r3 be Real, PM,PM2 be non empty MetrStruct, x be Element of PM, x2 be Element of PM2;

      assume that

       A1: d <= a and

       A2: a <= b and

       A3: b <= e and

       A4: PM = ( Closed-Interval-MSpace (a,b)) and

       A5: PM2 = ( Closed-Interval-MSpace (d,e)) and

       A6: x = x2;

      a <= e by A2, A3, XXREAL_0: 2;

      then

       A7: a in [.d, e.] by A1, XXREAL_1: 1;

      let z be object;

      assume z in ( Ball (x,r3));

      then z in { y where y be Element of PM : ( dist (x,y)) < r3 } by METRIC_1: 17;

      then

      consider y be Element of PM such that

       A8: y = z & ( dist (x,y)) < r3;

      the carrier of PM = [.a, b.] by A2, A4, TOPMETR: 10;

      then

       A9: y in [.a, b.];

      

       A10: d <= b by A1, A2, XXREAL_0: 2;

      then b in [.d, e.] by A3, XXREAL_1: 1;

      then [.a, b.] c= [.d, e.] by A7, XXREAL_2:def 12;

      then

      reconsider y3 = y as Element of PM2 by A3, A5, A10, A9, TOPMETR: 10, XXREAL_0: 2;

      

       A11: ( dist (x,y)) = (the distance of PM . (x,y)) by METRIC_1:def 1;

      

       A12: (the distance of PM . (x,y)) = ( real_dist . (x,y)) by A4, METRIC_1:def 13, TOPMETR:def 1;

      ( real_dist . (x,y)) = (the distance of PM2 . (x2,y3)) by A5, A6, METRIC_1:def 13, TOPMETR:def 1

      .= ( dist (x2,y3)) by METRIC_1:def 1;

      then z in { y2 where y2 be Element of PM2 : ( dist (x2,y2)) < r3 } by A8, A12, A11;

      hence thesis by METRIC_1: 17;

    end;

    theorem :: JGRAPH_5:4

    for a,b be Real, B be Subset of I[01] st 0 <= a & a <= b & b <= 1 & B = [.a, b.] holds ( Closed-Interval-TSpace (a,b)) = ( I[01] | B) by TOPMETR: 20, TOPMETR: 23;

    theorem :: JGRAPH_5:5

    

     Th5: for X be TopStruct, Y,Z be non empty TopStruct, f be Function of X, Y, h be Function of Y, Z st h is being_homeomorphism & f is continuous holds (h * f) is continuous

    proof

      let X be TopStruct, Y,Z be non empty TopStruct, f be Function of X, Y, h be Function of Y, Z;

      assume that

       A1: h is being_homeomorphism and

       A2: f is continuous;

      h is continuous by A1, TOPS_2:def 5;

      hence thesis by A2, TOPS_2: 46;

    end;

    theorem :: JGRAPH_5:6

    

     Th6: for X,Y,Z be TopStruct, f be Function of X, Y, h be Function of Y, Z st h is being_homeomorphism & f is one-to-one holds (h * f) is one-to-one

    proof

      let X,Y,Z be TopStruct, f be Function of X, Y, h be Function of Y, Z;

      assume that

       A1: h is being_homeomorphism and

       A2: f is one-to-one;

      h is one-to-one by A1, TOPS_2:def 5;

      hence thesis by A2;

    end;

    theorem :: JGRAPH_5:7

    

     Th7: for X be TopStruct, S,V be non empty TopStruct, B be non empty Subset of S, f be Function of X, (S | B), g be Function of S, V, h be Function of X, V st h = (g * f) & f is continuous & g is continuous holds h is continuous

    proof

      let X be TopStruct, S,V be non empty TopStruct, B be non empty Subset of S, f be Function of X, (S | B), g be Function of S, V, h be Function of X, V;

      assume that

       A1: h = (g * f) and

       A2: f is continuous and

       A3: g is continuous;

      now

        let P be Subset of V;

        

         A4: ((g * f) " P) = (f " (g " P)) by RELAT_1: 146;

        now

          assume P is closed;

          then

           A5: (g " P) is closed by A3, PRE_TOPC:def 6;

          

           A6: the carrier of (S | B) = B by PRE_TOPC: 8;

          then

          reconsider F = (B /\ (g " P)) as Subset of (S | B) by XBOOLE_1: 17;

          

           A7: (( rng f) /\ the carrier of (S | B)) = ( rng f) by XBOOLE_1: 28;

          ( [#] (S | B)) = B by PRE_TOPC:def 5;

          then

           A8: F is closed by A5, PRE_TOPC: 13;

          (h " P) = (f " (( rng f) /\ (g " P))) by A1, A4, RELAT_1: 133

          .= (f " (( rng f) /\ (the carrier of (S | B) /\ (g " P)))) by A7, XBOOLE_1: 16

          .= (f " F) by A6, RELAT_1: 133;

          hence (h " P) is closed by A2, A8, PRE_TOPC:def 6;

        end;

        hence P is closed implies (h " P) is closed;

      end;

      hence thesis by PRE_TOPC:def 6;

    end;

    theorem :: JGRAPH_5:8

    

     Th8: for a,b,d,e,s1,s2,t1,t2 be Real, h be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & (h . s1) = t1 & (h . s2) = t2 & (h . b) = e & d <= e & t1 <= t2 & s1 in [.a, b.] & s2 in [.a, b.] holds s1 <= s2

    proof

      let a,b,d,e,s1,s2,t1,t2 be Real, h be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (d,e));

      assume that

       A1: h is being_homeomorphism and

       A2: (h . s1) = t1 and

       A3: (h . s2) = t2 and

       A4: (h . b) = e and

       A5: d <= e and

       A6: t1 <= t2 and

       A7: s1 in [.a, b.] and

       A8: s2 in [.a, b.];

      

       A9: s1 <= b by A7, XXREAL_1: 1;

      reconsider C = [.d, e.] as non empty Subset of R^1 by A5, TOPMETR: 17, XXREAL_1: 1;

      

       A10: ( R^1 | C) = ( Closed-Interval-TSpace (d,e)) by A5, TOPMETR: 19;

      

       A11: a <= s1 by A7, XXREAL_1: 1;

      then

       A12: the carrier of ( Closed-Interval-TSpace (a,b)) = [.a, b.] by A9, TOPMETR: 18, XXREAL_0: 2;

      then

      reconsider B1 = [.s1, b.] as Subset of ( Closed-Interval-TSpace (a,b)) by A11, XXREAL_1: 34;

      

       A13: ( dom h) = ( [#] ( Closed-Interval-TSpace (a,b))) by A1, TOPS_2:def 5

      .= [.a, b.] by A11, A9, TOPMETR: 18, XXREAL_0: 2;

      

       A14: a <= s2 by A8, XXREAL_1: 1;

      then

      reconsider B = [.s2, s1.] as Subset of ( Closed-Interval-TSpace (a,b)) by A9, A12, XXREAL_1: 34;

      reconsider Bb = [.s2, s1.] as Subset of ( Closed-Interval-TSpace (a,b)) by A14, A9, A12, XXREAL_1: 34;

      reconsider f3 = (h | Bb) as Function of (( Closed-Interval-TSpace (a,b)) | B), ( Closed-Interval-TSpace (d,e)) by PRE_TOPC: 9;

      assume

       A15: s1 > s2;

      then

       A16: ( Closed-Interval-TSpace (s2,s1)) = (( Closed-Interval-TSpace (a,b)) | B) by A14, A9, TOPMETR: 23;

      then f3 is Function of ( Closed-Interval-TSpace (s2,s1)), R^1 by A10, JORDAN6: 3;

      then

      reconsider f = (h | B) as Function of ( Closed-Interval-TSpace (s2,s1)), R^1 ;

      s2 in B by A15, XXREAL_1: 1;

      then

       A17: (f . s2) = t2 by A3, FUNCT_1: 49;

      set t = ((t1 + t2) / 2);

      

       A18: the carrier of ( Closed-Interval-TSpace (d,e)) = [.d, e.] by A5, TOPMETR: 18;

      h is one-to-one by A1, TOPS_2:def 5;

      then t1 <> t2 by A2, A3, A7, A8, A13, A15, FUNCT_1:def 4;

      then

       A19: t1 < t2 by A6, XXREAL_0: 1;

      then (t1 + t1) < (t1 + t2) by XREAL_1: 8;

      then

       A20: ((2 * t1) / 2) < t by XREAL_1: 74;

      ( dom f) = the carrier of ( Closed-Interval-TSpace (s2,s1)) by FUNCT_2:def 1;

      then ( dom f) = [.s2, s1.] by A15, TOPMETR: 18;

      then s2 in ( dom f) by A15, XXREAL_1: 1;

      then t2 in ( rng f3) by A17, FUNCT_1:def 3;

      then

       A21: t2 <= e by A18, XXREAL_1: 1;

      (t1 + t2) < (t2 + t2) by A19, XREAL_1: 8;

      then

       A22: ((2 * t2) / 2) > t by XREAL_1: 74;

      then

       A23: e > t by A21, XXREAL_0: 2;

      reconsider B1b = [.s1, b.] as Subset of ( Closed-Interval-TSpace (a,b)) by A11, A12, XXREAL_1: 34;

      reconsider f4 = (h | B1b) as Function of (( Closed-Interval-TSpace (a,b)) | B1), ( Closed-Interval-TSpace (d,e)) by PRE_TOPC: 9;

      

       A24: ( Closed-Interval-TSpace (s1,b)) = (( Closed-Interval-TSpace (a,b)) | B1) by A11, A9, TOPMETR: 23;

      then f4 is Function of ( Closed-Interval-TSpace (s1,b)), R^1 by A10, JORDAN6: 3;

      then

      reconsider f1 = (h | B1) as Function of ( Closed-Interval-TSpace (s1,b)), R^1 ;

      

       A25: h is continuous by A1, TOPS_2:def 5;

      then f4 is continuous by TOPMETR: 7;

      then

       A26: f1 is continuous by A10, A24, JORDAN6: 3;

      b in B1 by A9, XXREAL_1: 1;

      then

       A27: (f1 . b) = e by A4, FUNCT_1: 49;

      s1 in B1 by A9, XXREAL_1: 1;

      then

       A28: (f1 . s1) = t1 by A2, FUNCT_1: 49;

      s1 < b by A2, A4, A9, A19, A21, XXREAL_0: 1;

      then

      consider r1 be Real such that

       A29: (f1 . r1) = t and

       A30: s1 < r1 and

       A31: r1 < b by A20, A26, A28, A27, A23, TOPREAL5: 6;

      

       A32: r1 in B1 by A30, A31, XXREAL_1: 1;

      s1 in B by A15, XXREAL_1: 1;

      then

       A33: (f . s1) = t1 by A2, FUNCT_1: 49;

      f3 is continuous by A25, TOPMETR: 7;

      then f is continuous by A10, A16, JORDAN6: 3;

      then

      consider r be Real such that

       A34: (f . r) = t and

       A35: s2 < r and

       A36: r < s1 by A15, A17, A33, A20, A22, TOPREAL5: 7;

      

       A37: a < r by A14, A35, XXREAL_0: 2;

      a < r1 by A11, A30, XXREAL_0: 2;

      then

       A38: r1 in [.a, b.] by A31, XXREAL_1: 1;

      

       A39: h is one-to-one by A1, TOPS_2:def 5;

      r < b by A9, A36, XXREAL_0: 2;

      then

       A40: r in [.a, b.] by A37, XXREAL_1: 1;

      r in [.s2, s1.] by A35, A36, XXREAL_1: 1;

      

      then (h . r) = t by A34, FUNCT_1: 49

      .= (h . r1) by A29, A32, FUNCT_1: 49;

      hence contradiction by A13, A39, A36, A40, A30, A38, FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_5:9

    

     Th9: for a,b,d,e,s1,s2,t1,t2 be Real, h be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & (h . s1) = t1 & (h . s2) = t2 & (h . b) = d & e >= d & t1 >= t2 & s1 in [.a, b.] & s2 in [.a, b.] holds s1 <= s2

    proof

      let a,b,d,e,s1,s2,t1,t2 be Real, h be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (d,e));

      assume that

       A1: h is being_homeomorphism and

       A2: (h . s1) = t1 and

       A3: (h . s2) = t2 and

       A4: (h . b) = d and

       A5: e >= d and

       A6: t1 >= t2 and

       A7: s1 in [.a, b.] and

       A8: s2 in [.a, b.];

      

       A9: s1 <= b by A7, XXREAL_1: 1;

      reconsider C = [.d, e.] as non empty Subset of R^1 by A5, TOPMETR: 17, XXREAL_1: 1;

      

       A10: ( R^1 | C) = ( Closed-Interval-TSpace (d,e)) by A5, TOPMETR: 19;

      

       A11: a <= s1 by A7, XXREAL_1: 1;

      then

       A12: the carrier of ( Closed-Interval-TSpace (a,b)) = [.a, b.] by A9, TOPMETR: 18, XXREAL_0: 2;

      then

      reconsider B1 = [.s1, b.] as Subset of ( Closed-Interval-TSpace (a,b)) by A11, XXREAL_1: 34;

      

       A13: ( dom h) = ( [#] ( Closed-Interval-TSpace (a,b))) by A1, TOPS_2:def 5

      .= [.a, b.] by A11, A9, TOPMETR: 18, XXREAL_0: 2;

      

       A14: a <= s2 by A8, XXREAL_1: 1;

      then

      reconsider B = [.s2, s1.] as Subset of ( Closed-Interval-TSpace (a,b)) by A9, A12, XXREAL_1: 34;

      reconsider Bb = [.s2, s1.] as Subset of ( Closed-Interval-TSpace (a,b)) by A14, A9, A12, XXREAL_1: 34;

      reconsider f3 = (h | Bb) as Function of (( Closed-Interval-TSpace (a,b)) | B), ( Closed-Interval-TSpace (d,e)) by PRE_TOPC: 9;

      assume

       A15: s1 > s2;

      then

       A16: ( Closed-Interval-TSpace (s2,s1)) = (( Closed-Interval-TSpace (a,b)) | B) by A14, A9, TOPMETR: 23;

      then f3 is Function of ( Closed-Interval-TSpace (s2,s1)), R^1 by A10, JORDAN6: 3;

      then

      reconsider f = (h | B) as Function of ( Closed-Interval-TSpace (s2,s1)), R^1 ;

      s2 in B by A15, XXREAL_1: 1;

      then

       A17: (f . s2) = t2 by A3, FUNCT_1: 49;

      set t = ((t1 + t2) / 2);

      

       A18: the carrier of ( Closed-Interval-TSpace (d,e)) = [.d, e.] by A5, TOPMETR: 18;

      h is one-to-one by A1, TOPS_2:def 5;

      then t1 <> t2 by A2, A3, A7, A8, A13, A15, FUNCT_1:def 4;

      then

       A19: t1 > t2 by A6, XXREAL_0: 1;

      then (t1 + t1) > (t1 + t2) by XREAL_1: 8;

      then

       A20: ((2 * t1) / 2) > t by XREAL_1: 74;

      ( dom f) = the carrier of ( Closed-Interval-TSpace (s2,s1)) by FUNCT_2:def 1;

      then ( dom f) = [.s2, s1.] by A15, TOPMETR: 18;

      then s2 in ( dom f) by A15, XXREAL_1: 1;

      then t2 in ( rng f3) by A17, FUNCT_1:def 3;

      then

       A21: d <= t2 by A18, XXREAL_1: 1;

      (t1 + t2) > (t2 + t2) by A19, XREAL_1: 8;

      then

       A22: ((2 * t2) / 2) < t by XREAL_1: 74;

      then

       A23: d < t by A21, XXREAL_0: 2;

      reconsider B1b = [.s1, b.] as Subset of ( Closed-Interval-TSpace (a,b)) by A11, A12, XXREAL_1: 34;

      reconsider f4 = (h | B1b) as Function of (( Closed-Interval-TSpace (a,b)) | B1), ( Closed-Interval-TSpace (d,e)) by PRE_TOPC: 9;

      

       A24: ( Closed-Interval-TSpace (s1,b)) = (( Closed-Interval-TSpace (a,b)) | B1) by A11, A9, TOPMETR: 23;

      then f4 is Function of ( Closed-Interval-TSpace (s1,b)), R^1 by A10, JORDAN6: 3;

      then

      reconsider f1 = (h | B1) as Function of ( Closed-Interval-TSpace (s1,b)), R^1 ;

      

       A25: h is continuous by A1, TOPS_2:def 5;

      then f4 is continuous by TOPMETR: 7;

      then

       A26: f1 is continuous by A10, A24, JORDAN6: 3;

      b in B1 by A9, XXREAL_1: 1;

      then

       A27: (f1 . b) = d by A4, FUNCT_1: 49;

      s1 in B1 by A9, XXREAL_1: 1;

      then

       A28: (f1 . s1) = t1 by A2, FUNCT_1: 49;

      s1 < b by A2, A4, A9, A19, A21, XXREAL_0: 1;

      then

      consider r1 be Real such that

       A29: (f1 . r1) = t and

       A30: s1 < r1 and

       A31: r1 < b by A20, A26, A28, A27, A23, TOPREAL5: 7;

      

       A32: r1 in B1 by A30, A31, XXREAL_1: 1;

      s1 in B by A15, XXREAL_1: 1;

      then

       A33: (f . s1) = t1 by A2, FUNCT_1: 49;

      f3 is continuous by A25, TOPMETR: 7;

      then f is continuous by A10, A16, JORDAN6: 3;

      then

      consider r be Real such that

       A34: (f . r) = t and

       A35: s2 < r and

       A36: r < s1 by A15, A17, A33, A20, A22, TOPREAL5: 6;

      

       A37: a < r by A14, A35, XXREAL_0: 2;

      a < r1 by A11, A30, XXREAL_0: 2;

      then

       A38: r1 in [.a, b.] by A31, XXREAL_1: 1;

      

       A39: h is one-to-one by A1, TOPS_2:def 5;

      r < b by A9, A36, XXREAL_0: 2;

      then

       A40: r in [.a, b.] by A37, XXREAL_1: 1;

      r in [.s2, s1.] by A35, A36, XXREAL_1: 1;

      

      then (h . r) = t by A34, FUNCT_1: 49

      .= (h . r1) by A29, A32, FUNCT_1: 49;

      hence contradiction by A13, A39, A36, A40, A30, A38, FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_5:10

    for n be Element of NAT holds ( - ( 0. ( TOP-REAL n))) = ( 0. ( TOP-REAL n))

    proof

      let n be Element of NAT ;

      (( 0. ( TOP-REAL n)) + ( 0. ( TOP-REAL n))) = ( 0. ( TOP-REAL n)) by RLVECT_1: 4;

      hence thesis by RLVECT_1: 6;

    end;

    begin

    theorem :: JGRAPH_5:11

    

     Th11: for f,g be Function of I[01] , ( TOP-REAL 2), a,b,c,d be Real, O,I be Point of I[01] st O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & a <> b & c <> d & ((f . O) `1 ) = a & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & ((f . I) `1 ) = b & c <= ((f . I) `2 ) & ((f . I) `2 ) <= d & ((g . O) `2 ) = c & a <= ((g . O) `1 ) & ((g . O) `1 ) <= b & ((g . I) `2 ) = d & a <= ((g . I) `1 ) & ((g . I) `1 ) <= b & (for r be Point of I[01] holds (a >= ((f . r) `1 ) or ((f . r) `1 ) >= b or c >= ((f . r) `2 ) or ((f . r) `2 ) >= d) & (a >= ((g . r) `1 ) or ((g . r) `1 ) >= b or c >= ((g . r) `2 ) or ((g . r) `2 ) >= d)) holds ( rng f) meets ( rng g)

    proof

      let f,g be Function of I[01] , ( TOP-REAL 2), a,b,c,d be Real, O,I be Point of I[01] ;

      assume that

       A1: O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one and

       A2: a <> b and

       A3: c <> d and

       A4: ((f . O) `1 ) = a and

       A5: c <= ((f . O) `2 ) & ((f . O) `2 ) <= d and

       A6: ((f . I) `1 ) = b & c <= ((f . I) `2 ) & ((f . I) `2 ) <= d & ((g . O) `2 ) = c and

       A7: a <= ((g . O) `1 ) & ((g . O) `1 ) <= b and

       A8: ((g . I) `2 ) = d & a <= ((g . I) `1 ) & (((g . I) `1 ) <= b & for r be Point of I[01] holds (a >= ((f . r) `1 ) or ((f . r) `1 ) >= b or c >= ((f . r) `2 ) or ((f . r) `2 ) >= d) & (a >= ((g . r) `1 ) or ((g . r) `1 ) >= b or c >= ((g . r) `2 ) or ((g . r) `2 ) >= d));

      c <= d by A5, XXREAL_0: 2;

      then

       A9: c < d by A3, XXREAL_0: 1;

      a <= b by A7, XXREAL_0: 2;

      then a < b by A2, XXREAL_0: 1;

      hence thesis by A1, A4, A5, A6, A7, A8, A9, JGRAPH_2: 45;

    end;

    

     Lm1: 0 in [. 0 , 1.] by XXREAL_1: 1;

    

     Lm2: 1 in [. 0 , 1.] by XXREAL_1: 1;

    theorem :: JGRAPH_5:12

    

     Th12: for f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one holds ex f2 be Function of I[01] , ( TOP-REAL 2) st (f2 . 0 ) = (f . 1) & (f2 . 1) = (f . 0 ) & ( rng f2) = ( rng f) & f2 is continuous & f2 is one-to-one

    proof

      let f be Function of I[01] , ( TOP-REAL 2);

      

       A1: I[01] is compact by HEINE: 4, TOPMETR: 20;

      

       A2: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

      reconsider P = ( rng f) as non empty Subset of ( TOP-REAL 2) by Lm1, BORSUK_1: 40, FUNCT_1: 3;

      (f . 1) in ( rng f) & (f . 0 ) in ( rng f) by A2, Lm1, Lm2, BORSUK_1: 40, FUNCT_1: 3;

      then

      reconsider p1 = (f . 0 ), p2 = (f . 1) as Point of ( TOP-REAL 2);

      assume f is continuous one-to-one;

      then ex f1 be Function of I[01] , (( TOP-REAL 2) | P) st f1 = f & f1 is being_homeomorphism by A1, JGRAPH_1: 46;

      then P is_an_arc_of (p1,p2) by TOPREAL1:def 1;

      then P is_an_arc_of (p2,p1) by JORDAN5B: 14;

      then

      consider f3 be Function of I[01] , (( TOP-REAL 2) | P) such that

       A3: f3 is being_homeomorphism and

       A4: (f3 . 0 ) = p2 & (f3 . 1) = p1 by TOPREAL1:def 1;

      

       A5: ex f4 be Function of I[01] , ( TOP-REAL 2) st f3 = f4 & f4 is continuous & f4 is one-to-one by A3, JORDAN7: 15;

      ( rng f3) = ( [#] (( TOP-REAL 2) | P)) by A3, TOPS_2:def 5

      .= P by PRE_TOPC:def 5;

      hence thesis by A4, A5;

    end;

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

    theorem :: JGRAPH_5:13

    

     Th13: for f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] st O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

      assume

       A1: O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0;

      then ex g2 be Function of I[01] , ( TOP-REAL 2) st (g2 . 0 ) = (g . 1) & (g2 . 1) = (g . 0 ) & ( rng g2) = ( rng g) & g2 is continuous one-to-one by Th12;

      hence thesis by A1, JGRAPH_3: 44;

    end;

    theorem :: JGRAPH_5:14

    

     Th14: for f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] st O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYN & (g . I) in KYP & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

      assume

       A1: O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYN & (g . I) in KYP & ( rng f) c= C0 & ( rng g) c= C0;

      

       A2: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      reconsider gg = (( Sq_Circ " ) * g) as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 13, JGRAPH_3: 29;

      reconsider ff = (( Sq_Circ " ) * f) as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 13, JGRAPH_3: 29;

      

       A3: ( dom gg) = the carrier of I[01] by FUNCT_2:def 1;

      

       A4: ( dom ff) = the carrier of I[01] by FUNCT_2:def 1;

      

       A5: ((ff . O) `1 ) = ( - 1) & ((ff . I) `1 ) = 1 & ((gg . O) `2 ) = ( - 1) & ((gg . I) `2 ) = 1

      proof

        reconsider pz = (gg . O) as Point of ( TOP-REAL 2);

        reconsider py = (ff . I) as Point of ( TOP-REAL 2);

        reconsider px = (ff . O) as Point of ( TOP-REAL 2);

        set q = px;

        

         A6: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        reconsider pu = (gg . I) as Point of ( TOP-REAL 2);

        

         A7: ( |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| `1 ) = ((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) by EUCLID: 52;

        consider p2 be Point of ( TOP-REAL 2) such that

         A8: (f . I) = p2 and

         A9: |.p2.| = 1 and

         A10: (p2 `2 ) <= (p2 `1 ) and

         A11: (p2 `2 ) >= ( - (p2 `1 )) by A1;

        

         A12: (ff . I) = (( Sq_Circ " ) . (f . I)) by A4, FUNCT_1: 12;

        then

         A13: p2 = ( Sq_Circ . py) by A8, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A14: p2 <> ( 0. ( TOP-REAL 2)) by A9, TOPRNS_1: 23;

        then

         A15: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A10, A11, JGRAPH_3: 28;

        then

         A16: (py `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A12, A8, EUCLID: 52;

        (((p2 `2 ) / (p2 `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A17: ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

        

         A18: (py `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A12, A8, A15, EUCLID: 52;

         A19:

        now

          assume (py `1 ) = 0 & (py `2 ) = 0 ;

          then (p2 `1 ) = 0 & (p2 `2 ) = 0 by A16, A18, A17, XCMPLX_1: 6;

          hence contradiction by A14, EUCLID: 53, EUCLID: 54;

        end;

        

         A20: (p2 `2 ) <= (p2 `1 ) & (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A10, A11, A17, XREAL_1: 64;

        then ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) & ( - (py `1 )) <= (py `2 ) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A12, A8, A15, A16, A17, EUCLID: 52, XREAL_1: 64;

        then

         A21: ( Sq_Circ . py) = |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| by A16, A18, A19, JGRAPH_2: 3, JGRAPH_3:def 1;

        

         A22: (((py `2 ) / (py `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A23: ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) > 0 by SQUARE_1: 25;

         A24:

        now

          assume

           A25: (py `1 ) = ( - 1);

          ( - (p2 `2 )) <= ( - ( - (p2 `1 ))) by A11, XREAL_1: 24;

          then ( - (p2 `2 )) < 0 by A13, A21, A7, A22, A25, SQUARE_1: 25, XREAL_1: 141;

          then ( - ( - (p2 `2 ))) > ( - 0 );

          hence contradiction by A10, A13, A21, A23, A25, EUCLID: 52;

        end;

        ( |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| `2 ) = ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) by EUCLID: 52;

        

        then ( |.p2.| ^2 ) = ((((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 ) + (((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 )) by A13, A21, A7, JGRAPH_3: 1

        .= ((((py `1 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 )) + (((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((py `1 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 )) + (((py `2 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 ))) by A22, SQUARE_1:def 2

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 )))) by A22, SQUARE_1:def 2

        .= ((((py `1 ) ^2 ) + ((py `2 ) ^2 )) / (1 + (((py `2 ) / (py `1 )) ^2 ))) by XCMPLX_1: 62;

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) / (1 + (((py `2 ) / (py `1 )) ^2 ))) * (1 + (((py `2 ) / (py `1 )) ^2 ))) = (1 * (1 + (((py `2 ) / (py `1 )) ^2 ))) by A9;

        then (((py `1 ) ^2 ) + ((py `2 ) ^2 )) = (1 + (((py `2 ) / (py `1 )) ^2 )) by A22, XCMPLX_1: 87;

        then

         A26: ((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) = (((py `2 ) ^2 ) / ((py `1 ) ^2 )) by XCMPLX_1: 76;

        (py `1 ) <> 0 by A16, A18, A17, A19, A20, XREAL_1: 64;

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) * ((py `1 ) ^2 )) = ((py `2 ) ^2 ) by A26, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A27: ((((py `1 ) ^2 ) - 1) * (((py `1 ) ^2 ) + ((py `2 ) ^2 ))) = 0 ;

        (((py `1 ) ^2 ) + ((py `2 ) ^2 )) <> 0 by A19, COMPLEX1: 1;

        then (((py `1 ) - 1) * ((py `1 ) + 1)) = 0 by A27, XCMPLX_1: 6;

        then

         A28: ((py `1 ) - 1) = 0 or ((py `1 ) + 1) = 0 by XCMPLX_1: 6;

        

         A29: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

        consider p1 be Point of ( TOP-REAL 2) such that

         A30: (f . O) = p1 and

         A31: |.p1.| = 1 and

         A32: (p1 `2 ) >= (p1 `1 ) and

         A33: (p1 `2 ) <= ( - (p1 `1 )) by A1;

        (((p1 `2 ) / (p1 `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A34: ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

        

         A35: (ff . O) = (( Sq_Circ " ) . (f . O)) by A4, FUNCT_1: 12;

        then

         A36: p1 = ( Sq_Circ . px) by A30, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A37: p1 <> ( 0. ( TOP-REAL 2)) by A31, TOPRNS_1: 23;

        then (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A32, A33, JGRAPH_3: 28;

        then

         A38: (px `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) & (px `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A35, A30, EUCLID: 52;

         A39:

        now

          assume (px `1 ) = 0 & (px `2 ) = 0 ;

          then (p1 `1 ) = 0 & (p1 `2 ) = 0 by A38, A34, XCMPLX_1: 6;

          hence contradiction by A37, EUCLID: 53, EUCLID: 54;

        end;

        (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A32, A33, A34, XREAL_1: 64;

        then

         A40: (p1 `2 ) <= (p1 `1 ) & (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A38, A34, XREAL_1: 64;

        then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A38, A34, XREAL_1: 64;

        then

         A41: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A39, JGRAPH_2: 3, JGRAPH_3:def 1;

        

         A42: (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A43: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;

         A44:

        now

          assume

           A45: (px `1 ) = 1;

          ( - (p1 `2 )) >= ( - ( - (p1 `1 ))) by A33, XREAL_1: 24;

          then ( - (p1 `2 )) > 0 by A36, A41, A6, A43, A45, XREAL_1: 139;

          then ( - ( - (p1 `2 ))) < ( - 0 );

          hence contradiction by A32, A36, A41, A43, A45, EUCLID: 52;

        end;

        ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        

        then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A36, A41, A6, JGRAPH_3: 1

        .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A42, SQUARE_1:def 2

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A42, SQUARE_1:def 2

        .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62;

        then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A31;

        then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) = (1 + (((q `2 ) / (q `1 )) ^2 )) by A42, XCMPLX_1: 87;

        then

         A46: ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) = (((px `2 ) ^2 ) / ((px `1 ) ^2 )) by XCMPLX_1: 76;

        (px `1 ) <> 0 by A38, A34, A39, A40, XREAL_1: 64;

        then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) = ((px `2 ) ^2 ) by A46, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A47: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) = 0 ;

        

         A48: ( |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| `2 ) = ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by EUCLID: 52;

        consider p4 be Point of ( TOP-REAL 2) such that

         A49: (g . I) = p4 and

         A50: |.p4.| = 1 and

         A51: (p4 `2 ) >= (p4 `1 ) and

         A52: (p4 `2 ) >= ( - (p4 `1 )) by A1;

        (((p4 `1 ) / (p4 `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A53: ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

        

         A54: ( - (p4 `2 )) <= ( - ( - (p4 `1 ))) by A52, XREAL_1: 24;

        then

         A55: (p4 `1 ) <= (p4 `2 ) & (( - (p4 `2 )) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A51, A53, XREAL_1: 64;

        

         A56: (gg . I) = (( Sq_Circ " ) . (g . I)) by A3, FUNCT_1: 12;

        then

         A57: p4 = ( Sq_Circ . pu) by A49, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A58: p4 <> ( 0. ( TOP-REAL 2)) by A50, TOPRNS_1: 23;

        then

         A59: (( Sq_Circ " ) . p4) = |[((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))), ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))))]| by A51, A54, JGRAPH_3: 30;

        then

         A60: (pu `2 ) = ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A56, A49, EUCLID: 52;

        

         A61: (pu `1 ) = ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A56, A49, A59, EUCLID: 52;

         A62:

        now

          assume (pu `2 ) = 0 & (pu `1 ) = 0 ;

          then (p4 `2 ) = 0 & (p4 `1 ) = 0 by A60, A61, A53, XCMPLX_1: 6;

          hence contradiction by A58, EUCLID: 53, EUCLID: 54;

        end;

        ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) & ( - (pu `2 )) <= (pu `1 ) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A56, A49, A59, A60, A53, A55, EUCLID: 52, XREAL_1: 64;

        then

         A63: ( Sq_Circ . pu) = |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| by A60, A61, A62, JGRAPH_2: 3, JGRAPH_3: 4;

        

         A64: (((pu `1 ) / (pu `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A65: ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) > 0 by SQUARE_1: 25;

         A66:

        now

          assume

           A67: (pu `2 ) = ( - 1);

          then ( - (p4 `1 )) < 0 by A52, A57, A63, A48, A64, SQUARE_1: 25, XREAL_1: 141;

          then ( - ( - (p4 `1 ))) > ( - 0 );

          hence contradiction by A51, A57, A63, A65, A67, EUCLID: 52;

        end;

        ( |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| `1 ) = ((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by EUCLID: 52;

        

        then ( |.p4.| ^2 ) = ((((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 ) + (((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 )) by A57, A63, A48, JGRAPH_3: 1

        .= ((((pu `2 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 )) + (((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((pu `2 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 )) + (((pu `1 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 ))) by A64, SQUARE_1:def 2

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by A64, SQUARE_1:def 2

        .= ((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) by XCMPLX_1: 62;

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) = (1 * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) by A50;

        then (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) = (1 + (((pu `1 ) / (pu `2 )) ^2 )) by A64, XCMPLX_1: 87;

        then

         A68: ((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) = (((pu `1 ) ^2 ) / ((pu `2 ) ^2 )) by XCMPLX_1: 76;

        (pu `2 ) <> 0 by A60, A61, A53, A62, A55, XREAL_1: 64;

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) * ((pu `2 ) ^2 )) = ((pu `1 ) ^2 ) by A68, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A69: ((((pu `2 ) ^2 ) - 1) * (((pu `2 ) ^2 ) + ((pu `1 ) ^2 ))) = 0 ;

        (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) <> 0 by A62, COMPLEX1: 1;

        then (((pu `2 ) - 1) * ((pu `2 ) + 1)) = 0 by A69, XCMPLX_1: 6;

        then

         A70: ((pu `2 ) - 1) = 0 or ((pu `2 ) + 1) = 0 by XCMPLX_1: 6;

        consider p3 be Point of ( TOP-REAL 2) such that

         A71: (g . O) = p3 and

         A72: |.p3.| = 1 and

         A73: (p3 `2 ) <= (p3 `1 ) and

         A74: (p3 `2 ) <= ( - (p3 `1 )) by A1;

        

         A75: p3 <> ( 0. ( TOP-REAL 2)) by A72, TOPRNS_1: 23;

        

         A76: (gg . O) = (( Sq_Circ " ) . (g . O)) by A3, FUNCT_1: 12;

        then

         A77: p3 = ( Sq_Circ . pz) by A71, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A78: ( - (p3 `2 )) >= ( - ( - (p3 `1 ))) by A74, XREAL_1: 24;

        then

         A79: (( Sq_Circ " ) . p3) = |[((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A73, A75, JGRAPH_3: 30;

        then

         A80: (pz `2 ) = ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A76, A71, EUCLID: 52;

        (((p3 `1 ) / (p3 `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A81: ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

        

         A82: (pz `1 ) = ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A76, A71, A79, EUCLID: 52;

         A83:

        now

          assume (pz `2 ) = 0 & (pz `1 ) = 0 ;

          then (p3 `2 ) = 0 & (p3 `1 ) = 0 by A80, A82, A81, XCMPLX_1: 6;

          hence contradiction by A75, EUCLID: 53, EUCLID: 54;

        end;

        (p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A73, A78, A81, XREAL_1: 64;

        then

         A84: (p3 `1 ) <= (p3 `2 ) & (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A80, A82, A81, XREAL_1: 64;

        then ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A76, A71, A79, A80, A81, EUCLID: 52, XREAL_1: 64;

        then

         A85: ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A80, A82, A83, JGRAPH_2: 3, JGRAPH_3: 4;

        

         A86: (((pz `1 ) / (pz `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A87: ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) > 0 by SQUARE_1: 25;

         A88:

        now

          assume

           A89: (pz `2 ) = 1;

          then ( - (p3 `1 )) > 0 by A74, A77, A85, A29, A87, XREAL_1: 139;

          then ( - ( - (p3 `1 ))) < ( - 0 );

          hence contradiction by A73, A77, A85, A87, A89, EUCLID: 52;

        end;

        ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

        

        then ( |.p3.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A77, A85, A29, JGRAPH_3: 1

        .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A86, SQUARE_1:def 2

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A86, SQUARE_1:def 2

        .= ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by XCMPLX_1: 62;

        then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) = (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A72;

        then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) = (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A86, XCMPLX_1: 87;

        then

         A90: ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) = (((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) by XCMPLX_1: 76;

        (pz `2 ) <> 0 by A80, A82, A81, A83, A84, XREAL_1: 64;

        then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) = ((pz `1 ) ^2 ) by A90, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A91: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) = 0 ;

        (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A83, COMPLEX1: 1;

        then (((pz `2 ) - 1) * ((pz `2 ) + 1)) = 0 by A91, XCMPLX_1: 6;

        then

         A92: ((pz `2 ) - 1) = 0 or ((pz `2 ) + 1) = 0 by XCMPLX_1: 6;

        (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A39, COMPLEX1: 1;

        then (((px `1 ) - 1) * ((px `1 ) + 1)) = 0 by A47, XCMPLX_1: 6;

        then ((px `1 ) - 1) = 0 or ((px `1 ) + 1) = 0 by XCMPLX_1: 6;

        hence thesis by A44, A28, A24, A92, A88, A70, A66;

      end;

      

       A93: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      

       A94: for r be Point of I[01] holds (( - 1) >= ((ff . r) `1 ) or ((ff . r) `1 ) >= 1 or ( - 1) >= ((ff . r) `2 ) or ((ff . r) `2 ) >= 1) & (( - 1) >= ((gg . r) `1 ) or ((gg . r) `1 ) >= 1 or ( - 1) >= ((gg . r) `2 ) or ((gg . r) `2 ) >= 1)

      proof

        let r be Point of I[01] ;

        (f . r) in ( rng f) by A93, FUNCT_1: 3;

        then (f . r) in C0 by A1;

        then

        consider p1 be Point of ( TOP-REAL 2) such that

         A95: (f . r) = p1 and

         A96: |.p1.| >= 1 by A1;

        (g . r) in ( rng g) by A2, FUNCT_1: 3;

        then (g . r) in C0 by A1;

        then

        consider p2 be Point of ( TOP-REAL 2) such that

         A97: (g . r) = p2 and

         A98: |.p2.| >= 1 by A1;

        

         A99: (gg . r) = (( Sq_Circ " ) . (g . r)) by A3, FUNCT_1: 12;

         A100:

        now

          per cases ;

            case p2 = ( 0. ( TOP-REAL 2));

            hence contradiction by A98, TOPRNS_1: 23;

          end;

            case

             A101: p2 <> ( 0. ( TOP-REAL 2)) & ((p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 )));

            reconsider px = (gg . r) as Point of ( TOP-REAL 2);

            

             A102: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A101, JGRAPH_3: 28;

            then

             A103: (px `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A99, A97, EUCLID: 52;

            set q = px;

            

             A104: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

            ( |.p2.| ^2 ) >= |.p2.| by A98, XREAL_1: 151;

            then

             A105: ( |.p2.| ^2 ) >= 1 by A98, XXREAL_0: 2;

            

             A106: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

            (((p2 `2 ) / (p2 `1 )) ^2 ) >= 0 by XREAL_1: 63;

            then

             A107: ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

            

             A108: (px `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A99, A97, A102, EUCLID: 52;

             A109:

            now

              assume (px `1 ) = 0 & (px `2 ) = 0 ;

              then (p2 `1 ) = 0 & (p2 `2 ) = 0 by A103, A108, A107, XCMPLX_1: 6;

              hence contradiction by A101, EUCLID: 53, EUCLID: 54;

            end;

            (p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A101, A107, XREAL_1: 64;

            then

             A110: (p2 `2 ) <= (p2 `1 ) & (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A103, A108, A107, XREAL_1: 64;

            then

             A111: (px `1 ) <> 0 by A103, A108, A107, A109, XREAL_1: 64;

            ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A99, A97, A102, A103, A107, A110, EUCLID: 52, XREAL_1: 64;

            then

             A112: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A103, A108, A109, JGRAPH_2: 3, JGRAPH_3:def 1;

            (( Sq_Circ " ) . p2) = q by A3, A97, FUNCT_1: 12;

            then

             A113: p2 = ( Sq_Circ . px) by FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

            

             A114: (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

            ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            

            then ( |.p2.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A113, A112, JGRAPH_3: 1

            .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

            .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A114, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A114, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62;

            then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) >= (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A114, A105, XREAL_1: 64;

            then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) >= (1 + (((q `2 ) / (q `1 )) ^2 )) by A114, XCMPLX_1: 87;

            then (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

            then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) >= ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by XREAL_1: 9;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) >= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A104, XREAL_1: 64;

            then ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) >= ((px `2 ) ^2 ) by A111, XCMPLX_1: 6, XCMPLX_1: 87;

            then (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + (((px `1 ) ^2 ) * (((px `2 ) ^2 ) - 1))) - ((px `2 ) ^2 )) >= (((px `2 ) ^2 ) - ((px `2 ) ^2 )) by XREAL_1: 9;

            then

             A115: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) >= 0 ;

            (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A109, COMPLEX1: 1;

            then (((px `1 ) - 1) * ((px `1 ) + 1)) >= 0 by A104, A115, A106, XREAL_1: 132;

            hence ( - 1) >= ((gg . r) `1 ) or ((gg . r) `1 ) >= 1 or ( - 1) >= ((gg . r) `2 ) or ((gg . r) `2 ) >= 1 by XREAL_1: 95;

          end;

            case

             A116: p2 <> ( 0. ( TOP-REAL 2)) & not ((p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 )));

            reconsider pz = (gg . r) as Point of ( TOP-REAL 2);

            

             A117: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A116, JGRAPH_3: 28;

            then

             A118: (pz `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A99, A97, EUCLID: 52;

            (((p2 `1 ) / (p2 `2 )) ^2 ) >= 0 by XREAL_1: 63;

            then

             A119: ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

             A120:

            now

              assume that

               A121: (pz `2 ) = 0 and (pz `1 ) = 0 ;

              (p2 `2 ) = 0 by A118, A119, A121, XCMPLX_1: 6;

              hence contradiction by A116;

            end;

            

             A122: (pz `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A99, A97, A117, EUCLID: 52;

            (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 ) or (p2 `1 ) >= (p2 `2 ) & (p2 `1 ) <= ( - (p2 `2 )) by A116, JGRAPH_2: 13;

            then (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 ) or (p2 `1 ) >= (p2 `2 ) & ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= (( - (p2 `2 )) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A119, XREAL_1: 64;

            then

             A123: (p2 `1 ) <= (p2 `2 ) & (( - (p2 `2 )) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A118, A122, A119, XREAL_1: 64;

            then

             A124: (pz `2 ) <> 0 by A118, A122, A119, A120, XREAL_1: 64;

            ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A99, A97, A117, A118, A119, A123, EUCLID: 52, XREAL_1: 64;

            then

             A125: ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A118, A122, A120, JGRAPH_2: 3, JGRAPH_3: 4;

            

             A126: (((pz `1 ) / (pz `2 )) ^2 ) >= 0 by XREAL_1: 63;

            ( |.p2.| ^2 ) >= |.p2.| by A98, XREAL_1: 151;

            then

             A127: ( |.p2.| ^2 ) >= 1 by A98, XXREAL_0: 2;

            

             A128: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

            

             A129: ((pz `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A130: ((pz `2 ) ^2 ) >= 0 by XREAL_1: 63;

            p2 = ( Sq_Circ . pz) & ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A99, A97, EUCLID: 52, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

            

            then ( |.p2.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A125, A128, JGRAPH_3: 1

            .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

            .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A126, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A126, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by XCMPLX_1: 62;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) >= (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A126, A127, XREAL_1: 64;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) >= (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A126, XCMPLX_1: 87;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) >= (1 + (((pz `1 ) ^2 ) / ((pz `2 ) ^2 ))) by XCMPLX_1: 76;

            then ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) >= ((1 + (((pz `1 ) ^2 ) / ((pz `2 ) ^2 ))) - 1) by XREAL_1: 9;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) >= ((((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) * ((pz `2 ) ^2 )) by A130, XREAL_1: 64;

            then ((((pz `2 ) ^2 ) + (((pz `1 ) ^2 ) - 1)) * ((pz `2 ) ^2 )) >= ((pz `1 ) ^2 ) by A124, XCMPLX_1: 6, XCMPLX_1: 87;

            then (((((pz `2 ) ^2 ) * ((pz `2 ) ^2 )) + (((pz `2 ) ^2 ) * (((pz `1 ) ^2 ) - 1))) - ((pz `1 ) ^2 )) >= (((pz `1 ) ^2 ) - ((pz `1 ) ^2 )) by XREAL_1: 9;

            then

             A131: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) >= 0 ;

            (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A120, COMPLEX1: 1;

            then (((pz `2 ) - 1) * ((pz `2 ) + 1)) >= 0 by A130, A131, A129, XREAL_1: 132;

            hence ( - 1) >= ((gg . r) `1 ) or ((gg . r) `1 ) >= 1 or ( - 1) >= ((gg . r) `2 ) or ((gg . r) `2 ) >= 1 by XREAL_1: 95;

          end;

        end;

        

         A132: (ff . r) = (( Sq_Circ " ) . (f . r)) by A4, FUNCT_1: 12;

        now

          per cases ;

            case p1 = ( 0. ( TOP-REAL 2));

            hence contradiction by A96, TOPRNS_1: 23;

          end;

            case

             A133: p1 <> ( 0. ( TOP-REAL 2)) & ((p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 )));

            reconsider px = (ff . r) as Point of ( TOP-REAL 2);

            

             A134: (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A133, JGRAPH_3: 28;

            then

             A135: (px `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A132, A95, EUCLID: 52;

            (((p1 `2 ) / (p1 `1 )) ^2 ) >= 0 by XREAL_1: 63;

            then

             A136: ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

            

             A137: (px `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A132, A95, A134, EUCLID: 52;

             A138:

            now

              assume (px `1 ) = 0 & (px `2 ) = 0 ;

              then (p1 `1 ) = 0 & (p1 `2 ) = 0 by A135, A137, A136, XCMPLX_1: 6;

              hence contradiction by A133, EUCLID: 53, EUCLID: 54;

            end;

            (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A133, A136, XREAL_1: 64;

            then

             A139: (p1 `2 ) <= (p1 `1 ) & (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A135, A137, A136, XREAL_1: 64;

            then

             A140: (px `1 ) <> 0 by A135, A137, A136, A138, XREAL_1: 64;

            ( |.p1.| ^2 ) >= |.p1.| by A96, XREAL_1: 151;

            then

             A141: ( |.p1.| ^2 ) >= 1 by A96, XXREAL_0: 2;

            

             A142: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

            set q = px;

            

             A143: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            

             A144: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A145: (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

            ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A132, A95, A134, A135, A136, A139, EUCLID: 52, XREAL_1: 64;

            then

             A146: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A135, A137, A138, JGRAPH_2: 3, JGRAPH_3:def 1;

            p1 = ( Sq_Circ . px) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A132, A95, EUCLID: 52, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

            

            then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A146, A143, JGRAPH_3: 1

            .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

            .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A145, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A145, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62;

            then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) >= (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A145, A141, XREAL_1: 64;

            then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) >= (1 + (((q `2 ) / (q `1 )) ^2 )) by A145, XCMPLX_1: 87;

            then (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

            then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) >= ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by XREAL_1: 9;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) >= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A142, XREAL_1: 64;

            then ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) >= ((px `2 ) ^2 ) by A140, XCMPLX_1: 6, XCMPLX_1: 87;

            then (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + (((px `1 ) ^2 ) * (((px `2 ) ^2 ) - 1))) - ((px `2 ) ^2 )) >= (((px `2 ) ^2 ) - ((px `2 ) ^2 )) by XREAL_1: 9;

            then

             A147: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) >= 0 ;

            (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A138, COMPLEX1: 1;

            then (((px `1 ) - 1) * ((px `1 ) + 1)) >= 0 by A142, A147, A144, XREAL_1: 132;

            hence ( - 1) >= ((ff . r) `1 ) or ((ff . r) `1 ) >= 1 or ( - 1) >= ((ff . r) `2 ) or ((ff . r) `2 ) >= 1 by XREAL_1: 95;

          end;

            case

             A148: p1 <> ( 0. ( TOP-REAL 2)) & not ((p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 )));

            reconsider pz = (ff . r) as Point of ( TOP-REAL 2);

            

             A149: (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 ))))]| by A148, JGRAPH_3: 28;

            then

             A150: (pz `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A132, A95, EUCLID: 52;

            (((p1 `1 ) / (p1 `2 )) ^2 ) >= 0 by XREAL_1: 63;

            then

             A151: ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

             A152:

            now

              assume that

               A153: (pz `2 ) = 0 and (pz `1 ) = 0 ;

              (p1 `2 ) = 0 by A150, A151, A153, XCMPLX_1: 6;

              hence contradiction by A148;

            end;

            

             A154: (pz `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A132, A95, A149, EUCLID: 52;

            (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 ) or (p1 `1 ) >= (p1 `2 ) & (p1 `1 ) <= ( - (p1 `2 )) by A148, JGRAPH_2: 13;

            then (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 ) or (p1 `1 ) >= (p1 `2 ) & ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= (( - (p1 `2 )) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A151, XREAL_1: 64;

            then

             A155: (p1 `1 ) <= (p1 `2 ) & (( - (p1 `2 )) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A150, A154, A151, XREAL_1: 64;

            then

             A156: (pz `2 ) <> 0 by A150, A154, A151, A152, XREAL_1: 64;

            ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A132, A95, A149, A150, A151, A155, EUCLID: 52, XREAL_1: 64;

            then

             A157: ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A150, A154, A152, JGRAPH_2: 3, JGRAPH_3: 4;

            

             A158: (((pz `1 ) / (pz `2 )) ^2 ) >= 0 by XREAL_1: 63;

            ( |.p1.| ^2 ) >= |.p1.| by A96, XREAL_1: 151;

            then

             A159: ( |.p1.| ^2 ) >= 1 by A96, XXREAL_0: 2;

            

             A160: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

            

             A161: ((pz `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A162: ((pz `2 ) ^2 ) >= 0 by XREAL_1: 63;

            p1 = ( Sq_Circ . pz) & ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A132, A95, EUCLID: 52, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

            

            then ( |.p1.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A157, A160, JGRAPH_3: 1

            .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

            .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A158, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A158, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by XCMPLX_1: 62;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) >= (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A158, A159, XREAL_1: 64;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) >= (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A158, XCMPLX_1: 87;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) >= (1 + (((pz `1 ) ^2 ) / ((pz `2 ) ^2 ))) by XCMPLX_1: 76;

            then ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) >= ((1 + (((pz `1 ) ^2 ) / ((pz `2 ) ^2 ))) - 1) by XREAL_1: 9;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) >= ((((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) * ((pz `2 ) ^2 )) by A162, XREAL_1: 64;

            then ((((pz `2 ) ^2 ) + (((pz `1 ) ^2 ) - 1)) * ((pz `2 ) ^2 )) >= ((pz `1 ) ^2 ) by A156, XCMPLX_1: 6, XCMPLX_1: 87;

            then (((((pz `2 ) ^2 ) * ((pz `2 ) ^2 )) + (((pz `2 ) ^2 ) * (((pz `1 ) ^2 ) - 1))) - ((pz `1 ) ^2 )) >= (((pz `1 ) ^2 ) - ((pz `1 ) ^2 )) by XREAL_1: 9;

            then

             A163: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) >= 0 ;

            (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A152, COMPLEX1: 1;

            then (((pz `2 ) - 1) * ((pz `2 ) + 1)) >= 0 by A162, A163, A161, XREAL_1: 132;

            hence ( - 1) >= ((ff . r) `1 ) or ((ff . r) `1 ) >= 1 or ( - 1) >= ((ff . r) `2 ) or ((ff . r) `2 ) >= 1 by XREAL_1: 95;

          end;

        end;

        hence thesis by A100;

      end;

      ( - 1) <= ((ff . O) `2 ) & ((ff . O) `2 ) <= 1 & ( - 1) <= ((ff . I) `2 ) & ((ff . I) `2 ) <= 1 & ( - 1) <= ((gg . O) `1 ) & ((gg . O) `1 ) <= 1 & ( - 1) <= ((gg . I) `1 ) & ((gg . I) `1 ) <= 1

      proof

        reconsider pz = (gg . O) as Point of ( TOP-REAL 2);

        reconsider py = (ff . I) as Point of ( TOP-REAL 2);

        reconsider px = (ff . O) as Point of ( TOP-REAL 2);

        set q = px;

        

         A164: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        

         A165: (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

        consider p1 be Point of ( TOP-REAL 2) such that

         A166: (f . O) = p1 and

         A167: |.p1.| = 1 and

         A168: (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 )) by A1;

        

         A169: (ff . O) = (( Sq_Circ " ) . (f . O)) by A4, FUNCT_1: 12;

        then

         A170: p1 = ( Sq_Circ . px) by A166, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        (((p1 `2 ) / (p1 `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A171: ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

        

         A172: p1 <> ( 0. ( TOP-REAL 2)) by A167, TOPRNS_1: 23;

        then (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A168, JGRAPH_3: 28;

        then

         A173: (px `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) & (px `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A169, A166, EUCLID: 52;

         A174:

        now

          assume (px `1 ) = 0 & (px `2 ) = 0 ;

          then (p1 `1 ) = 0 & (p1 `2 ) = 0 by A173, A171, XCMPLX_1: 6;

          hence contradiction by A172, EUCLID: 53, EUCLID: 54;

        end;

        (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A168, A171, XREAL_1: 64;

        then

         A175: (p1 `2 ) <= (p1 `1 ) & (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A173, A171, XREAL_1: 64;

        then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A173, A171, XREAL_1: 64;

        then ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A174, JGRAPH_2: 3, JGRAPH_3:def 1;

        

        then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A170, A164, JGRAPH_3: 1

        .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A165, SQUARE_1:def 2

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A165, SQUARE_1:def 2

        .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62;

        then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A167;

        then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) = (1 + (((q `2 ) / (q `1 )) ^2 )) by A165, XCMPLX_1: 87;

        then

         A176: ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) = (((px `2 ) ^2 ) / ((px `1 ) ^2 )) by XCMPLX_1: 76;

        (px `1 ) <> 0 by A173, A171, A174, A175, XREAL_1: 64;

        then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) = ((px `2 ) ^2 ) by A176, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A177: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) = 0 ;

        (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A174, COMPLEX1: 1;

        then (((px `1 ) - 1) * ((px `1 ) + 1)) = 0 by A177, XCMPLX_1: 6;

        then ((px `1 ) - 1) = 0 or ((px `1 ) + 1) = 0 by XCMPLX_1: 6;

        then (px `1 ) = 1 or (px `1 ) = ( 0 - 1);

        hence ( - 1) <= ((ff . O) `2 ) & ((ff . O) `2 ) <= 1 by A173, A171, A175, XREAL_1: 64;

        

         A178: (((py `2 ) / (py `1 )) ^2 ) >= 0 by XREAL_1: 63;

        reconsider pu = (gg . I) as Point of ( TOP-REAL 2);

        

         A179: ( |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| `1 ) = ((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) & ( |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| `2 ) = ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) by EUCLID: 52;

        

         A180: (((pz `1 ) / (pz `2 )) ^2 ) >= 0 by XREAL_1: 63;

        consider p2 be Point of ( TOP-REAL 2) such that

         A181: (f . I) = p2 and

         A182: |.p2.| = 1 and

         A183: (p2 `2 ) <= (p2 `1 ) & (p2 `2 ) >= ( - (p2 `1 )) by A1;

        

         A184: (ff . I) = (( Sq_Circ " ) . (f . I)) by A4, FUNCT_1: 12;

        then

         A185: p2 = ( Sq_Circ . py) by A181, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A186: p2 <> ( 0. ( TOP-REAL 2)) by A182, TOPRNS_1: 23;

        then

         A187: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A183, JGRAPH_3: 28;

        then

         A188: (py `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A184, A181, EUCLID: 52;

        

         A189: (py `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A184, A181, A187, EUCLID: 52;

        (((p2 `2 ) / (p2 `1 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A190: ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) > 0 by SQUARE_1: 25;

         A191:

        now

          assume (py `1 ) = 0 & (py `2 ) = 0 ;

          then (p2 `1 ) = 0 & (p2 `2 ) = 0 by A188, A189, A190, XCMPLX_1: 6;

          hence contradiction by A186, EUCLID: 53, EUCLID: 54;

        end;

        

         A192: (p2 `2 ) <= (p2 `1 ) & (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A183, A190, XREAL_1: 64;

        then

         A193: ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) & ( - (py `1 )) <= (py `2 ) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A184, A181, A187, A188, A190, EUCLID: 52, XREAL_1: 64;

        then ( Sq_Circ . py) = |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| by A188, A189, A191, JGRAPH_2: 3, JGRAPH_3:def 1;

        

        then ( |.p2.| ^2 ) = ((((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 ) + (((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 )) by A185, A179, JGRAPH_3: 1

        .= ((((py `1 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 )) + (((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((py `1 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 )) + (((py `2 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 ))) by A178, SQUARE_1:def 2

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 )))) by A178, SQUARE_1:def 2

        .= ((((py `1 ) ^2 ) + ((py `2 ) ^2 )) / (1 + (((py `2 ) / (py `1 )) ^2 ))) by XCMPLX_1: 62;

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) / (1 + (((py `2 ) / (py `1 )) ^2 ))) * (1 + (((py `2 ) / (py `1 )) ^2 ))) = (1 * (1 + (((py `2 ) / (py `1 )) ^2 ))) by A182;

        then (((py `1 ) ^2 ) + ((py `2 ) ^2 )) = (1 + (((py `2 ) / (py `1 )) ^2 )) by A178, XCMPLX_1: 87;

        then

         A194: ((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) = (((py `2 ) ^2 ) / ((py `1 ) ^2 )) by XCMPLX_1: 76;

        (py `1 ) <> 0 by A188, A189, A190, A191, A192, XREAL_1: 64;

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) * ((py `1 ) ^2 )) = ((py `2 ) ^2 ) by A194, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A195: ((((py `1 ) ^2 ) - 1) * (((py `1 ) ^2 ) + ((py `2 ) ^2 ))) = 0 ;

        (((py `1 ) ^2 ) + ((py `2 ) ^2 )) <> 0 by A191, COMPLEX1: 1;

        then (((py `1 ) - 1) * ((py `1 ) + 1)) = 0 by A195, XCMPLX_1: 6;

        then ((py `1 ) - 1) = 0 or ((py `1 ) + 1) = 0 by XCMPLX_1: 6;

        hence ( - 1) <= ((ff . I) `2 ) & ((ff . I) `2 ) <= 1 by A188, A189, A193;

        

         A196: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) & ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

        consider p3 be Point of ( TOP-REAL 2) such that

         A197: (g . O) = p3 and

         A198: |.p3.| = 1 and

         A199: (p3 `2 ) <= (p3 `1 ) and

         A200: (p3 `2 ) <= ( - (p3 `1 )) by A1;

        

         A201: p3 <> ( 0. ( TOP-REAL 2)) by A198, TOPRNS_1: 23;

        

         A202: (gg . O) = (( Sq_Circ " ) . (g . O)) by A3, FUNCT_1: 12;

        then

         A203: p3 = ( Sq_Circ . pz) by A197, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A204: ( - (p3 `2 )) >= ( - ( - (p3 `1 ))) by A200, XREAL_1: 24;

        then

         A205: (( Sq_Circ " ) . p3) = |[((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A199, A201, JGRAPH_3: 30;

        then

         A206: (pz `2 ) = ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A202, A197, EUCLID: 52;

        

         A207: (pz `1 ) = ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A202, A197, A205, EUCLID: 52;

        (((p3 `1 ) / (p3 `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A208: ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

         A209:

        now

          assume (pz `2 ) = 0 & (pz `1 ) = 0 ;

          then (p3 `2 ) = 0 & (p3 `1 ) = 0 by A206, A207, A208, XCMPLX_1: 6;

          hence contradiction by A201, EUCLID: 53, EUCLID: 54;

        end;

        (p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A199, A204, A208, XREAL_1: 64;

        then

         A210: (p3 `1 ) <= (p3 `2 ) & (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A206, A207, A208, XREAL_1: 64;

        then

         A211: ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A202, A197, A205, A206, A208, EUCLID: 52, XREAL_1: 64;

        then ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A206, A207, A209, JGRAPH_2: 3, JGRAPH_3: 4;

        

        then ( |.p3.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A203, A196, JGRAPH_3: 1

        .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((pz `2 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 )) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A180, SQUARE_1:def 2

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A180, SQUARE_1:def 2

        .= ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by XCMPLX_1: 62;

        then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) = (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A198;

        then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) = (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A180, XCMPLX_1: 87;

        then

         A212: ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) = (((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) by XCMPLX_1: 76;

        (pz `2 ) <> 0 by A206, A207, A208, A209, A210, XREAL_1: 64;

        then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) = ((pz `1 ) ^2 ) by A212, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A213: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) = 0 ;

        (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A209, COMPLEX1: 1;

        then (((pz `2 ) - 1) * ((pz `2 ) + 1)) = 0 by A213, XCMPLX_1: 6;

        then ((pz `2 ) - 1) = 0 or ((pz `2 ) + 1) = 0 by XCMPLX_1: 6;

        hence ( - 1) <= ((gg . O) `1 ) & ((gg . O) `1 ) <= 1 by A206, A207, A211;

        

         A214: ( |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| `2 ) = ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) & ( |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| `1 ) = ((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by EUCLID: 52;

        

         A215: (((pu `1 ) / (pu `2 )) ^2 ) >= 0 by XREAL_1: 63;

        consider p4 be Point of ( TOP-REAL 2) such that

         A216: (g . I) = p4 and

         A217: |.p4.| = 1 and

         A218: (p4 `2 ) >= (p4 `1 ) and

         A219: (p4 `2 ) >= ( - (p4 `1 )) by A1;

        

         A220: ( - (p4 `2 )) <= ( - ( - (p4 `1 ))) by A219, XREAL_1: 24;

        

         A221: (gg . I) = (( Sq_Circ " ) . (g . I)) by A3, FUNCT_1: 12;

        then

         A222: p4 = ( Sq_Circ . pu) by A216, FUNCT_1: 32, JGRAPH_3: 22, JGRAPH_3: 43;

        

         A223: p4 <> ( 0. ( TOP-REAL 2)) by A217, TOPRNS_1: 23;

        then

         A224: (( Sq_Circ " ) . p4) = |[((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))), ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))))]| by A218, A220, JGRAPH_3: 30;

        then

         A225: (pu `2 ) = ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A221, A216, EUCLID: 52;

        

         A226: (pu `1 ) = ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A221, A216, A224, EUCLID: 52;

        (((p4 `1 ) / (p4 `2 )) ^2 ) >= 0 by XREAL_1: 63;

        then

         A227: ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))) > 0 by SQUARE_1: 25;

         A228:

        now

          assume (pu `2 ) = 0 & (pu `1 ) = 0 ;

          then (p4 `2 ) = 0 & (p4 `1 ) = 0 by A225, A226, A227, XCMPLX_1: 6;

          hence contradiction by A223, EUCLID: 53, EUCLID: 54;

        end;

        

         A229: (p4 `1 ) <= (p4 `2 ) & (( - (p4 `2 )) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A218, A220, A227, XREAL_1: 64;

        then

         A230: ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) & ( - (pu `2 )) <= (pu `1 ) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A221, A216, A224, A225, A227, EUCLID: 52, XREAL_1: 64;

        then ( Sq_Circ . pu) = |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| by A225, A226, A228, JGRAPH_2: 3, JGRAPH_3: 4;

        

        then ( |.p4.| ^2 ) = ((((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 ) + (((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 )) by A222, A214, JGRAPH_3: 1

        .= ((((pu `2 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 )) + (((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

        .= ((((pu `2 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 )) + (((pu `1 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 ))) by A215, SQUARE_1:def 2

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by A215, SQUARE_1:def 2

        .= ((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) by XCMPLX_1: 62;

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) = (1 * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) by A217;

        then (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) = (1 + (((pu `1 ) / (pu `2 )) ^2 )) by A215, XCMPLX_1: 87;

        then

         A231: ((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) = (((pu `1 ) ^2 ) / ((pu `2 ) ^2 )) by XCMPLX_1: 76;

        (pu `2 ) <> 0 by A225, A226, A227, A228, A229, XREAL_1: 64;

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) * ((pu `2 ) ^2 )) = ((pu `1 ) ^2 ) by A231, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A232: ((((pu `2 ) ^2 ) - 1) * (((pu `2 ) ^2 ) + ((pu `1 ) ^2 ))) = 0 ;

        (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) <> 0 by A228, COMPLEX1: 1;

        then (((pu `2 ) - 1) * ((pu `2 ) + 1)) = 0 by A232, XCMPLX_1: 6;

        then ((pu `2 ) - 1) = 0 or ((pu `2 ) + 1) = 0 by XCMPLX_1: 6;

        hence thesis by A225, A226, A230;

      end;

      then ( rng ff) meets ( rng gg) by A1, A5, A94, Th11, JGRAPH_3: 22, JGRAPH_3: 42;

      then

      consider y be object such that

       A233: y in ( rng ff) and

       A234: y in ( rng gg) by XBOOLE_0: 3;

      consider x1 be object such that

       A235: x1 in ( dom ff) and

       A236: y = (ff . x1) by A233, FUNCT_1:def 3;

      consider x2 be object such that

       A237: x2 in ( dom gg) and

       A238: y = (gg . x2) by A234, FUNCT_1:def 3;

      

       A239: ( dom ( Sq_Circ " )) = the carrier of ( TOP-REAL 2) & (gg . x2) = (( Sq_Circ " ) . (g . x2)) by A237, FUNCT_1: 12, FUNCT_2:def 1, JGRAPH_3: 29;

      x1 in ( dom f) by A235, FUNCT_1: 11;

      then

       A240: (f . x1) in ( rng f) by FUNCT_1:def 3;

      x2 in ( dom g) by A237, FUNCT_1: 11;

      then

       A241: (g . x2) in ( rng g) by FUNCT_1:def 3;

      (ff . x1) = (( Sq_Circ " ) . (f . x1)) by A235, FUNCT_1: 12;

      then (f . x1) = (g . x2) by A236, A238, A240, A241, A239, FUNCT_1:def 4, JGRAPH_3: 22;

      hence thesis by A240, A241, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_5:15

    

     Th15: for f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] st O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

      assume

       A1: O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0;

      then ex g2 be Function of I[01] , ( TOP-REAL 2) st (g2 . 0 ) = (g . 1) & (g2 . 1) = (g . 0 ) & ( rng g2) = ( rng g) & g2 is continuous one-to-one by Th12;

      hence thesis by A1, Th14;

    end;

    theorem :: JGRAPH_5:16

    

     Th16: for f,g be Function of I[01] , ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st C0 = { q : |.q.| >= 1 } & f is continuous one-to-one & g is continuous one-to-one & (f . 0 ) = |[( - 1), 0 ]| & (f . 1) = |[1, 0 ]| & (g . 1) = |[ 0 , 1]| & (g . 0 ) = |[ 0 , ( - 1)]| & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      reconsider I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

      reconsider O = 0 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

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

      let f,g be Function of I[01] , ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: C0 = { q : |.q.| >= 1 } & f is continuous one-to-one & g is continuous one-to-one & (f . 0 ) = |[( - 1), 0 ]| & (f . 1) = |[1, 0 ]| & (g . 1) = |[ 0 , 1]| & (g . 0 ) = |[ 0 , ( - 1)]| & ( rng f) c= C0 & ( rng g) c= C0;

      { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      then

      reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

      

       A2: ( |[ 0 , 1]| `1 ) = 0 by EUCLID: 52;

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

      { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      then

      reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

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

      { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      then

      reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

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

      { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      then

      reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

      

       A3: ( |[ 0 , ( - 1)]| `1 ) = 0 by EUCLID: 52;

      ( |[ 0 , ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;

      

      then

       A4: |. |[ 0 , ( - 1)]|.| = ( sqrt (( 0 ^2 ) + (( - 1) ^2 ))) by A3, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      ( |[ 0 , ( - 1)]| `2 ) <= ( - ( |[ 0 , ( - 1)]| `1 )) by A3, EUCLID: 52;

      then

       A5: (g . O) in KYN by A1, A3, A4;

      

       A6: ( |[( - 1), 0 ]| `1 ) = ( - 1) by EUCLID: 52;

      then

       A7: ( |[( - 1), 0 ]| `2 ) <= ( - ( |[( - 1), 0 ]| `1 )) by EUCLID: 52;

      ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

      

      then

       A8: |. |[ 0 , 1]|.| = ( sqrt (( 0 ^2 ) + (1 ^2 ))) by A2, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      ( |[ 0 , 1]| `2 ) >= ( - ( |[ 0 , 1]| `1 )) by A2, EUCLID: 52;

      then

       A9: (g . I) in KYP by A1, A2, A8;

      

       A10: ( |[1, 0 ]| `1 ) = 1 & ( |[1, 0 ]| `2 ) = 0 by EUCLID: 52;

      

      then |. |[1, 0 ]|.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      then

       A11: (f . I) in KXP by A1, A10;

      

       A12: ( |[( - 1), 0 ]| `2 ) = 0 by EUCLID: 52;

      

      then |. |[( - 1), 0 ]|.| = ( sqrt ((( - 1) ^2 ) + ( 0 ^2 ))) by A6, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      then (f . O) in KXN by A1, A6, A12, A7;

      hence thesis by A1, A11, A5, A9, Th14;

    end;

    theorem :: JGRAPH_5:17

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st C0 = { p : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & (ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h is being_homeomorphism & (h .: C0) c= C0 & (h . p1) = |[( - 1), 0 ]| & (h . p2) = |[ 0 , 1]| & (h . p3) = |[1, 0 ]| & (h . p4) = |[ 0 , ( - 1)]|) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: C0 = { p : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h is being_homeomorphism & (h .: C0) c= C0 & (h . p1) = |[( - 1), 0 ]| & (h . p2) = |[ 0 , 1]| & (h . p3) = |[1, 0 ]| & (h . p4) = |[ 0 , ( - 1)]|;

      then

      consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

       A2: h is being_homeomorphism and

       A3: (h .: C0) c= C0 and

       A4: (h . p1) = |[( - 1), 0 ]| and

       A5: (h . p2) = |[ 0 , 1]| and

       A6: (h . p3) = |[1, 0 ]| and

       A7: (h . p4) = |[ 0 , ( - 1)]|;

      let f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A8: f is continuous one-to-one & g is continuous one-to-one and

       A9: (f . 0 ) = p1 and

       A10: (f . 1) = p3 and

       A11: (g . 0 ) = p4 and

       A12: (g . 1) = p2 and

       A13: ( rng f) c= C0 and

       A14: ( rng g) c= C0;

      reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

       0 in ( dom f2) by Lm1, BORSUK_1: 40, FUNCT_2:def 1;

      then

       A15: (f2 . 0 ) = |[( - 1), 0 ]| by A4, A9, FUNCT_1: 12;

      reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

       0 in ( dom g2) by Lm1, BORSUK_1: 40, FUNCT_2:def 1;

      then

       A16: (g2 . 0 ) = |[ 0 , ( - 1)]| by A7, A11, FUNCT_1: 12;

      1 in ( dom g2) by Lm2, BORSUK_1: 40, FUNCT_2:def 1;

      then

       A17: (g2 . 1) = |[ 0 , 1]| by A5, A12, FUNCT_1: 12;

      

       A18: ( rng f2) c= C0

      proof

        let y be object;

        

         A19: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        assume y in ( rng f2);

        then

        consider x be object such that

         A20: x in ( dom f2) and

         A21: y = (f2 . x) by FUNCT_1:def 3;

        x in ( dom f) by A20, FUNCT_1: 11;

        then

         A22: (f . x) in ( rng f) by FUNCT_1:def 3;

        y = (h . (f . x)) by A20, A21, FUNCT_1: 12;

        then y in (h .: C0) by A13, A22, A19, FUNCT_1:def 6;

        hence thesis by A3;

      end;

      

       A23: ( rng g2) c= C0

      proof

        let y be object;

        

         A24: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        assume y in ( rng g2);

        then

        consider x be object such that

         A25: x in ( dom g2) and

         A26: y = (g2 . x) by FUNCT_1:def 3;

        x in ( dom g) by A25, FUNCT_1: 11;

        then

         A27: (g . x) in ( rng g) by FUNCT_1:def 3;

        y = (h . (g . x)) by A25, A26, FUNCT_1: 12;

        then y in (h .: C0) by A14, A27, A24, FUNCT_1:def 6;

        hence thesis by A3;

      end;

      1 in ( dom f2) by Lm2, BORSUK_1: 40, FUNCT_2:def 1;

      then

       A28: (f2 . 1) = |[1, 0 ]| by A6, A10, FUNCT_1: 12;

      h is continuous & h is one-to-one by A2, TOPS_2:def 5;

      then ( rng f2) meets ( rng g2) by A1, A8, A15, A28, A16, A17, A18, A23, Th16;

      then

      consider q5 be object such that

       A29: q5 in ( rng f2) and

       A30: q5 in ( rng g2) by XBOOLE_0: 3;

      consider x be object such that

       A31: x in ( dom f2) and

       A32: q5 = (f2 . x) by A29, FUNCT_1:def 3;

      x in ( dom f) by A31, FUNCT_1: 11;

      then

       A33: (f . x) in ( rng f) by FUNCT_1:def 3;

      consider u be object such that

       A34: u in ( dom g2) and

       A35: q5 = (g2 . u) by A30, FUNCT_1:def 3;

      

       A36: q5 = (h . (g . u)) & (g . u) in ( dom h) by A34, A35, FUNCT_1: 11, FUNCT_1: 12;

      

       A37: h is one-to-one by A2, TOPS_2:def 5;

      u in ( dom g) by A34, FUNCT_1: 11;

      then

       A38: (g . u) in ( rng g) by FUNCT_1:def 3;

      q5 = (h . (f . x)) & (f . x) in ( dom h) by A31, A32, FUNCT_1: 11, FUNCT_1: 12;

      then (f . x) = (g . u) by A37, A36, FUNCT_1:def 4;

      hence thesis by A33, A38, XBOOLE_0: 3;

    end;

    begin

    theorem :: JGRAPH_5:18

    

     Th18: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q `2 ) > 0 holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphN ) . q) holds (p `2 ) > 0

    proof

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

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q `2 ) > 0 ;

      now

        per cases ;

          case ((q `1 ) / |.q.|) >= cn;

          hence thesis by A2, A3, JGRAPH_4: 75;

        end;

          case ((q `1 ) / |.q.|) < cn;

          hence thesis by A1, A3, JGRAPH_4: 76;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:19

    for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q `2 ) >= 0 holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphN ) . q) holds (p `2 ) >= 0

    proof

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

      assume that

       A1: ( - 1) < cn & cn < 1 and

       A2: (q `2 ) >= 0 ;

      now

        per cases by A2;

          case (q `2 ) > 0 ;

          hence thesis by A1, Th18;

        end;

          case (q `2 ) = 0 ;

          hence thesis by JGRAPH_4: 49;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:20

    

     Th20: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q `2 ) >= 0 & ((q `1 ) / |.q.|) < cn & |.q.| <> 0 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: cn < 1 and

       A3: (q `2 ) >= 0 & ((q `1 ) / |.q.|) < cn and

       A4: |.q.| <> 0 ;

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

      assume

       A5: p = ((cn -FanMorphN ) . q);

      now

        per cases ;

          case

           A6: (q `2 ) = 0 ;

          

          then ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ( 0 ^2 )) by JGRAPH_3: 1

          .= ((q `1 ) ^2 );

          then

           A7: |.q.| = (q `1 ) or |.q.| = ( - (q `1 )) by SQUARE_1: 40;

          q = p by A5, A6, JGRAPH_4: 49;

          hence thesis by A2, A3, A4, A7, XCMPLX_1: 60;

        end;

          case (q `2 ) <> 0 ;

          hence thesis by A1, A3, A5, JGRAPH_4: 76;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:21

    

     Th21: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q1 `2 ) >= 0 & (q2 `2 ) >= 0 & |.q1.| <> 0 & |.q2.| <> 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 & cn < 1 and

       A2: (q1 `2 ) >= 0 and

       A3: (q2 `2 ) >= 0 and

       A4: |.q1.| <> 0 and

       A5: |.q2.| <> 0 and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      now

        per cases by A2;

          case

           A7: (q1 `2 ) > 0 ;

          now

            per cases by A3;

              case (q2 `2 ) > 0 ;

              hence thesis by A1, A6, A7, JGRAPH_4: 79;

            end;

              case

               A8: (q2 `2 ) = 0 ;

               A9:

              now

                ( |.q1.| ^2 ) = (((q1 `1 ) ^2 ) + ((q1 `2 ) ^2 )) by JGRAPH_3: 1;

                then (( |.q1.| ^2 ) - ((q1 `1 ) ^2 )) >= 0 by XREAL_1: 63;

                then ((( |.q1.| ^2 ) - ((q1 `1 ) ^2 )) + ((q1 `1 ) ^2 )) >= ( 0 + ((q1 `1 ) ^2 )) by XREAL_1: 7;

                then ( - |.q1.|) <= (q1 `1 ) by SQUARE_1: 47;

                then

                 A10: (( - |.q1.|) / |.q1.|) <= ((q1 `1 ) / |.q1.|) by XREAL_1: 72;

                assume |.q2.| = ( - (q2 `1 ));

                then 1 = (( - (q2 `1 )) / |.q2.|) by A5, XCMPLX_1: 60;

                then ((q1 `1 ) / |.q1.|) < ( - 1) by A6, XCMPLX_1: 190;

                hence contradiction by A4, A10, XCMPLX_1: 197;

              end;

              ( |.q2.| ^2 ) = (((q2 `1 ) ^2 ) + ( 0 ^2 )) by A8, JGRAPH_3: 1

              .= ((q2 `1 ) ^2 );

              then |.q2.| = (q2 `1 ) or |.q2.| = ( - (q2 `1 )) by SQUARE_1: 40;

              then

               A11: ((q2 `1 ) / |.q2.|) = 1 by A5, A9, XCMPLX_1: 60;

              thus 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 p1,p2 be Point of ( TOP-REAL 2);

                assume that

                 A12: p1 = ((cn -FanMorphN ) . q1) and

                 A13: p2 = ((cn -FanMorphN ) . q2);

                

                 A14: |.p1.| = |.q1.| by A12, JGRAPH_4: 66;

                

                 A15: ( |.p1.| ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A16: (p1 `2 ) > 0 by A1, A7, A12, Th18;

                 A17:

                now

                  assume 1 = ((p1 `1 ) / |.p1.|);

                  then (1 * |.p1.|) = (p1 `1 ) by A4, A14, XCMPLX_1: 87;

                  hence contradiction by A15, A16, XCMPLX_1: 6;

                end;

                

                 A18: p2 = q2 by A8, A13, JGRAPH_4: 49;

                (( |.p1.| ^2 ) - ((p1 `1 ) ^2 )) >= 0 by A15, XREAL_1: 63;

                then ((( |.p1.| ^2 ) - ((p1 `1 ) ^2 )) + ((p1 `1 ) ^2 )) >= ( 0 + ((p1 `1 ) ^2 )) by XREAL_1: 7;

                then (p1 `1 ) <= |.p1.| by SQUARE_1: 47;

                then ( |.p1.| / |.p1.|) >= ((p1 `1 ) / |.p1.|) by XREAL_1: 72;

                then 1 >= ((p1 `1 ) / |.p1.|) by A4, A14, XCMPLX_1: 60;

                hence thesis by A11, A18, A17, XXREAL_0: 1;

              end;

            end;

          end;

          hence thesis;

        end;

          case

           A19: (q1 `2 ) = 0 ;

           A20:

          now

            ( |.q2.| ^2 ) = (((q2 `1 ) ^2 ) + ((q2 `2 ) ^2 )) by JGRAPH_3: 1;

            then (( |.q2.| ^2 ) - ((q2 `1 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.q2.| ^2 ) - ((q2 `1 ) ^2 )) + ((q2 `1 ) ^2 )) >= ( 0 + ((q2 `1 ) ^2 )) by XREAL_1: 7;

            then (q2 `1 ) <= |.q2.| by SQUARE_1: 47;

            then

             A21: ( |.q2.| / |.q2.|) >= ((q2 `1 ) / |.q2.|) by XREAL_1: 72;

            assume |.q1.| = (q1 `1 );

            then ((q2 `1 ) / |.q2.|) > 1 by A4, A6, XCMPLX_1: 60;

            hence contradiction by A5, A21, XCMPLX_1: 60;

          end;

          ( |.q1.| ^2 ) = (((q1 `1 ) ^2 ) + ( 0 ^2 )) by A19, JGRAPH_3: 1

          .= ((q1 `1 ) ^2 );

          then |.q1.| = (q1 `1 ) or |.q1.| = ( - (q1 `1 )) by SQUARE_1: 40;

          then (( - (q1 `1 )) / |.q1.|) = 1 by A4, A20, XCMPLX_1: 60;

          then

           A22: ( - ((q1 `1 ) / |.q1.|)) = 1 by XCMPLX_1: 187;

          thus 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 p1,p2 be Point of ( TOP-REAL 2);

            assume that

             A23: p1 = ((cn -FanMorphN ) . q1) and

             A24: p2 = ((cn -FanMorphN ) . q2);

            

             A25: |.p2.| = |.q2.| by A24, JGRAPH_4: 66;

            

             A26: ( |.p2.| ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

            then (( |.p2.| ^2 ) - ((p2 `1 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.p2.| ^2 ) - ((p2 `1 ) ^2 )) + ((p2 `1 ) ^2 )) >= ( 0 + ((p2 `1 ) ^2 )) by XREAL_1: 7;

            then ( - |.p2.|) <= (p2 `1 ) by SQUARE_1: 47;

            then (( - |.p2.|) / |.p2.|) <= ((p2 `1 ) / |.p2.|) by XREAL_1: 72;

            then

             A27: ( - 1) <= ((p2 `1 ) / |.p2.|) by A5, A25, XCMPLX_1: 197;

             A28:

            now

              per cases ;

                case (q2 `2 ) = 0 ;

                then p2 = q2 by A24, JGRAPH_4: 49;

                hence ((p2 `1 ) / |.p2.|) > ( - 1) by A6, A22;

              end;

                case (q2 `2 ) <> 0 ;

                then

                 A29: (p2 `2 ) > 0 by A1, A3, A24, Th18;

                now

                  assume ( - 1) = ((p2 `1 ) / |.p2.|);

                  then (( - 1) * |.p2.|) = (p2 `1 ) by A5, A25, XCMPLX_1: 87;

                  then ( |.p2.| ^2 ) = ((p2 `1 ) ^2 );

                  hence contradiction by A26, A29, XCMPLX_1: 6;

                end;

                hence ((p2 `1 ) / |.p2.|) > ( - 1) by A27, XXREAL_0: 1;

              end;

            end;

            p1 = q1 by A19, A23, JGRAPH_4: 49;

            hence thesis by A22, A28;

          end;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:22

    

     Th22: for sn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (q `1 ) > 0 holds for p be Point of ( TOP-REAL 2) st p = ((sn -FanMorphE ) . q) holds (p `1 ) > 0

    proof

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

      assume that

       A1: ( - 1) < sn and

       A2: sn < 1 and

       A3: (q `1 ) > 0 ;

      now

        per cases ;

          case ((q `2 ) / |.q.|) >= sn;

          hence thesis by A2, A3, JGRAPH_4: 106;

        end;

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

          hence thesis by A1, A3, JGRAPH_4: 107;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:23

    for sn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (q `1 ) >= 0 & ((q `2 ) / |.q.|) < sn & |.q.| <> 0 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: sn < 1 and

       A3: (q `1 ) >= 0 & ((q `2 ) / |.q.|) < sn and

       A4: |.q.| <> 0 ;

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

      assume

       A5: p = ((sn -FanMorphE ) . q);

      now

        per cases ;

          case

           A6: (q `1 ) = 0 ;

          

          then ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ( 0 ^2 )) by JGRAPH_3: 1

          .= ((q `2 ) ^2 );

          then

           A7: |.q.| = (q `2 ) or |.q.| = ( - (q `2 )) by SQUARE_1: 40;

          q = p by A5, A6, JGRAPH_4: 82;

          hence thesis by A2, A3, A4, A7, XCMPLX_1: 60;

        end;

          case (q `1 ) <> 0 ;

          hence thesis by A1, A3, A5, JGRAPH_4: 107;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:24

    

     Th24: for sn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < sn & sn < 1 & (q1 `1 ) >= 0 & (q2 `1 ) >= 0 & |.q1.| <> 0 & |.q2.| <> 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 & sn < 1 and

       A2: (q1 `1 ) >= 0 and

       A3: (q2 `1 ) >= 0 and

       A4: |.q1.| <> 0 and

       A5: |.q2.| <> 0 and

       A6: ((q1 `2 ) / |.q1.|) < ((q2 `2 ) / |.q2.|);

      now

        per cases by A2;

          case

           A7: (q1 `1 ) > 0 ;

          now

            per cases by A3;

              case (q2 `1 ) > 0 ;

              hence thesis by A1, A6, A7, JGRAPH_4: 110;

            end;

              case

               A8: (q2 `1 ) = 0 ;

               A9:

              now

                ( |.q1.| ^2 ) = (((q1 `2 ) ^2 ) + ((q1 `1 ) ^2 )) by JGRAPH_3: 1;

                then (( |.q1.| ^2 ) - ((q1 `2 ) ^2 )) >= 0 by XREAL_1: 63;

                then ((( |.q1.| ^2 ) - ((q1 `2 ) ^2 )) + ((q1 `2 ) ^2 )) >= ( 0 + ((q1 `2 ) ^2 )) by XREAL_1: 7;

                then ( - |.q1.|) <= (q1 `2 ) by SQUARE_1: 47;

                then

                 A10: (( - |.q1.|) / |.q1.|) <= ((q1 `2 ) / |.q1.|) by XREAL_1: 72;

                assume |.q2.| = ( - (q2 `2 ));

                then 1 = (( - (q2 `2 )) / |.q2.|) by A5, XCMPLX_1: 60;

                then ((q1 `2 ) / |.q1.|) < ( - 1) by A6, XCMPLX_1: 190;

                hence contradiction by A4, A10, XCMPLX_1: 197;

              end;

              ( |.q2.| ^2 ) = (((q2 `2 ) ^2 ) + ( 0 ^2 )) by A8, JGRAPH_3: 1

              .= ((q2 `2 ) ^2 );

              then |.q2.| = (q2 `2 ) or |.q2.| = ( - (q2 `2 )) by SQUARE_1: 40;

              then

               A11: ((q2 `2 ) / |.q2.|) = 1 by A5, A9, XCMPLX_1: 60;

              thus 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 p1,p2 be Point of ( TOP-REAL 2);

                assume that

                 A12: p1 = ((sn -FanMorphE ) . q1) and

                 A13: p2 = ((sn -FanMorphE ) . q2);

                

                 A14: |.p1.| = |.q1.| by A12, JGRAPH_4: 97;

                

                 A15: ( |.p1.| ^2 ) = (((p1 `2 ) ^2 ) + ((p1 `1 ) ^2 )) by JGRAPH_3: 1;

                

                 A16: (p1 `1 ) > 0 by A1, A7, A12, Th22;

                 A17:

                now

                  assume 1 = ((p1 `2 ) / |.p1.|);

                  then (1 * |.p1.|) = (p1 `2 ) by A4, A14, XCMPLX_1: 87;

                  hence contradiction by A15, A16, XCMPLX_1: 6;

                end;

                

                 A18: p2 = q2 by A8, A13, JGRAPH_4: 82;

                (( |.p1.| ^2 ) - ((p1 `2 ) ^2 )) >= 0 by A15, XREAL_1: 63;

                then ((( |.p1.| ^2 ) - ((p1 `2 ) ^2 )) + ((p1 `2 ) ^2 )) >= ( 0 + ((p1 `2 ) ^2 )) by XREAL_1: 7;

                then (p1 `2 ) <= |.p1.| by SQUARE_1: 47;

                then ( |.p1.| / |.p1.|) >= ((p1 `2 ) / |.p1.|) by XREAL_1: 72;

                then 1 >= ((p1 `2 ) / |.p1.|) by A4, A14, XCMPLX_1: 60;

                hence thesis by A11, A18, A17, XXREAL_0: 1;

              end;

            end;

          end;

          hence thesis;

        end;

          case

           A19: (q1 `1 ) = 0 ;

           A20:

          now

            ( |.q2.| ^2 ) = (((q2 `2 ) ^2 ) + ((q2 `1 ) ^2 )) by JGRAPH_3: 1;

            then (( |.q2.| ^2 ) - ((q2 `2 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.q2.| ^2 ) - ((q2 `2 ) ^2 )) + ((q2 `2 ) ^2 )) >= ( 0 + ((q2 `2 ) ^2 )) by XREAL_1: 7;

            then (q2 `2 ) <= |.q2.| by SQUARE_1: 47;

            then

             A21: ( |.q2.| / |.q2.|) >= ((q2 `2 ) / |.q2.|) by XREAL_1: 72;

            assume |.q1.| = (q1 `2 );

            then ((q2 `2 ) / |.q2.|) > 1 by A4, A6, XCMPLX_1: 60;

            hence contradiction by A5, A21, XCMPLX_1: 60;

          end;

          ( |.q1.| ^2 ) = (((q1 `2 ) ^2 ) + ( 0 ^2 )) by A19, JGRAPH_3: 1

          .= ((q1 `2 ) ^2 );

          then |.q1.| = (q1 `2 ) or |.q1.| = ( - (q1 `2 )) by SQUARE_1: 40;

          then (( - (q1 `2 )) / |.q1.|) = 1 by A4, A20, XCMPLX_1: 60;

          then

           A22: ( - ((q1 `2 ) / |.q1.|)) = 1 by XCMPLX_1: 187;

          thus 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 p1,p2 be Point of ( TOP-REAL 2);

            assume that

             A23: p1 = ((sn -FanMorphE ) . q1) and

             A24: p2 = ((sn -FanMorphE ) . q2);

            

             A25: |.p2.| = |.q2.| by A24, JGRAPH_4: 97;

            

             A26: ( |.p2.| ^2 ) = (((p2 `2 ) ^2 ) + ((p2 `1 ) ^2 )) by JGRAPH_3: 1;

            then (( |.p2.| ^2 ) - ((p2 `2 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.p2.| ^2 ) - ((p2 `2 ) ^2 )) + ((p2 `2 ) ^2 )) >= ( 0 + ((p2 `2 ) ^2 )) by XREAL_1: 7;

            then ( - |.p2.|) <= (p2 `2 ) by SQUARE_1: 47;

            then (( - |.p2.|) / |.p2.|) <= ((p2 `2 ) / |.p2.|) by XREAL_1: 72;

            then

             A27: ( - 1) <= ((p2 `2 ) / |.p2.|) by A5, A25, XCMPLX_1: 197;

             A28:

            now

              per cases ;

                case (q2 `1 ) = 0 ;

                then p2 = q2 by A24, JGRAPH_4: 82;

                hence ((p2 `2 ) / |.p2.|) > ( - 1) by A6, A22;

              end;

                case (q2 `1 ) <> 0 ;

                then

                 A29: (p2 `1 ) > 0 by A1, A3, A24, Th22;

                now

                  assume ( - 1) = ((p2 `2 ) / |.p2.|);

                  then (( - 1) * |.p2.|) = (p2 `2 ) by A5, A25, XCMPLX_1: 87;

                  then ( |.p2.| ^2 ) = ((p2 `2 ) ^2 );

                  hence contradiction by A26, A29, XCMPLX_1: 6;

                end;

                hence ((p2 `2 ) / |.p2.|) > ( - 1) by A27, XXREAL_0: 1;

              end;

            end;

            p1 = q1 by A19, A23, JGRAPH_4: 82;

            hence thesis by A22, A28;

          end;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:25

    

     Th25: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q `2 ) < 0 holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphS ) . q) holds (p `2 ) < 0

    proof

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

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q `2 ) < 0 ;

      now

        per cases ;

          case ((q `1 ) / |.q.|) >= cn;

          hence thesis by A2, A3, JGRAPH_4: 137;

        end;

          case ((q `1 ) / |.q.|) < cn;

          hence thesis by A1, A3, JGRAPH_4: 138;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:26

    

     Th26: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & 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: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q `2 ) < 0 and

       A4: ((q `1 ) / |.q.|) > cn;

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

      assume

       A5: p = ((cn -FanMorphS ) . q);

      now

        set q1 = ( |.p.| * |[cn, ( - ( sqrt (1 - (cn ^2 ))))]|);

        set p1 = ((1 / |.p.|) * p);

        set p2 = ((cn -FanMorphS ) . q1);

        ( |[ 0 , ( - 1)]| `1 ) = 0 & ( |[ 0 , ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;

        

        then

         A6: ( |.p.| * |[ 0 , ( - 1)]|) = |[( |.p.| * 0 ), ( |.p.| * ( - 1))]| by EUCLID: 57

        .= |[ 0 , ( - |.p.|)]|;

        

         A7: ( |[cn, ( - ( sqrt (1 - (cn ^2 ))))]| `1 ) = cn & ( |[cn, ( - ( sqrt (1 - (cn ^2 ))))]| `2 ) = ( - ( sqrt (1 - (cn ^2 )))) by EUCLID: 52;

        then

         A8: q1 = |[( |.p.| * cn), ( |.p.| * ( - ( sqrt (1 - (cn ^2 )))))]| by EUCLID: 57;

        then

         A9: (q1 `1 ) = ( |.p.| * cn) by EUCLID: 52;

        assume

         A10: (p `1 ) = 0 ;

        

        then ( |.p.| ^2 ) = (((p `2 ) ^2 ) + ( 0 ^2 )) by JGRAPH_3: 1

        .= ((p `2 ) ^2 );

        then

         A11: (p `2 ) = |.p.| or (p `2 ) = ( - |.p.|) by SQUARE_1: 40;

        then

         A12: |.p.| <> 0 by A2, A3, A4, A5, JGRAPH_4: 137;

        

         A13: (q1 `2 ) = ( - (( sqrt (1 - (cn ^2 ))) * |.p.|)) by A8, EUCLID: 52;

        (1 ^2 ) > (cn ^2 ) by A1, A2, SQUARE_1: 50;

        then

         A14: (1 - (cn ^2 )) > 0 by XREAL_1: 50;

        then ( sqrt (1 - (cn ^2 ))) > 0 by SQUARE_1: 25;

        then ( - ( - (( sqrt (1 - (cn ^2 ))) * |.p.|))) > 0 by A12, XREAL_1: 129;

        then

         A15: (q1 `2 ) < 0 by A13;

        

         A16: ( |.p.| * p1) = (( |.p.| * (1 / |.p.|)) * p) by RLVECT_1:def 7

        .= (1 * p) by A12, XCMPLX_1: 106

        .= p by RLVECT_1:def 8;

        

         A17: p1 = |[((1 / |.p.|) * (p `1 )), ((1 / |.p.|) * (p `2 ))]| by EUCLID: 57;

        

        then (p1 `2 ) = ( - ( |.p.| * (1 / |.p.|))) by A2, A3, A4, A5, A11, EUCLID: 52, JGRAPH_4: 137

        .= ( - 1) by A12, XCMPLX_1: 106;

        then

         A18: p = ( |.p.| * |[ 0 , ( - 1)]|) by A10, A16, A17, EUCLID: 52;

        

         A19: |.q1.| = ( |. |.p.|.| * |. |[cn, ( - ( sqrt (1 - (cn ^2 ))))]|.|) by TOPRNS_1: 7

        .= ( |. |.p.|.| * ( sqrt ((cn ^2 ) + (( - ( sqrt (1 - (cn ^2 )))) ^2 )))) by A7, JGRAPH_3: 1

        .= ( |. |.p.|.| * ( sqrt ((cn ^2 ) + (( sqrt (1 - (cn ^2 ))) ^2 ))))

        .= ( |. |.p.|.| * ( sqrt ((cn ^2 ) + (1 - (cn ^2 ))))) by A14, SQUARE_1:def 2

        .= |.p.| by ABSVALUE:def 1, SQUARE_1: 18;

        then

         A20: |.p2.| = |.p.| by JGRAPH_4: 128;

        

         A21: ((q1 `1 ) / |.q1.|) = cn by A12, A9, A19, XCMPLX_1: 89;

        then

         A22: (p2 `1 ) = 0 by A15, JGRAPH_4: 142;

        

        then ( |.p2.| ^2 ) = (((p2 `2 ) ^2 ) + ( 0 ^2 )) by JGRAPH_3: 1

        .= ((p2 `2 ) ^2 );

        then (p2 `2 ) = |.p2.| or (p2 `2 ) = ( - |.p2.|) by SQUARE_1: 40;

        then

         A23: p2 = |[ 0 , ( - |.p.|)]| by A15, A21, A22, A20, EUCLID: 53, JGRAPH_4: 142;

        (cn -FanMorphS ) is one-to-one & ( dom (cn -FanMorphS )) = the carrier of ( TOP-REAL 2) by A1, A2, FUNCT_2:def 1, JGRAPH_4: 133;

        then q1 = q by A5, A18, A23, A6, FUNCT_1:def 4;

        hence contradiction by A4, A12, A9, A19, XCMPLX_1: 89;

      end;

      hence thesis by A2, A3, A4, A5, JGRAPH_4: 137;

    end;

    theorem :: JGRAPH_5:27

    

     Th27: for cn be Real, q1,q2 be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q1 `2 ) <= 0 & (q2 `2 ) <= 0 & |.q1.| <> 0 & |.q2.| <> 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 & cn < 1 and

       A2: (q1 `2 ) <= 0 and

       A3: (q2 `2 ) <= 0 and

       A4: |.q1.| <> 0 and

       A5: |.q2.| <> 0 and

       A6: ((q1 `1 ) / |.q1.|) < ((q2 `1 ) / |.q2.|);

      now

        per cases by A2;

          case

           A7: (q1 `2 ) < 0 ;

          now

            per cases by A3;

              case (q2 `2 ) < 0 ;

              hence thesis by A1, A6, A7, JGRAPH_4: 141;

            end;

              case

               A8: (q2 `2 ) = 0 ;

               A9:

              now

                ( |.q1.| ^2 ) = (((q1 `1 ) ^2 ) + ((q1 `2 ) ^2 )) by JGRAPH_3: 1;

                then (( |.q1.| ^2 ) - ((q1 `1 ) ^2 )) >= 0 by XREAL_1: 63;

                then ((( |.q1.| ^2 ) - ((q1 `1 ) ^2 )) + ((q1 `1 ) ^2 )) >= ( 0 + ((q1 `1 ) ^2 )) by XREAL_1: 7;

                then ( - |.q1.|) <= (q1 `1 ) by SQUARE_1: 47;

                then

                 A10: (( - |.q1.|) / |.q1.|) <= ((q1 `1 ) / |.q1.|) by XREAL_1: 72;

                assume |.q2.| = ( - (q2 `1 ));

                then 1 = (( - (q2 `1 )) / |.q2.|) by A5, XCMPLX_1: 60;

                then ((q1 `1 ) / |.q1.|) < ( - 1) by A6, XCMPLX_1: 190;

                hence contradiction by A4, A10, XCMPLX_1: 197;

              end;

              ( |.q2.| ^2 ) = (((q2 `1 ) ^2 ) + ( 0 ^2 )) by A8, JGRAPH_3: 1

              .= ((q2 `1 ) ^2 );

              then |.q2.| = (q2 `1 ) or |.q2.| = ( - (q2 `1 )) by SQUARE_1: 40;

              then

               A11: ((q2 `1 ) / |.q2.|) = 1 by A5, A9, XCMPLX_1: 60;

              thus 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 p1,p2 be Point of ( TOP-REAL 2);

                assume that

                 A12: p1 = ((cn -FanMorphS ) . q1) and

                 A13: p2 = ((cn -FanMorphS ) . q2);

                

                 A14: |.p1.| = |.q1.| by A12, JGRAPH_4: 128;

                

                 A15: ( |.p1.| ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

                

                 A16: (p1 `2 ) < 0 by A1, A7, A12, Th25;

                 A17:

                now

                  assume 1 = ((p1 `1 ) / |.p1.|);

                  then (1 * |.p1.|) = (p1 `1 ) by A4, A14, XCMPLX_1: 87;

                  hence contradiction by A15, A16, XCMPLX_1: 6;

                end;

                

                 A18: p2 = q2 by A8, A13, JGRAPH_4: 113;

                (( |.p1.| ^2 ) - ((p1 `1 ) ^2 )) >= 0 by A15, XREAL_1: 63;

                then ((( |.p1.| ^2 ) - ((p1 `1 ) ^2 )) + ((p1 `1 ) ^2 )) >= ( 0 + ((p1 `1 ) ^2 )) by XREAL_1: 7;

                then (p1 `1 ) <= |.p1.| by SQUARE_1: 47;

                then ( |.p1.| / |.p1.|) >= ((p1 `1 ) / |.p1.|) by XREAL_1: 72;

                then 1 >= ((p1 `1 ) / |.p1.|) by A4, A14, XCMPLX_1: 60;

                hence thesis by A11, A18, A17, XXREAL_0: 1;

              end;

            end;

          end;

          hence thesis;

        end;

          case

           A19: (q1 `2 ) = 0 ;

           A20:

          now

            ( |.q2.| ^2 ) = (((q2 `1 ) ^2 ) + ((q2 `2 ) ^2 )) by JGRAPH_3: 1;

            then (( |.q2.| ^2 ) - ((q2 `1 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.q2.| ^2 ) - ((q2 `1 ) ^2 )) + ((q2 `1 ) ^2 )) >= ( 0 + ((q2 `1 ) ^2 )) by XREAL_1: 7;

            then (q2 `1 ) <= |.q2.| by SQUARE_1: 47;

            then

             A21: ( |.q2.| / |.q2.|) >= ((q2 `1 ) / |.q2.|) by XREAL_1: 72;

            assume |.q1.| = (q1 `1 );

            then ((q2 `1 ) / |.q2.|) > 1 by A4, A6, XCMPLX_1: 60;

            hence contradiction by A5, A21, XCMPLX_1: 60;

          end;

          ( |.q1.| ^2 ) = (((q1 `1 ) ^2 ) + ( 0 ^2 )) by A19, JGRAPH_3: 1

          .= ((q1 `1 ) ^2 );

          then |.q1.| = (q1 `1 ) or |.q1.| = ( - (q1 `1 )) by SQUARE_1: 40;

          then (( - (q1 `1 )) / |.q1.|) = 1 by A4, A20, XCMPLX_1: 60;

          then

           A22: ( - ((q1 `1 ) / |.q1.|)) = 1 by XCMPLX_1: 187;

          thus 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 p1,p2 be Point of ( TOP-REAL 2);

            assume that

             A23: p1 = ((cn -FanMorphS ) . q1) and

             A24: p2 = ((cn -FanMorphS ) . q2);

            

             A25: |.p2.| = |.q2.| by A24, JGRAPH_4: 128;

            

             A26: ( |.p2.| ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

            then (( |.p2.| ^2 ) - ((p2 `1 ) ^2 )) >= 0 by XREAL_1: 63;

            then ((( |.p2.| ^2 ) - ((p2 `1 ) ^2 )) + ((p2 `1 ) ^2 )) >= ( 0 + ((p2 `1 ) ^2 )) by XREAL_1: 7;

            then ( - |.p2.|) <= (p2 `1 ) by SQUARE_1: 47;

            then (( - |.p2.|) / |.p2.|) <= ((p2 `1 ) / |.p2.|) by XREAL_1: 72;

            then

             A27: ( - 1) <= ((p2 `1 ) / |.p2.|) by A5, A25, XCMPLX_1: 197;

             A28:

            now

              per cases ;

                case (q2 `2 ) = 0 ;

                then p2 = q2 by A24, JGRAPH_4: 113;

                hence ((p2 `1 ) / |.p2.|) > ( - 1) by A6, A22;

              end;

                case (q2 `2 ) <> 0 ;

                then

                 A29: (p2 `2 ) < 0 by A1, A3, A24, Th25;

                now

                  assume ( - 1) = ((p2 `1 ) / |.p2.|);

                  then (( - 1) * |.p2.|) = (p2 `1 ) by A5, A25, XCMPLX_1: 87;

                  then ( |.p2.| ^2 ) = ((p2 `1 ) ^2 );

                  hence contradiction by A26, A29, XCMPLX_1: 6;

                end;

                hence ((p2 `1 ) / |.p2.|) > ( - 1) by A27, XXREAL_0: 1;

              end;

            end;

            p1 = q1 by A19, A23, JGRAPH_4: 113;

            hence thesis by A22, A28;

          end;

        end;

      end;

      hence thesis;

    end;

    begin

     Lm3:

    now

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

      assume

       A1: P = { q : |.q.| = 1 };

      

       A2: [.( - 1), 1.] c= ( proj1 .: P)

      proof

        let y be object;

        assume y in [.( - 1), 1.];

        then y in { r where r be Real : ( - 1) <= r & r <= 1 } by RCOMP_1:def 1;

        then

        consider r be Real such that

         A3: y = r and

         A4: ( - 1) <= r & r <= 1;

        set q = |[r, ( sqrt (1 - (r ^2 )))]|;

        (1 ^2 ) >= (r ^2 ) by A4, SQUARE_1: 49;

        then

         A5: (1 - (r ^2 )) >= 0 by XREAL_1: 48;

        (q `1 ) = r & (q `2 ) = ( sqrt (1 - (r ^2 ))) by EUCLID: 52;

        

        then |.q.| = ( sqrt ((r ^2 ) + (( sqrt (1 - (r ^2 ))) ^2 ))) by JGRAPH_3: 1

        .= ( sqrt ((r ^2 ) + (1 - (r ^2 )))) by A5, SQUARE_1:def 2

        .= 1 by SQUARE_1: 18;

        then

         A6: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & q in P by A1, FUNCT_2:def 1;

        ( proj1 . q) = (q `1 ) by PSCOMP_1:def 5

        .= r by EUCLID: 52;

        hence thesis by A3, A6, FUNCT_1:def 6;

      end;

      ( proj1 .: P) c= [.( - 1), 1.]

      proof

        let y be object;

        assume y in ( proj1 .: P);

        then

        consider x be object such that

         A7: x in ( dom proj1 ) and

         A8: x in P and

         A9: y = ( proj1 . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A7;

        ex q2 be Point of ( TOP-REAL 2) st q2 = x & |.q2.| = 1 by A1, A8;

        then

         A10: (((q `1 ) ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by JGRAPH_3: 1;

         0 <= ((q `2 ) ^2 ) by XREAL_1: 63;

        then ((1 - ((q `1 ) ^2 )) + ((q `1 ) ^2 )) >= ( 0 + ((q `1 ) ^2 )) by A10, XREAL_1: 7;

        then

         A11: ( - 1) <= (q `1 ) & (q `1 ) <= 1 by SQUARE_1: 51;

        y = (q `1 ) by A9, PSCOMP_1:def 5;

        hence thesis by A11, XXREAL_1: 1;

      end;

      hence ( proj1 .: P) = [.( - 1), 1.] by A2, XBOOLE_0:def 10;

      

       A12: [.( - 1), 1.] c= ( proj2 .: P)

      proof

        let y be object;

        assume y in [.( - 1), 1.];

        then y in { r where r be Real : ( - 1) <= r & r <= 1 } by RCOMP_1:def 1;

        then

        consider r be Real such that

         A13: y = r and

         A14: ( - 1) <= r & r <= 1;

        set q = |[( sqrt (1 - (r ^2 ))), r]|;

        (1 ^2 ) >= (r ^2 ) by A14, SQUARE_1: 49;

        then

         A15: (1 - (r ^2 )) >= 0 by XREAL_1: 48;

        (q `2 ) = r & (q `1 ) = ( sqrt (1 - (r ^2 ))) by EUCLID: 52;

        

        then |.q.| = ( sqrt ((( sqrt (1 - (r ^2 ))) ^2 ) + (r ^2 ))) by JGRAPH_3: 1

        .= ( sqrt ((1 - (r ^2 )) + (r ^2 ))) by A15, SQUARE_1:def 2

        .= 1 by SQUARE_1: 18;

        then

         A16: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & q in P by A1, FUNCT_2:def 1;

        ( proj2 . q) = (q `2 ) by PSCOMP_1:def 6

        .= r by EUCLID: 52;

        hence thesis by A13, A16, FUNCT_1:def 6;

      end;

      ( proj2 .: P) c= [.( - 1), 1.]

      proof

        let y be object;

        assume y in ( proj2 .: P);

        then

        consider x be object such that

         A17: x in ( dom proj2 ) and

         A18: x in P and

         A19: y = ( proj2 . x) by FUNCT_1:def 6;

        reconsider q = x as Point of ( TOP-REAL 2) by A17;

        ex q2 be Point of ( TOP-REAL 2) st q2 = x & |.q2.| = 1 by A1, A18;

        then

         A20: (((q `1 ) ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by JGRAPH_3: 1;

         0 <= ((q `1 ) ^2 ) by XREAL_1: 63;

        then ((1 - ((q `2 ) ^2 )) + ((q `2 ) ^2 )) >= ( 0 + ((q `2 ) ^2 )) by A20, XREAL_1: 7;

        then

         A21: ( - 1) <= (q `2 ) & (q `2 ) <= 1 by SQUARE_1: 51;

        y = (q `2 ) by A19, PSCOMP_1:def 6;

        hence thesis by A21, XXREAL_1: 1;

      end;

      hence ( proj2 .: P) = [.( - 1), 1.] by A12, XBOOLE_0:def 10;

    end;

    

     Lm4: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q : |.q.| = 1 } holds ( W-bound P) = ( - 1)

    proof

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

      assume P = { q : |.q.| = 1 };

      then ( proj1 .: P) = [.( - 1), 1.] by Lm3;

      then (( proj1 | P) .: P) = [.( - 1), 1.] by RELAT_1: 129;

      then the carrier of (( TOP-REAL 2) | P) = P & ( lower_bound (( proj1 | P) .: P)) = ( - 1) by JORDAN5A: 19, PRE_TOPC: 8;

      then ( lower_bound ( proj1 | P)) = ( - 1) by PSCOMP_1:def 1;

      hence thesis by PSCOMP_1:def 7;

    end;

    theorem :: JGRAPH_5:28

    

     Th28: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q : |.q.| = 1 } holds ( W-bound P) = ( - 1) & ( E-bound P) = 1 & ( S-bound P) = ( - 1) & ( N-bound P) = 1

    proof

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

      

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

      assume

       A2: P = { q : |.q.| = 1 };

      hence ( W-bound P) = ( - 1) by Lm4;

      ( proj1 .: P) = [.( - 1), 1.] by A2, Lm3;

      then (( proj1 | P) .: P) = [.( - 1), 1.] by RELAT_1: 129;

      then ( upper_bound (( proj1 | P) .: the carrier of (( TOP-REAL 2) | P))) = 1 by A1, JORDAN5A: 19;

      then ( upper_bound ( proj1 | P)) = 1 by PSCOMP_1:def 2;

      hence ( E-bound P) = 1 by PSCOMP_1:def 9;

      ( proj2 .: P) = [.( - 1), 1.] by A2, Lm3;

      then

       A3: (( proj2 | P) .: P) = [.( - 1), 1.] by RELAT_1: 129;

      then ( lower_bound (( proj2 | P) .: P)) = ( - 1) by JORDAN5A: 19;

      then ( lower_bound ( proj2 | P)) = ( - 1) by A1, PSCOMP_1:def 1;

      hence ( S-bound P) = ( - 1) by PSCOMP_1:def 10;

      ( upper_bound (( proj2 | P) .: P)) = 1 by A3, JORDAN5A: 19;

      then ( upper_bound ( proj2 | P)) = 1 by A1, PSCOMP_1:def 2;

      hence thesis by PSCOMP_1:def 8;

    end;

    theorem :: JGRAPH_5:29

    

     Th29: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q : |.q.| = 1 } holds ( W-min P) = |[( - 1), 0 ]|

    proof

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

      

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

      assume

       A2: P = { q : |.q.| = 1 };

      then

       A3: ( W-bound P) = ( - 1) by Lm4;

      ( proj2 .: P) = [.( - 1), 1.] by A2, Lm3;

      then

       A4: (( proj2 | P) .: P) = [.( - 1), 1.] by RELAT_1: 129;

      then ( upper_bound (( proj2 | P) .: P)) = 1 by JORDAN5A: 19;

      then ( upper_bound ( proj2 | P)) = 1 by A1, PSCOMP_1:def 2;

      then ( N-bound P) = 1 by PSCOMP_1:def 8;

      then

       A5: ( NW-corner P) = |[( - 1), 1]| by A3, PSCOMP_1:def 12;

      ( lower_bound (( proj2 | P) .: P)) = ( - 1) by A4, JORDAN5A: 19;

      then ( lower_bound ( proj2 | P)) = ( - 1) by A1, PSCOMP_1:def 1;

      then ( S-bound P) = ( - 1) by PSCOMP_1:def 10;

      then

       A6: ( SW-corner P) = |[( - 1), ( - 1)]| by A3, PSCOMP_1:def 11;

      

       A7: (( LSeg (( SW-corner P),( NW-corner P))) /\ P) c= { |[( - 1), 0 ]|}

      proof

        let x be object;

        assume

         A8: x in (( LSeg (( SW-corner P),( NW-corner P))) /\ P);

        then

         A9: x in { (((1 - l) * ( SW-corner P)) + (l * ( NW-corner P))) where l be Real : 0 <= l & l <= 1 } by XBOOLE_0:def 4;

        x in P by A8, XBOOLE_0:def 4;

        then

         A10: ex q2 be Point of ( TOP-REAL 2) st q2 = x & |.q2.| = 1 by A2;

        consider l be Real such that

         A11: x = (((1 - l) * ( SW-corner P)) + (l * ( NW-corner P))) and 0 <= l and l <= 1 by A9;

        reconsider q3 = x as Point of ( TOP-REAL 2) by A11;

        x = ( |[((1 - l) * ( - 1)), ((1 - l) * ( - 1))]| + (l * |[( - 1), 1]|)) by A6, A5, A11, EUCLID: 58;

        then x = ( |[((1 - l) * ( - 1)), ((1 - l) * ( - 1))]| + |[(l * ( - 1)), (l * 1)]|) by EUCLID: 58;

        then

         A12: x = |[(((1 - l) * ( - 1)) + (l * ( - 1))), (((1 - l) * ( - 1)) + (l * 1))]| by EUCLID: 56;

        then (q3 `1 ) = ( - 1) by EUCLID: 52;

        

        then

         A13: 1 = ( sqrt ((( - 1) ^2 ) + ((q3 `2 ) ^2 ))) by A10, JGRAPH_3: 1

        .= ( sqrt (1 + ((q3 `2 ) ^2 )));

        now

          assume ((q3 `2 ) ^2 ) > 0 ;

          then 1 < (1 + ((q3 `2 ) ^2 )) by XREAL_1: 29;

          hence contradiction by A13, SQUARE_1: 18, SQUARE_1: 27;

        end;

        then ((q3 `2 ) ^2 ) = 0 by XREAL_1: 63;

        then

         A14: (q3 `2 ) = 0 by XCMPLX_1: 6;

        (q3 `2 ) = (((1 - l) * ( - 1)) + l) by A12, EUCLID: 52;

        hence thesis by A12, A14, TARSKI:def 1;

      end;

       { |[( - 1), 0 ]|} c= (( LSeg (( SW-corner P),( NW-corner P))) /\ P)

      proof

        set q = |[( - 1), 0 ]|;

        let x be object;

        assume x in { |[( - 1), 0 ]|};

        then

         A15: x = |[( - 1), 0 ]| by TARSKI:def 1;

        (q `2 ) = 0 & (q `1 ) = ( - 1) by EUCLID: 52;

        

        then |.q.| = ( sqrt ((( - 1) ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

        .= 1 by SQUARE_1: 18;

        then

         A16: x in P by A2, A15;

        q = |[(((1 / 2) * ( - 1)) + ((1 / 2) * ( - 1))), (((1 / 2) * ( - 1)) + ((1 / 2) * 1))]|;

        then q = ( |[((1 / 2) * ( - 1)), ((1 / 2) * ( - 1))]| + |[((1 / 2) * ( - 1)), ((1 / 2) * 1)]|) by EUCLID: 56;

        then q = ( |[((1 / 2) * ( - 1)), ((1 / 2) * ( - 1))]| + ((1 / 2) * |[( - 1), 1]|)) by EUCLID: 58;

        then q = (((1 / 2) * |[( - 1), ( - 1)]|) + ((1 - (1 / 2)) * |[( - 1), 1]|)) by EUCLID: 58;

        then x in ( LSeg (( SW-corner P),( NW-corner P))) by A6, A5, A15;

        hence thesis by A16, XBOOLE_0:def 4;

      end;

      then (( LSeg (( SW-corner P),( NW-corner P))) /\ P) = { |[( - 1), 0 ]|} by A7, XBOOLE_0:def 10;

      then

       A17: ( W-most P) = { |[( - 1), 0 ]|} by PSCOMP_1:def 15;

      (( proj2 | ( W-most P)) .: the carrier of (( TOP-REAL 2) | ( W-most P))) = (( proj2 | ( W-most P)) .: ( W-most P)) by PRE_TOPC: 8

      .= ( Im ( proj2 , |[( - 1), 0 ]|)) by A17, RELAT_1: 129

      .= {( proj2 . |[( - 1), 0 ]|)} by SETWISEO: 8

      .= {( |[( - 1), 0 ]| `2 )} by PSCOMP_1:def 6

      .= { 0 } by EUCLID: 52;

      then ( lower_bound (( proj2 | ( W-most P)) .: the carrier of (( TOP-REAL 2) | ( W-most P)))) = 0 by SEQ_4: 9;

      then ( lower_bound ( proj2 | ( W-most P))) = 0 by PSCOMP_1:def 1;

      hence thesis by A3, PSCOMP_1:def 19;

    end;

    theorem :: JGRAPH_5:30

    

     Th30: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q : |.q.| = 1 } holds ( E-max P) = |[1, 0 ]|

    proof

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

      

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

      assume

       A2: P = { q : |.q.| = 1 };

      then

       A3: ( E-bound P) = 1 by Th28;

      ( proj2 .: P) = [.( - 1), 1.] by A2, Lm3;

      then

       A4: (( proj2 | P) .: P) = [.( - 1), 1.] by RELAT_1: 129;

      then ( upper_bound (( proj2 | P) .: P)) = 1 by JORDAN5A: 19;

      then ( upper_bound ( proj2 | P)) = 1 by A1, PSCOMP_1:def 2;

      then ( N-bound P) = 1 by PSCOMP_1:def 8;

      then

       A5: ( NE-corner P) = |[1, 1]| by A3, PSCOMP_1:def 13;

      ( lower_bound (( proj2 | P) .: P)) = ( - 1) by A4, JORDAN5A: 19;

      then ( lower_bound ( proj2 | P)) = ( - 1) by A1, PSCOMP_1:def 1;

      then ( S-bound P) = ( - 1) by PSCOMP_1:def 10;

      then

       A6: ( SE-corner P) = |[1, ( - 1)]| by A3, PSCOMP_1:def 14;

      

       A7: (( LSeg (( SE-corner P),( NE-corner P))) /\ P) c= { |[1, 0 ]|}

      proof

        let x be object;

        assume

         A8: x in (( LSeg (( SE-corner P),( NE-corner P))) /\ P);

        then

         A9: x in { (((1 - l) * ( SE-corner P)) + (l * ( NE-corner P))) where l be Real : 0 <= l & l <= 1 } by XBOOLE_0:def 4;

        x in P by A8, XBOOLE_0:def 4;

        then

         A10: ex q2 be Point of ( TOP-REAL 2) st q2 = x & |.q2.| = 1 by A2;

        consider l be Real such that

         A11: x = (((1 - l) * ( SE-corner P)) + (l * ( NE-corner P))) and 0 <= l and l <= 1 by A9;

        reconsider q3 = x as Point of ( TOP-REAL 2) by A11;

        x = ( |[((1 - l) * 1), ((1 - l) * ( - 1))]| + (l * |[1, 1]|)) by A6, A5, A11, EUCLID: 58;

        then x = ( |[((1 - l) * 1), ((1 - l) * ( - 1))]| + |[(l * 1), (l * 1)]|) by EUCLID: 58;

        then

         A12: x = |[(((1 - l) + l) * 1), (((1 - l) * ( - 1)) + (l * 1))]| by EUCLID: 56;

        then

         A13: (q3 `1 ) = 1 by EUCLID: 52;

        now

          assume ((q3 `2 ) ^2 ) > 0 ;

          then (1 ^2 ) < (1 + ((q3 `2 ) ^2 )) by XREAL_1: 29;

          hence contradiction by A13, A10, JGRAPH_3: 1;

        end;

        then ((q3 `2 ) ^2 ) = 0 by XREAL_1: 63;

        then

         A14: (q3 `2 ) = 0 by XCMPLX_1: 6;

        (q3 `2 ) = (((1 - l) * ( - 1)) + l) by A12, EUCLID: 52;

        hence thesis by A12, A14, TARSKI:def 1;

      end;

       { |[1, 0 ]|} c= (( LSeg (( SE-corner P),( NE-corner P))) /\ P)

      proof

        set q = |[1, 0 ]|;

        let x be object;

        assume x in { |[1, 0 ]|};

        then

         A15: x = |[1, 0 ]| by TARSKI:def 1;

        (q `2 ) = 0 & (q `1 ) = 1 by EUCLID: 52;

        

        then |.q.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by JGRAPH_3: 1

        .= 1 by SQUARE_1: 18;

        then

         A16: x in P by A2, A15;

        q = |[(((1 / 2) * 1) + ((1 / 2) * 1)), (((1 / 2) * ( - 1)) + ((1 / 2) * 1))]|;

        then q = ( |[((1 / 2) * 1), ((1 / 2) * ( - 1))]| + |[((1 / 2) * 1), ((1 / 2) * 1)]|) by EUCLID: 56;

        then q = ( |[((1 / 2) * 1), ((1 / 2) * ( - 1))]| + ((1 / 2) * |[1, 1]|)) by EUCLID: 58;

        then q = (((1 / 2) * |[1, ( - 1)]|) + ((1 - (1 / 2)) * |[1, 1]|)) by EUCLID: 58;

        then x in ( LSeg (( SE-corner P),( NE-corner P))) by A6, A5, A15;

        hence thesis by A16, XBOOLE_0:def 4;

      end;

      then (( LSeg (( SE-corner P),( NE-corner P))) /\ P) = { |[1, 0 ]|} by A7, XBOOLE_0:def 10;

      then

       A17: ( E-most P) = { |[1, 0 ]|} by PSCOMP_1:def 17;

      (( proj2 | ( E-most P)) .: the carrier of (( TOP-REAL 2) | ( E-most P))) = (( proj2 | ( E-most P)) .: ( E-most P)) by PRE_TOPC: 8

      .= ( Im ( proj2 , |[1, 0 ]|)) by A17, RELAT_1: 129

      .= {( proj2 . |[1, 0 ]|)} by SETWISEO: 8

      .= {( |[1, 0 ]| `2 )} by PSCOMP_1:def 6

      .= { 0 } by EUCLID: 52;

      then ( upper_bound (( proj2 | ( E-most P)) .: the carrier of (( TOP-REAL 2) | ( E-most P)))) = 0 by SEQ_4: 9;

      then ( upper_bound ( proj2 | ( E-most P))) = 0 by PSCOMP_1:def 2;

      hence thesis by A3, PSCOMP_1:def 23;

    end;

    theorem :: JGRAPH_5:31

    for f be Function of ( TOP-REAL 2), R^1 st (for p be Point of ( TOP-REAL 2) holds (f . p) = ( proj1 . p)) holds f is continuous

    proof

      let f be Function of ( TOP-REAL 2), R^1 ;

      assume

       A1: for p be Point of ( TOP-REAL 2) holds (f . p) = ( proj1 . p);

      reconsider f as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

      (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then f is continuous by A1, JGRAPH_2: 29;

      hence thesis by PRE_TOPC: 32;

    end;

    theorem :: JGRAPH_5:32

    

     Th32: for f be Function of ( TOP-REAL 2), R^1 st (for p be Point of ( TOP-REAL 2) holds (f . p) = ( proj2 . p)) holds f is continuous

    proof

      let f be Function of ( TOP-REAL 2), R^1 ;

      assume

       A1: for p be Point of ( TOP-REAL 2) holds (f . p) = ( proj2 . p);

      reconsider f as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

      (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then f is continuous by A1, JGRAPH_2: 30;

      hence thesis by PRE_TOPC: 32;

    end;

    theorem :: JGRAPH_5:33

    

     Th33: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 } holds ( Upper_Arc P) c= P & ( Lower_Arc P) c= P

    proof

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

      assume P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 };

      then P is being_simple_closed_curve by JGRAPH_3: 26;

      hence thesis by JORDAN6: 61;

    end;

    theorem :: JGRAPH_5:34

    

     Th34: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 } holds ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 }

    proof

      reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

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

      set P4 = ( Lower_Arc P);

      set P1 = ( Upper_Arc P), P2 = ( Lower_Arc P), Q = ( Vertical_Line 0 );

      set p8 = ( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line 0 )));

      set pj = ( Last_Point (( Lower_Arc P),( E-max P),( W-min P),( Vertical_Line 0 )));

      

       A1: ( LSeg ( |[ 0 , ( - 1)]|, |[ 0 , 1]|)) c= Q

      proof

        let x be object;

        assume x in ( LSeg ( |[ 0 , ( - 1)]|, |[ 0 , 1]|));

        then

        consider l be Real such that

         A2: x = (((1 - l) * |[ 0 , ( - 1)]|) + (l * |[ 0 , 1]|)) and 0 <= l and l <= 1;

        ((((1 - l) * |[ 0 , ( - 1)]|) + (l * |[ 0 , 1]|)) `1 ) = ((((1 - l) * |[ 0 , ( - 1)]|) `1 ) + ((l * |[ 0 , 1]|) `1 )) by TOPREAL3: 2

        .= (((1 - l) * ( |[ 0 , ( - 1)]| `1 )) + ((l * |[ 0 , 1]|) `1 )) by TOPREAL3: 4

        .= (((1 - l) * ( |[ 0 , ( - 1)]| `1 )) + (l * ( |[ 0 , 1]| `1 ))) by TOPREAL3: 4

        .= (((1 - l) * 0 ) + (l * ( |[ 0 , 1]| `1 ))) by EUCLID: 52

        .= (((1 - l) * 0 ) + (l * 0 )) by EUCLID: 52

        .= 0 ;

        hence thesis by A2;

      end;

      reconsider R = ( Upper_Arc P) as non empty Subset of ( TOP-REAL 2);

      assume

       A3: P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 };

      then

       A4: P is being_simple_closed_curve by JGRAPH_3: 26;

      then

       A5: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by JORDAN6:def 8;

      then

      consider f be Function of I[01] , (( TOP-REAL 2) | R) such that

       A6: f is being_homeomorphism and

       A7: (f . 0 ) = ( W-min P) and

       A8: (f . 1) = ( E-max P) by TOPREAL1:def 1;

      

       A9: ( dom f) = the carrier of I[01] & ( dom h2) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A10: ex P2 be non empty Subset of ( TOP-REAL 2) st P2 is_an_arc_of (( E-max P),( W-min P)) & (( Upper_Arc P) /\ P2) = {( W-min P), ( E-max P)} & (( Upper_Arc P) \/ P2) = P & (( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) > (( Last_Point (P2,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) by A4, JORDAN6:def 8;

      then

       A11: ( Upper_Arc P) c= P by XBOOLE_1: 7;

      

       A12: ( rng f) = ( [#] (( TOP-REAL 2) | R)) by A6, TOPS_2:def 5

      .= R by PRE_TOPC:def 5;

      

       A13: ( S-bound P) = ( - 1) & ( N-bound P) = 1 by A3, Th28;

      

       A14: ( Vertical_Line 0 ) is closed by JORDAN6: 30;

      

       A15: for p be Point of ( TOP-REAL 2) holds (h2 . p) = ( proj2 . p);

      

       A16: ( W-bound P) = ( - 1) & ( E-bound P) = 1 by A3, Th28;

      then

       A17: P1 meets Q by A4, A13, A1, JORDAN6: 69, XBOOLE_1: 64;

      

       A18: P2 meets Q by A4, A16, A13, A1, JORDAN6: 70, XBOOLE_1: 64;

      

       A19: (( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) > (( Last_Point (P4,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) by A4, JORDAN6:def 9;

      ( Upper_Arc P) is closed by A5, JORDAN6: 11;

      then (P1 /\ Q) is closed by A14, TOPS_1: 8;

      then

       A20: p8 in (P1 /\ Q) by A5, A17, JORDAN5C:def 1;

      then p8 in P1 by XBOOLE_0:def 4;

      then

      consider x8 be object such that

       A21: x8 in ( dom f) and

       A22: p8 = (f . x8) by A12, FUNCT_1:def 3;

      ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

      then x8 in { r where r be Real : 0 <= r & r <= 1 } by A21, RCOMP_1:def 1;

      then

      consider r8 be Real such that

       A23: x8 = r8 and

       A24: 0 <= r8 and

       A25: r8 <= 1;

      

       A26: ( Vertical_Line 0 ) is closed by JORDAN6: 30;

      (P1 /\ Q) c= { |[ 0 , ( - 1)]|, |[ 0 , 1]|}

      proof

        let x be object;

        assume

         A27: x in (P1 /\ Q);

        then x in P1 by XBOOLE_0:def 4;

        then x in P by A10, XBOOLE_0:def 3;

        then

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

         A28: q = x and

         A29: |.q.| = 1 by A3;

        x in Q by A27, XBOOLE_0:def 4;

        then

         A30: ex p be Point of ( TOP-REAL 2) st p = x & (p `1 ) = 0 ;

        then (( 0 ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by A28, A29, JGRAPH_3: 1;

        then (q `2 ) = 1 or (q `2 ) = ( - 1) by SQUARE_1: 41;

        then x = |[ 0 , ( - 1)]| or x = |[ 0 , 1]| by A30, A28, EUCLID: 53;

        hence thesis by TARSKI:def 2;

      end;

      then p8 = |[ 0 , ( - 1)]| or p8 = |[ 0 , 1]| by A20, TARSKI:def 2;

      then

       A31: (p8 `2 ) = ( - 1) or (p8 `2 ) = 1 by EUCLID: 52;

       A32:

      now

        assume r8 = 0 ;

        then p8 = |[( - 1), 0 ]| by A3, A7, A22, A23, Th29;

        hence contradiction by A31, EUCLID: 52;

      end;

      

       A33: ( rng (h2 * f)) c= the carrier of R^1 ;

      

       A34: the carrier of (( TOP-REAL 2) | R) = R by PRE_TOPC: 8;

      then ( rng f) c= the carrier of ( TOP-REAL 2) by XBOOLE_1: 1;

      then ( dom (h2 * f)) = the carrier of I[01] by A9, RELAT_1: 27;

      then

      reconsider g0 = (h2 * f) as Function of I[01] , R^1 by A33, FUNCT_2: 2;

      

       A35: f is one-to-one by A6, TOPS_2:def 5;

      

       A36: f is continuous by A6, TOPS_2:def 5;

      

       A37: (ex p be Point of ( TOP-REAL 2), t be Real st 0 < t & t < 1 & (f . t) = p & (p `2 ) > 0 ) implies for q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) holds (q `2 ) >= 0

      proof

        given p be Point of ( TOP-REAL 2), t be Real such that

         A38: 0 < t and

         A39: t < 1 and

         A40: (f . t) = p and

         A41: (p `2 ) > 0 ;

        now

          assume ex q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) & (q `2 ) < 0 ;

          then

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

           A42: q in ( Upper_Arc P) and

           A43: (q `2 ) < 0 ;

          ( rng f) = ( [#] (( TOP-REAL 2) | R)) by A6, TOPS_2:def 5

          .= R by PRE_TOPC:def 5;

          then

          consider x be object such that

           A44: x in ( dom f) and

           A45: q = (f . x) by A42, FUNCT_1:def 3;

          

           A46: ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

          then

           A47: x in { r where r be Real : 0 <= r & r <= 1 } by A44, RCOMP_1:def 1;

          t in { v where v be Real : 0 <= v & v <= 1 } by A38, A39;

          then

           A48: t in [. 0 , 1.] by RCOMP_1:def 1;

          

          then

           A49: ((h2 * f) . t) = (h2 . p) by A40, A46, FUNCT_1: 13

          .= (p `2 ) by PSCOMP_1:def 6;

          consider r be Real such that

           A50: x = r and

           A51: 0 <= r and

           A52: r <= 1 by A47;

          

           A53: ((h2 * f) . r) = (h2 . q) by A44, A45, A50, FUNCT_1: 13

          .= (q `2 ) by PSCOMP_1:def 6;

          now

            per cases by XXREAL_0: 1;

              case

               A54: r < t;

              then

              reconsider B = [.r, t.] as non empty Subset of I[01] by A44, A50, A48, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g0 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A55: ((q `2 ) * (p `2 )) < 0 by A41, A43, XREAL_1: 132;

              t in { r4 where r4 be Real : r <= r4 & r4 <= t } by A54;

              then t in B by RCOMP_1:def 1;

              then

               A56: (p `2 ) = (g . t) by A49, FUNCT_1: 49;

              r in { r4 where r4 be Real : r <= r4 & r4 <= t } by A54;

              then r in B by RCOMP_1:def 1;

              then

               A57: (q `2 ) = (g . r) by A53, FUNCT_1: 49;

              g0 is continuous by A36, A15, Th7, Th32;

              then

               A58: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (r,t)) = ( I[01] | B) by A39, A51, A54, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A59: (g . r1) = 0 and

               A60: r < r1 and

               A61: r1 < t by A54, A58, A55, A57, A56, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : r <= r4 & r4 <= t } by A60, A61;

              then

               A62: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A39, A61, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A51, A60;

              then

               A63: r1 in ( dom f) by A46, RCOMP_1:def 1;

              then (f . r1) in ( rng f) by FUNCT_1:def 3;

              then (f . r1) in R by A34;

              then (f . r1) in P by A11;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A64: q3 = (f . r1) and

               A65: |.q3.| = 1 by A3;

              

               A66: (q3 `2 ) = (h2 . (f . r1)) by A64, PSCOMP_1:def 6

              .= (g0 . r1) by A63, FUNCT_1: 13

              .= 0 by A59, A62, FUNCT_1: 49;

              

              then

               A67: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A65, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A67, SQUARE_1: 41;

                  case

                   A68: (q3 `1 ) = 1;

                  

                   A69: 1 in ( dom f) by A46, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A66, A68, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A8, A35, A39, A61, A63, A64, A69, FUNCT_1:def 4;

                end;

                  case

                   A70: (q3 `1 ) = ( - 1);

                  

                   A71: 0 in ( dom f) by A46, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A66, A70, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A7, A35, A51, A60, A63, A64, A71, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case

               A72: t < r;

              then

              reconsider B = [.t, r.] as non empty Subset of I[01] by A44, A50, A48, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g0 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A73: ((q `2 ) * (p `2 )) < 0 by A41, A43, XREAL_1: 132;

              t in { r4 where r4 be Real : t <= r4 & r4 <= r } by A72;

              then t in B by RCOMP_1:def 1;

              then

               A74: (p `2 ) = (g . t) by A49, FUNCT_1: 49;

              r in { r4 where r4 be Real : t <= r4 & r4 <= r } by A72;

              then r in B by RCOMP_1:def 1;

              then

               A75: (q `2 ) = (g . r) by A53, FUNCT_1: 49;

              g0 is continuous by A36, A15, Th7, Th32;

              then

               A76: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (t,r)) = ( I[01] | B) by A38, A52, A72, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A77: (g . r1) = 0 and

               A78: t < r1 and

               A79: r1 < r by A72, A76, A73, A75, A74, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : t <= r4 & r4 <= r } by A78, A79;

              then

               A80: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A52, A79, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A38, A78;

              then

               A81: r1 in ( dom f) by A46, RCOMP_1:def 1;

              then (f . r1) in ( rng f) by FUNCT_1:def 3;

              then (f . r1) in R by A34;

              then (f . r1) in P by A11;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A82: q3 = (f . r1) and

               A83: |.q3.| = 1 by A3;

              

               A84: (q3 `2 ) = (h2 . (f . r1)) by A82, PSCOMP_1:def 6

              .= ((h2 * f) . r1) by A81, FUNCT_1: 13

              .= 0 by A77, A80, FUNCT_1: 49;

              

              then

               A85: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A83, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A85, SQUARE_1: 41;

                  case

                   A86: (q3 `1 ) = 1;

                  

                   A87: 1 in ( dom f) by A46, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A84, A86, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A8, A35, A52, A79, A81, A82, A87, FUNCT_1:def 4;

                end;

                  case

                   A88: (q3 `1 ) = ( - 1);

                  

                   A89: 0 in ( dom f) by A46, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A84, A88, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A7, A35, A38, A78, A81, A82, A89, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case t = r;

              hence contradiction by A41, A43, A53, A49;

            end;

          end;

          hence contradiction;

        end;

        hence thesis;

      end;

      reconsider R = ( Lower_Arc P) as non empty Subset of ( TOP-REAL 2);

      

       A90: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A4, JORDAN6:def 9;

      then

      consider f2 be Function of I[01] , (( TOP-REAL 2) | R) such that

       A91: f2 is being_homeomorphism and

       A92: (f2 . 0 ) = ( E-max P) and

       A93: (f2 . 1) = ( W-min P) by TOPREAL1:def 1;

      

       A94: ( dom f2) = the carrier of I[01] & ( dom h2) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A95: ( rng (h2 * f2)) c= the carrier of R^1 ;

      

       A96: the carrier of (( TOP-REAL 2) | R) = R by PRE_TOPC: 8;

      then ( rng f2) c= the carrier of ( TOP-REAL 2) by XBOOLE_1: 1;

      then ( dom (h2 * f2)) = the carrier of I[01] by A94, RELAT_1: 27;

      then

      reconsider g1 = (h2 * f2) as Function of I[01] , R^1 by A95, FUNCT_2: 2;

      

       A97: f2 is one-to-one by A91, TOPS_2:def 5;

      

       A98: (( Upper_Arc P) \/ P4) = P by A4, JORDAN6:def 9;

      then

       A99: ( Lower_Arc P) c= P by XBOOLE_1: 7;

      

       A100: (P2 /\ Q) c= { |[ 0 , ( - 1)]|, |[ 0 , 1]|}

      proof

        let x be object;

        assume

         A101: x in (P2 /\ Q);

        then x in P2 by XBOOLE_0:def 4;

        then x in P by A98, XBOOLE_0:def 3;

        then

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

         A102: q = x and

         A103: |.q.| = 1 by A3;

        x in Q by A101, XBOOLE_0:def 4;

        then

         A104: ex p be Point of ( TOP-REAL 2) st p = x & (p `1 ) = 0 ;

        then (( 0 ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by A102, A103, JGRAPH_3: 1;

        then (q `2 ) = 1 or (q `2 ) = ( - 1) by SQUARE_1: 41;

        then x = |[ 0 , ( - 1)]| or x = |[ 0 , 1]| by A104, A102, EUCLID: 53;

        hence thesis by TARSKI:def 2;

      end;

      

       A105: for p be Point of ( TOP-REAL 2) holds (h2 . p) = ( proj2 . p);

      

       A106: f2 is continuous by A91, TOPS_2:def 5;

      

       A107: (ex p be Point of ( TOP-REAL 2), t be Real st 0 < t & t < 1 & (f2 . t) = p & (p `2 ) > 0 ) implies for q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) holds (q `2 ) >= 0

      proof

        given p be Point of ( TOP-REAL 2), t be Real such that

         A108: 0 < t and

         A109: t < 1 and

         A110: (f2 . t) = p and

         A111: (p `2 ) > 0 ;

        now

          assume ex q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) & (q `2 ) < 0 ;

          then

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

           A112: q in ( Lower_Arc P) and

           A113: (q `2 ) < 0 ;

          ( rng f2) = ( [#] (( TOP-REAL 2) | R)) by A91, TOPS_2:def 5

          .= R by PRE_TOPC:def 5;

          then

          consider x be object such that

           A114: x in ( dom f2) and

           A115: q = (f2 . x) by A112, FUNCT_1:def 3;

          

           A116: ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

          then

           A117: x in { r where r be Real : 0 <= r & r <= 1 } by A114, RCOMP_1:def 1;

          t in { v where v be Real : 0 <= v & v <= 1 } by A108, A109;

          then

           A118: t in [. 0 , 1.] by RCOMP_1:def 1;

          

          then

           A119: ((h2 * f2) . t) = (h2 . p) by A110, A116, FUNCT_1: 13

          .= (p `2 ) by PSCOMP_1:def 6;

          consider r be Real such that

           A120: x = r and

           A121: 0 <= r and

           A122: r <= 1 by A117;

          

           A123: ((h2 * f2) . r) = (h2 . q) by A114, A115, A120, FUNCT_1: 13

          .= (q `2 ) by PSCOMP_1:def 6;

          now

            per cases by XXREAL_0: 1;

              case

               A124: r < t;

              then

              reconsider B = [.r, t.] as non empty Subset of I[01] by A114, A120, A118, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g1 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A125: ((q `2 ) * (p `2 )) < 0 by A111, A113, XREAL_1: 132;

              t in { r4 where r4 be Real : r <= r4 & r4 <= t } by A124;

              then t in B by RCOMP_1:def 1;

              then

               A126: (p `2 ) = (g . t) by A119, FUNCT_1: 49;

              r in { r4 where r4 be Real : r <= r4 & r4 <= t } by A124;

              then r in B by RCOMP_1:def 1;

              then

               A127: (q `2 ) = (g . r) by A123, FUNCT_1: 49;

              g1 is continuous by A106, A105, Th7, Th32;

              then

               A128: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (r,t)) = ( I[01] | B) by A109, A121, A124, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A129: (g . r1) = 0 and

               A130: r < r1 and

               A131: r1 < t by A124, A128, A125, A127, A126, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : r <= r4 & r4 <= t } by A130, A131;

              then

               A132: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A109, A131, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A121, A130;

              then

               A133: r1 in ( dom f2) by A116, RCOMP_1:def 1;

              then (f2 . r1) in ( rng f2) by FUNCT_1:def 3;

              then (f2 . r1) in R by A96;

              then (f2 . r1) in P by A99;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A134: q3 = (f2 . r1) and

               A135: |.q3.| = 1 by A3;

              

               A136: (q3 `2 ) = (h2 . (f2 . r1)) by A134, PSCOMP_1:def 6

              .= ((h2 * f2) . r1) by A133, FUNCT_1: 13

              .= 0 by A129, A132, FUNCT_1: 49;

              

              then

               A137: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A135, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A137, SQUARE_1: 41;

                  case

                   A138: (q3 `1 ) = 1;

                  

                   A139: 0 in ( dom f2) by A116, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A136, A138, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A92, A97, A121, A130, A133, A134, A139, FUNCT_1:def 4;

                end;

                  case

                   A140: (q3 `1 ) = ( - 1);

                  

                   A141: 1 in ( dom f2) by A116, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A136, A140, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A93, A97, A109, A131, A133, A134, A141, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case

               A142: t < r;

              then

              reconsider B = [.t, r.] as non empty Subset of I[01] by A114, A120, A118, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g1 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A143: ((q `2 ) * (p `2 )) < 0 by A111, A113, XREAL_1: 132;

              t in { r4 where r4 be Real : t <= r4 & r4 <= r } by A142;

              then t in B by RCOMP_1:def 1;

              then

               A144: (p `2 ) = (g . t) by A119, FUNCT_1: 49;

              r in { r4 where r4 be Real : t <= r4 & r4 <= r } by A142;

              then r in B by RCOMP_1:def 1;

              then

               A145: (q `2 ) = (g . r) by A123, FUNCT_1: 49;

              g1 is continuous by A106, A105, Th7, Th32;

              then

               A146: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (t,r)) = ( I[01] | B) by A108, A122, A142, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A147: (g . r1) = 0 and

               A148: t < r1 and

               A149: r1 < r by A142, A146, A143, A145, A144, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : t <= r4 & r4 <= r } by A148, A149;

              then

               A150: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A122, A149, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A108, A148;

              then

               A151: r1 in ( dom f2) by A116, RCOMP_1:def 1;

              then (f2 . r1) in ( rng f2) by FUNCT_1:def 3;

              then (f2 . r1) in R by A96;

              then (f2 . r1) in P by A99;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A152: q3 = (f2 . r1) and

               A153: |.q3.| = 1 by A3;

              

               A154: (q3 `2 ) = (h2 . (f2 . r1)) by A152, PSCOMP_1:def 6

              .= (g1 . r1) by A151, FUNCT_1: 13

              .= 0 by A147, A150, FUNCT_1: 49;

              

              then

               A155: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A153, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A155, SQUARE_1: 41;

                  case

                   A156: (q3 `1 ) = 1;

                  

                   A157: 0 in ( dom f2) by A116, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A154, A156, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A92, A97, A108, A148, A151, A152, A157, FUNCT_1:def 4;

                end;

                  case

                   A158: (q3 `1 ) = ( - 1);

                  

                   A159: 1 in ( dom f2) by A116, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A154, A158, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A93, A97, A122, A149, A151, A152, A159, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case t = r;

              hence contradiction by A111, A113, A123, A119;

            end;

          end;

          hence contradiction;

        end;

        hence thesis;

      end;

      ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A160: ( W-min P) in ( Upper_Arc P) by A10, XBOOLE_0:def 4;

      

       A161: ( W-bound P) = ( - 1) & ( E-bound P) = 1 by A3, Th28;

      now

        assume r8 = 1;

        then p8 = |[1, 0 ]| by A3, A8, A22, A23, Th30;

        hence contradiction by A31, EUCLID: 52;

      end;

      then

       A162: 1 > r8 by A25, XXREAL_0: 1;

      ( Lower_Arc P) is closed by A90, JORDAN6: 11;

      then (P2 /\ Q) is closed by A26, TOPS_1: 8;

      then pj in (P2 /\ Q) by A90, A18, JORDAN5C:def 2;

      then

       A163: pj = |[ 0 , ( - 1)]| or pj = |[ 0 , 1]| by A100, TARSKI:def 2;

      ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A164: ( E-max P) in ( Upper_Arc P) by A10, XBOOLE_0:def 4;

      

       A165: { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } c= ( Upper_Arc P)

      proof

        let x be object;

        assume x in { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 };

        then

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

         A166: p = x and

         A167: p in P and

         A168: (p `2 ) >= 0 ;

        now

          per cases by A168;

            case

             A169: (p `2 ) = 0 ;

            ex p8 be Point of ( TOP-REAL 2) st p8 = p & |.p8.| = 1 by A3, A167;

            

            then 1 = ( sqrt (((p `1 ) ^2 ) + ((p `2 ) ^2 ))) by JGRAPH_3: 1

            .= |.(p `1 ).| by A169, COMPLEX1: 72;

            then p = |[(p `1 ), (p `2 )]| & ((p `1 ) ^2 ) = (1 ^2 ) by COMPLEX1: 75, EUCLID: 53;

            then p = |[1, 0 ]| or p = |[( - 1), 0 ]| by A169, SQUARE_1: 41;

            hence thesis by A3, A164, A160, A166, Th29, Th30;

          end;

            case

             A170: (p `2 ) > 0 ;

            now

              assume not x in ( Upper_Arc P);

              then

               A171: x in ( Lower_Arc P) by A98, A166, A167, XBOOLE_0:def 3;

              ( rng f2) = ( [#] (( TOP-REAL 2) | R)) by A91, TOPS_2:def 5

              .= R by PRE_TOPC:def 5;

              then

              consider x2 be object such that

               A172: x2 in ( dom f2) and

               A173: p = (f2 . x2) by A166, A171, FUNCT_1:def 3;

              ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

              then x2 in { r where r be Real : 0 <= r & r <= 1 } by A172, RCOMP_1:def 1;

              then

              consider t2 be Real such that

               A174: x2 = t2 and

               A175: 0 <= t2 and

               A176: t2 <= 1;

              

               A177: ( |[ 0 , ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;

              now

                assume t2 = 1;

                then p = |[( - 1), 0 ]| by A3, A93, A173, A174, Th29;

                hence contradiction by A170, EUCLID: 52;

              end;

              then

               A178: t2 < 1 by A176, XXREAL_0: 1;

               A179:

              now

                assume t2 = 0 ;

                then p = |[1, 0 ]| by A3, A92, A173, A174, Th30;

                hence contradiction by A170, EUCLID: 52;

              end;

              ( |[ 0 , ( - 1)]| `1 ) = 0 by EUCLID: 52;

              

              then |. |[ 0 , ( - 1)]|.| = ( sqrt (( 0 ^2 ) + (( - 1) ^2 ))) by A177, JGRAPH_3: 1

              .= 1 by SQUARE_1: 18;

              then

               A180: |[ 0 , ( - 1)]| in { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 };

              now

                per cases by A3, A98, A180, XBOOLE_0:def 3;

                  case |[ 0 , ( - 1)]| in ( Upper_Arc P);

                  hence contradiction by A19, A161, A31, A163, A22, A23, A24, A32, A162, A37, A177, EUCLID: 52;

                end;

                  case |[ 0 , ( - 1)]| in ( Lower_Arc P);

                  hence contradiction by A107, A170, A173, A174, A175, A179, A178, A177;

                end;

              end;

              hence contradiction;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      ( Upper_Arc P) c= { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 }

      proof

        let x2 be object;

        assume

         A181: x2 in ( Upper_Arc P);

        then

        reconsider q3 = x2 as Point of ( TOP-REAL 2);

        (q3 `2 ) >= 0 by A19, A161, A31, A163, A22, A23, A24, A32, A162, A37, A181, EUCLID: 52;

        hence thesis by A11, A181;

      end;

      hence thesis by A165, XBOOLE_0:def 10;

    end;

    theorem :: JGRAPH_5:35

    

     Th35: for P be compact non empty Subset of ( TOP-REAL 2) st P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 } holds ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 }

    proof

      reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

      reconsider Q = ( Vertical_Line 0 ) as Subset of ( TOP-REAL 2);

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

      set P4 = ( Lower_Arc P);

      reconsider P1 = ( Lower_Arc P) as Subset of ( TOP-REAL 2);

      reconsider P2 = ( Upper_Arc P) as Subset of ( TOP-REAL 2);

      set pj = ( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line 0 )));

      set p8 = ( Last_Point (( Lower_Arc P),( E-max P),( W-min P),( Vertical_Line 0 )));

      

       A1: ( LSeg ( |[ 0 , ( - 1)]|, |[ 0 , 1]|)) c= Q

      proof

        let x be object;

        assume x in ( LSeg ( |[ 0 , ( - 1)]|, |[ 0 , 1]|));

        then

        consider l be Real such that

         A2: x = (((1 - l) * |[ 0 , ( - 1)]|) + (l * |[ 0 , 1]|)) and 0 <= l and l <= 1;

        ((((1 - l) * |[ 0 , ( - 1)]|) + (l * |[ 0 , 1]|)) `1 ) = ((((1 - l) * |[ 0 , ( - 1)]|) `1 ) + ((l * |[ 0 , 1]|) `1 )) by TOPREAL3: 2

        .= (((1 - l) * ( |[ 0 , ( - 1)]| `1 )) + ((l * |[ 0 , 1]|) `1 )) by TOPREAL3: 4

        .= (((1 - l) * ( |[ 0 , ( - 1)]| `1 )) + (l * ( |[ 0 , 1]| `1 ))) by TOPREAL3: 4

        .= (((1 - l) * 0 ) + (l * ( |[ 0 , 1]| `1 ))) by EUCLID: 52

        .= (((1 - l) * 0 ) + (l * 0 )) by EUCLID: 52

        .= 0 ;

        hence thesis by A2;

      end;

      assume

       A3: P = { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 };

      then

       A4: P is being_simple_closed_curve by JGRAPH_3: 26;

      then

       A5: (( Upper_Arc P) \/ P4) = P by JORDAN6:def 9;

      then

       A6: ( Lower_Arc P) c= P by XBOOLE_1: 7;

      

       A7: (P2 /\ Q) c= { |[ 0 , ( - 1)]|, |[ 0 , 1]|}

      proof

        let x be object;

        assume

         A8: x in (P2 /\ Q);

        then x in P2 by XBOOLE_0:def 4;

        then x in P by A5, XBOOLE_0:def 3;

        then

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

         A9: q = x and

         A10: |.q.| = 1 by A3;

        x in Q by A8, XBOOLE_0:def 4;

        then

         A11: ex p be Point of ( TOP-REAL 2) st p = x & (p `1 ) = 0 ;

        then (( 0 ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by A9, A10, JGRAPH_3: 1;

        then (q `2 ) = 1 or (q `2 ) = ( - 1) by SQUARE_1: 41;

        then x = |[ 0 , ( - 1)]| or x = |[ 0 , 1]| by A11, A9, EUCLID: 53;

        hence thesis by TARSKI:def 2;

      end;

      

       A12: for p be Point of ( TOP-REAL 2) holds (h2 . p) = ( proj2 . p);

      reconsider R = ( Lower_Arc P) as non empty Subset of ( TOP-REAL 2);

      

       A13: ( Vertical_Line 0 ) is closed by JORDAN6: 30;

      

       A14: ( Vertical_Line 0 ) is closed by JORDAN6: 30;

      

       A15: for p be Point of ( TOP-REAL 2) holds (h2 . p) = ( proj2 . p);

      

       A16: ( S-bound P) = ( - 1) & ( N-bound P) = 1 by A3, Th28;

      

       A17: ( W-bound P) = ( - 1) & ( E-bound P) = 1 by A3, Th28;

      then

       A18: P1 meets Q by A4, A16, A1, JORDAN6: 70, XBOOLE_1: 64;

      

       A19: P2 meets Q by A4, A17, A16, A1, JORDAN6: 69, XBOOLE_1: 64;

      

       A20: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by A4, JORDAN6:def 9;

      

       A21: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A4, JORDAN6:def 9;

      then

      consider f be Function of I[01] , (( TOP-REAL 2) | R) such that

       A22: f is being_homeomorphism and

       A23: (f . 0 ) = ( E-max P) and

       A24: (f . 1) = ( W-min P) by TOPREAL1:def 1;

      

       A25: ( dom f) = the carrier of I[01] & ( dom h2) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A26: ( rng f) = ( [#] (( TOP-REAL 2) | R)) by A22, TOPS_2:def 5

      .= R by PRE_TOPC:def 5;

      

       A27: ( Upper_Arc P) c= P by A5, XBOOLE_1: 7;

      

       A28: ( rng (h2 * f)) c= the carrier of R^1 ;

      

       A29: the carrier of (( TOP-REAL 2) | R) = R by PRE_TOPC: 8;

      then ( rng f) c= the carrier of ( TOP-REAL 2) by XBOOLE_1: 1;

      then ( dom (h2 * f)) = the carrier of I[01] by A25, RELAT_1: 27;

      then

      reconsider g0 = (h2 * f) as Function of I[01] , R^1 by A28, FUNCT_2: 2;

      

       A30: f is one-to-one by A22, TOPS_2:def 5;

      

       A31: f is continuous by A22, TOPS_2:def 5;

      

       A32: (ex p be Point of ( TOP-REAL 2), t be Real st 0 < t & t < 1 & (f . t) = p & (p `2 ) < 0 ) implies for q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) holds (q `2 ) <= 0

      proof

        given p be Point of ( TOP-REAL 2), t be Real such that

         A33: 0 < t and

         A34: t < 1 and

         A35: (f . t) = p and

         A36: (p `2 ) < 0 ;

        now

          assume ex q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) & (q `2 ) > 0 ;

          then

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

           A37: q in ( Lower_Arc P) and

           A38: (q `2 ) > 0 ;

          ( rng f) = ( [#] (( TOP-REAL 2) | R)) by A22, TOPS_2:def 5

          .= R by PRE_TOPC:def 5;

          then

          consider x be object such that

           A39: x in ( dom f) and

           A40: q = (f . x) by A37, FUNCT_1:def 3;

          

           A41: ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

          then

           A42: x in { r where r be Real : 0 <= r & r <= 1 } by A39, RCOMP_1:def 1;

          t in { v where v be Real : 0 <= v & v <= 1 } by A33, A34;

          then

           A43: t in [. 0 , 1.] by RCOMP_1:def 1;

          

          then

           A44: ((h2 * f) . t) = (h2 . p) by A35, A41, FUNCT_1: 13

          .= (p `2 ) by PSCOMP_1:def 6;

          consider r be Real such that

           A45: x = r and

           A46: 0 <= r and

           A47: r <= 1 by A42;

          

           A48: ((h2 * f) . r) = (h2 . q) by A39, A40, A45, FUNCT_1: 13

          .= (q `2 ) by PSCOMP_1:def 6;

          now

            per cases by XXREAL_0: 1;

              case

               A49: r < t;

              then

              reconsider B = [.r, t.] as non empty Subset of I[01] by A39, A45, A43, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g0 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A50: ((q `2 ) * (p `2 )) < 0 by A36, A38, XREAL_1: 132;

              t in { r4 where r4 be Real : r <= r4 & r4 <= t } by A49;

              then t in B by RCOMP_1:def 1;

              then

               A51: (p `2 ) = (g . t) by A44, FUNCT_1: 49;

              r in { r4 where r4 be Real : r <= r4 & r4 <= t } by A49;

              then r in B by RCOMP_1:def 1;

              then

               A52: (q `2 ) = (g . r) by A48, FUNCT_1: 49;

              g0 is continuous by A31, A12, Th7, Th32;

              then

               A53: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (r,t)) = ( I[01] | B) by A34, A46, A49, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A54: (g . r1) = 0 and

               A55: r < r1 and

               A56: r1 < t by A49, A53, A50, A52, A51, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : r <= r4 & r4 <= t } by A55, A56;

              then

               A57: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A34, A56, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A46, A55;

              then

               A58: r1 in ( dom f) by A41, RCOMP_1:def 1;

              then (f . r1) in ( rng f) by FUNCT_1:def 3;

              then (f . r1) in R by A29;

              then (f . r1) in P by A6;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A59: q3 = (f . r1) and

               A60: |.q3.| = 1 by A3;

              

               A61: (q3 `2 ) = (h2 . (f . r1)) by A59, PSCOMP_1:def 6

              .= ((h2 * f) . r1) by A58, FUNCT_1: 13

              .= 0 by A54, A57, FUNCT_1: 49;

              

              then

               A62: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A60, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A62, SQUARE_1: 41;

                  case

                   A63: (q3 `1 ) = 1;

                  

                   A64: 0 in ( dom f) by A41, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A61, A63, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A23, A30, A46, A55, A58, A59, A64, FUNCT_1:def 4;

                end;

                  case

                   A65: (q3 `1 ) = ( - 1);

                  

                   A66: 1 in ( dom f) by A41, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A61, A65, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A24, A30, A34, A56, A58, A59, A66, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case

               A67: t < r;

              then

              reconsider B = [.t, r.] as non empty Subset of I[01] by A39, A45, A43, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g0 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A68: ((q `2 ) * (p `2 )) < 0 by A36, A38, XREAL_1: 132;

              t in { r4 where r4 be Real : t <= r4 & r4 <= r } by A67;

              then t in B by RCOMP_1:def 1;

              then

               A69: (p `2 ) = (g . t) by A44, FUNCT_1: 49;

              r in { r4 where r4 be Real : t <= r4 & r4 <= r } by A67;

              then r in B by RCOMP_1:def 1;

              then

               A70: (q `2 ) = (g . r) by A48, FUNCT_1: 49;

              g0 is continuous by A31, A12, Th7, Th32;

              then

               A71: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (t,r)) = ( I[01] | B) by A33, A47, A67, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A72: (g . r1) = 0 and

               A73: t < r1 and

               A74: r1 < r by A67, A71, A68, A70, A69, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : t <= r4 & r4 <= r } by A73, A74;

              then

               A75: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A47, A74, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A33, A73;

              then

               A76: r1 in ( dom f) by A41, RCOMP_1:def 1;

              then (f . r1) in ( rng f) by FUNCT_1:def 3;

              then (f . r1) in R by A29;

              then (f . r1) in P by A6;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A77: q3 = (f . r1) and

               A78: |.q3.| = 1 by A3;

              

               A79: (q3 `2 ) = (h2 . (f . r1)) by A77, PSCOMP_1:def 6

              .= ((h2 * f) . r1) by A76, FUNCT_1: 13

              .= 0 by A72, A75, FUNCT_1: 49;

              

              then

               A80: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A78, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A80, SQUARE_1: 41;

                  case

                   A81: (q3 `1 ) = 1;

                  

                   A82: 0 in ( dom f) by A41, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A79, A81, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A23, A30, A33, A73, A76, A77, A82, FUNCT_1:def 4;

                end;

                  case

                   A83: (q3 `1 ) = ( - 1);

                  

                   A84: 1 in ( dom f) by A41, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A79, A83, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A24, A30, A47, A74, A76, A77, A84, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case t = r;

              hence contradiction by A36, A38, A48, A44;

            end;

          end;

          hence contradiction;

        end;

        hence thesis;

      end;

      reconsider R = ( Upper_Arc P) as non empty Subset of ( TOP-REAL 2);

      

       A85: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A4, JORDAN6:def 8;

      then

      consider f2 be Function of I[01] , (( TOP-REAL 2) | R) such that

       A86: f2 is being_homeomorphism and

       A87: (f2 . 0 ) = ( W-min P) and

       A88: (f2 . 1) = ( E-max P) by TOPREAL1:def 1;

      

       A89: ( dom f2) = the carrier of I[01] & ( dom h2) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A90: ( rng (h2 * f2)) c= the carrier of R^1 ;

      

       A91: the carrier of (( TOP-REAL 2) | R) = R by PRE_TOPC: 8;

      then ( rng f2) c= the carrier of ( TOP-REAL 2) by XBOOLE_1: 1;

      then ( dom (h2 * f2)) = the carrier of I[01] by A89, RELAT_1: 27;

      then

      reconsider g1 = (h2 * f2) as Function of I[01] , R^1 by A90, FUNCT_2: 2;

      

       A92: f2 is one-to-one by A86, TOPS_2:def 5;

      

       A93: f2 is continuous by A86, TOPS_2:def 5;

      

       A94: (ex p be Point of ( TOP-REAL 2), t be Real st 0 < t & t < 1 & (f2 . t) = p & (p `2 ) < 0 ) implies for q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) holds (q `2 ) <= 0

      proof

        given p be Point of ( TOP-REAL 2), t be Real such that

         A95: 0 < t and

         A96: t < 1 and

         A97: (f2 . t) = p and

         A98: (p `2 ) < 0 ;

        now

          assume ex q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) & (q `2 ) > 0 ;

          then

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

           A99: q in ( Upper_Arc P) and

           A100: (q `2 ) > 0 ;

          ( rng f2) = ( [#] (( TOP-REAL 2) | R)) by A86, TOPS_2:def 5

          .= R by PRE_TOPC:def 5;

          then

          consider x be object such that

           A101: x in ( dom f2) and

           A102: q = (f2 . x) by A99, FUNCT_1:def 3;

          

           A103: ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

          then

           A104: x in { r where r be Real : 0 <= r & r <= 1 } by A101, RCOMP_1:def 1;

          t in { v where v be Real : 0 <= v & v <= 1 } by A95, A96;

          then

           A105: t in [. 0 , 1.] by RCOMP_1:def 1;

          

          then

           A106: ((h2 * f2) . t) = (h2 . p) by A97, A103, FUNCT_1: 13

          .= (p `2 ) by PSCOMP_1:def 6;

          consider r be Real such that

           A107: x = r and

           A108: 0 <= r and

           A109: r <= 1 by A104;

          

           A110: ((h2 * f2) . r) = (h2 . q) by A101, A102, A107, FUNCT_1: 13

          .= (q `2 ) by PSCOMP_1:def 6;

          now

            per cases by XXREAL_0: 1;

              case

               A111: r < t;

              then

              reconsider B = [.r, t.] as non empty Subset of I[01] by A101, A107, A105, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g1 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A112: ((q `2 ) * (p `2 )) < 0 by A98, A100, XREAL_1: 132;

              t in { r4 where r4 be Real : r <= r4 & r4 <= t } by A111;

              then t in B by RCOMP_1:def 1;

              then

               A113: (p `2 ) = (g . t) by A106, FUNCT_1: 49;

              r in { r4 where r4 be Real : r <= r4 & r4 <= t } by A111;

              then r in B by RCOMP_1:def 1;

              then

               A114: (q `2 ) = (g . r) by A110, FUNCT_1: 49;

              g1 is continuous by A93, A15, Th7, Th32;

              then

               A115: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (r,t)) = ( I[01] | B) by A96, A108, A111, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A116: (g . r1) = 0 and

               A117: r < r1 and

               A118: r1 < t by A111, A115, A112, A114, A113, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : r <= r4 & r4 <= t } by A117, A118;

              then

               A119: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A96, A118, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A108, A117;

              then

               A120: r1 in ( dom f2) by A103, RCOMP_1:def 1;

              then (f2 . r1) in ( rng f2) by FUNCT_1:def 3;

              then (f2 . r1) in R by A91;

              then (f2 . r1) in P by A27;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A121: q3 = (f2 . r1) and

               A122: |.q3.| = 1 by A3;

              

               A123: (q3 `2 ) = (h2 . (f2 . r1)) by A121, PSCOMP_1:def 6

              .= ((h2 * f2) . r1) by A120, FUNCT_1: 13

              .= 0 by A116, A119, FUNCT_1: 49;

              

              then

               A124: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A122, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A124, SQUARE_1: 41;

                  case

                   A125: (q3 `1 ) = 1;

                  

                   A126: 1 in ( dom f2) by A103, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A123, A125, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A88, A92, A96, A118, A120, A121, A126, FUNCT_1:def 4;

                end;

                  case

                   A127: (q3 `1 ) = ( - 1);

                  

                   A128: 0 in ( dom f2) by A103, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A123, A127, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A87, A92, A108, A117, A120, A121, A128, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case

               A129: t < r;

              then

              reconsider B = [.t, r.] as non empty Subset of I[01] by A101, A107, A105, BORSUK_1: 40, XXREAL_1: 1, XXREAL_2:def 12;

              reconsider B0 = B as Subset of I[01] ;

              reconsider g = (g1 | B0) as Function of ( I[01] | B0), R^1 by PRE_TOPC: 9;

              

               A130: ((q `2 ) * (p `2 )) < 0 by A98, A100, XREAL_1: 132;

              t in { r4 where r4 be Real : t <= r4 & r4 <= r } by A129;

              then t in B by RCOMP_1:def 1;

              then

               A131: (p `2 ) = (g . t) by A106, FUNCT_1: 49;

              r in { r4 where r4 be Real : t <= r4 & r4 <= r } by A129;

              then r in B by RCOMP_1:def 1;

              then

               A132: (q `2 ) = (g . r) by A110, FUNCT_1: 49;

              g1 is continuous by A93, A15, Th7, Th32;

              then

               A133: g is continuous by TOPMETR: 7;

              ( Closed-Interval-TSpace (t,r)) = ( I[01] | B) by A95, A109, A129, TOPMETR: 20, TOPMETR: 23;

              then

              consider r1 be Real such that

               A134: (g . r1) = 0 and

               A135: t < r1 and

               A136: r1 < r by A129, A133, A130, A132, A131, TOPREAL5: 8;

              r1 in { r4 where r4 be Real : t <= r4 & r4 <= r } by A135, A136;

              then

               A137: r1 in B by RCOMP_1:def 1;

              r1 < 1 by A109, A136, XXREAL_0: 2;

              then r1 in { r2 where r2 be Real : 0 <= r2 & r2 <= 1 } by A95, A135;

              then

               A138: r1 in ( dom f2) by A103, RCOMP_1:def 1;

              then (f2 . r1) in ( rng f2) by FUNCT_1:def 3;

              then (f2 . r1) in R by A91;

              then (f2 . r1) in P by A27;

              then

              consider q3 be Point of ( TOP-REAL 2) such that

               A139: q3 = (f2 . r1) and

               A140: |.q3.| = 1 by A3;

              

               A141: (q3 `2 ) = (h2 . (f2 . r1)) by A139, PSCOMP_1:def 6

              .= ((h2 * f2) . r1) by A138, FUNCT_1: 13

              .= 0 by A134, A137, FUNCT_1: 49;

              

              then

               A142: (1 ^2 ) = (((q3 `1 ) ^2 ) + ( 0 ^2 )) by A140, JGRAPH_3: 1

              .= ((q3 `1 ) ^2 );

              now

                per cases by A142, SQUARE_1: 41;

                  case

                   A143: (q3 `1 ) = 1;

                  

                   A144: 1 in ( dom f2) by A103, XXREAL_1: 1;

                  q3 = |[1, 0 ]| by A141, A143, EUCLID: 53

                  .= ( E-max P) by A3, Th30;

                  hence contradiction by A88, A92, A109, A136, A138, A139, A144, FUNCT_1:def 4;

                end;

                  case

                   A145: (q3 `1 ) = ( - 1);

                  

                   A146: 0 in ( dom f2) by A103, XXREAL_1: 1;

                  q3 = |[( - 1), 0 ]| by A141, A145, EUCLID: 53

                  .= ( W-min P) by A3, Th29;

                  hence contradiction by A87, A92, A95, A135, A138, A139, A146, FUNCT_1:def 4;

                end;

              end;

              hence contradiction;

            end;

              case t = r;

              hence contradiction by A98, A100, A110, A106;

            end;

          end;

          hence contradiction;

        end;

        hence thesis;

      end;

      

       A147: ( W-bound P) = ( - 1) & ( E-bound P) = 1 by A3, Th28;

      ( Lower_Arc P) is closed by A21, JORDAN6: 11;

      then (P1 /\ Q) is closed by A13, TOPS_1: 8;

      then

       A148: p8 in (P1 /\ Q) by A21, A18, JORDAN5C:def 2;

      then p8 in P1 by XBOOLE_0:def 4;

      then

      consider x8 be object such that

       A149: x8 in ( dom f) and

       A150: p8 = (f . x8) by A26, FUNCT_1:def 3;

      ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

      then x8 in { r where r be Real : 0 <= r & r <= 1 } by A149, RCOMP_1:def 1;

      then

      consider r8 be Real such that

       A151: x8 = r8 and

       A152: 0 <= r8 and

       A153: r8 <= 1;

      (P1 /\ Q) c= { |[ 0 , ( - 1)]|, |[ 0 , 1]|}

      proof

        let x be object;

        assume

         A154: x in (P1 /\ Q);

        then x in P1 by XBOOLE_0:def 4;

        then x in P by A5, XBOOLE_0:def 3;

        then

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

         A155: q = x and

         A156: |.q.| = 1 by A3;

        x in Q by A154, XBOOLE_0:def 4;

        then

         A157: ex p be Point of ( TOP-REAL 2) st p = x & (p `1 ) = 0 ;

        then (( 0 ^2 ) + ((q `2 ) ^2 )) = (1 ^2 ) by A155, A156, JGRAPH_3: 1;

        then (q `2 ) = 1 or (q `2 ) = ( - 1) by SQUARE_1: 41;

        then x = |[ 0 , ( - 1)]| or x = |[ 0 , 1]| by A157, A155, EUCLID: 53;

        hence thesis by TARSKI:def 2;

      end;

      then p8 = |[ 0 , ( - 1)]| or p8 = |[ 0 , 1]| by A148, TARSKI:def 2;

      then

       A158: (p8 `2 ) = ( - 1) or (p8 `2 ) = 1 by EUCLID: 52;

       A159:

      now

        assume r8 = 0 ;

        then p8 = |[1, 0 ]| by A3, A23, A150, A151, Th30;

        hence contradiction by A158, EUCLID: 52;

      end;

      ( Upper_Arc P) is closed by A85, JORDAN6: 11;

      then (P2 /\ Q) is closed by A14, TOPS_1: 8;

      then pj in (P2 /\ Q) by A85, A19, JORDAN5C:def 1;

      then

       A160: pj = |[ 0 , ( - 1)]| or pj = |[ 0 , 1]| by A7, TARSKI:def 2;

      ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A161: ( W-min P) in ( Lower_Arc P) by A20, XBOOLE_0:def 4;

      

       A162: (( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) > (( Last_Point (P4,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) by A4, JORDAN6:def 9;

      now

        assume r8 = 1;

        then p8 = |[( - 1), 0 ]| by A3, A24, A150, A151, Th29;

        hence contradiction by A158, EUCLID: 52;

      end;

      then

       A163: 1 > r8 by A153, XXREAL_0: 1;

      ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A164: ( E-max P) in ( Lower_Arc P) by A20, XBOOLE_0:def 4;

      

       A165: { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } c= ( Lower_Arc P)

      proof

        let x be object;

        assume x in { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 };

        then

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

         A166: p = x and

         A167: p in P and

         A168: (p `2 ) <= 0 ;

        now

          per cases by A168;

            case

             A169: (p `2 ) = 0 ;

            ex p8 be Point of ( TOP-REAL 2) st p8 = p & |.p8.| = 1 by A3, A167;

            

            then 1 = ( sqrt (((p `1 ) ^2 ) + ((p `2 ) ^2 ))) by JGRAPH_3: 1

            .= |.(p `1 ).| by A169, COMPLEX1: 72;

            then p = |[(p `1 ), (p `2 )]| & ((p `1 ) ^2 ) = (1 ^2 ) by COMPLEX1: 75, EUCLID: 53;

            then p = |[1, 0 ]| or p = |[( - 1), 0 ]| by A169, SQUARE_1: 41;

            hence thesis by A3, A164, A161, A166, Th29, Th30;

          end;

            case

             A170: (p `2 ) < 0 ;

            now

              assume not x in ( Lower_Arc P);

              then

               A171: x in ( Upper_Arc P) by A5, A166, A167, XBOOLE_0:def 3;

              ( rng f2) = ( [#] (( TOP-REAL 2) | R)) by A86, TOPS_2:def 5

              .= R by PRE_TOPC:def 5;

              then

              consider x2 be object such that

               A172: x2 in ( dom f2) and

               A173: p = (f2 . x2) by A166, A171, FUNCT_1:def 3;

              ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

              then x2 in { r where r be Real : 0 <= r & r <= 1 } by A172, RCOMP_1:def 1;

              then

              consider t2 be Real such that

               A174: x2 = t2 and

               A175: 0 <= t2 and

               A176: t2 <= 1;

              

               A177: ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

              now

                assume t2 = 1;

                then p = |[1, 0 ]| by A3, A88, A173, A174, Th30;

                hence contradiction by A170, EUCLID: 52;

              end;

              then

               A178: t2 < 1 by A176, XXREAL_0: 1;

               A179:

              now

                assume t2 = 0 ;

                then p = |[( - 1), 0 ]| by A3, A87, A173, A174, Th29;

                hence contradiction by A170, EUCLID: 52;

              end;

              ( |[ 0 , 1]| `1 ) = 0 by EUCLID: 52;

              

              then |. |[ 0 , 1]|.| = ( sqrt (( 0 ^2 ) + (1 ^2 ))) by A177, JGRAPH_3: 1

              .= 1 by SQUARE_1: 18;

              then

               A180: |[ 0 , 1]| in { q where q be Point of ( TOP-REAL 2) : |.q.| = 1 };

              now

                per cases by A3, A5, A180, XBOOLE_0:def 3;

                  case |[ 0 , 1]| in ( Lower_Arc P);

                  hence contradiction by A162, A147, A158, A160, A150, A151, A152, A159, A163, A32, A177, EUCLID: 52;

                end;

                  case |[ 0 , 1]| in ( Upper_Arc P);

                  hence contradiction by A94, A170, A173, A174, A175, A179, A178, A177;

                end;

              end;

              hence contradiction;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      ( Lower_Arc P) c= { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 }

      proof

        let x2 be object;

        assume

         A181: x2 in ( Lower_Arc P);

        then

        reconsider q3 = x2 as Point of ( TOP-REAL 2);

        (q3 `2 ) <= 0 by A162, A147, A158, A160, A150, A151, A152, A159, A163, A32, A181, EUCLID: 52;

        hence thesis by A6, A181;

      end;

      hence thesis by A165, XBOOLE_0:def 10;

    end;

    theorem :: JGRAPH_5:36

    

     Th36: for a,b,d,e be Real st a <= b & e > 0 holds ex f be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st f is being_homeomorphism & for r be Real st r in [.a, b.] holds (f . r) = ((e * r) + d)

    proof

      let a,b,d,e be Real;

      assume that

       A1: a <= b and

       A2: e > 0 ;

      set S = ( Closed-Interval-TSpace (a,b));

      defpred P[ object, object] means (for r be Real st $1 = r holds $2 = ((e * r) + d));

      set X = the carrier of ( Closed-Interval-TSpace (a,b));

      

       A3: X = [.a, b.] by A1, TOPMETR: 18;

      then

      reconsider B = the carrier of S as Subset of R^1 by TOPMETR: 17;

      

       A4: ( R^1 | B) = S by A1, A3, TOPMETR: 19;

      set T = ( Closed-Interval-TSpace (((e * a) + d),((e * b) + d)));

      set Y = the carrier of ( Closed-Interval-TSpace (((e * a) + d),((e * b) + d)));

      

       A5: (e * a) <= (e * b) by A1, A2, XREAL_1: 64;

      then

       A6: Y = [.((e * a) + d), ((e * b) + d).] by TOPMETR: 18, XREAL_1: 7;

      then

      reconsider C = the carrier of T as Subset of R^1 by TOPMETR: 17;

      defpred P1[ object, object] means for r be Real st r = $1 holds $2 = ((e * r) + d);

      T = ( TopSpaceMetr ( Closed-Interval-MSpace (((e * a) + d),((e * b) + d)))) by TOPMETR:def 7;

      then

       A7: T is T_2 by PCOMPS_1: 34;

      

       A8: for x be object st x in X holds ex y be object st y in Y & P[x, y]

      proof

        let x be object;

        assume

         A9: x in X;

        then

        reconsider r1 = x as Real;

        set y1 = ((e * r1) + d);

        r1 <= b by A3, A9, XXREAL_1: 1;

        then (e * r1) <= (e * b) by A2, XREAL_1: 64;

        then

         A10: y1 <= ((e * b) + d) by XREAL_1: 7;

        a <= r1 by A3, A9, XXREAL_1: 1;

        then (e * a) <= (e * r1) by A2, XREAL_1: 64;

        then ((e * a) + d) <= y1 by XREAL_1: 7;

        then (for r be Real st x = r holds y1 = ((e * r) + d)) & y1 in Y by A6, A10, XXREAL_1: 1;

        hence thesis;

      end;

      ex f be Function of X, Y st for x be object st x in X holds P[x, (f . x)] from FUNCT_2:sch 1( A8);

      then

      consider f1 be Function of X, Y such that

       A11: for x be object st x in X holds P[x, (f1 . x)];

      reconsider f2 = f1 as Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (((e * a) + d),((e * b) + d)));

      

       A12: for r be Real st r in [.a, b.] holds (f2 . r) = ((e * r) + d) by A3, A11;

      

       A13: ( dom f2) = the carrier of S by FUNCT_2:def 1;

      ( [#] T) c= ( rng f2)

      proof

        let y be object;

        assume

         A14: y in ( [#] T);

        then

        reconsider ry = y as Real;

        ry <= ((e * b) + d) by A6, A14, XXREAL_1: 1;

        then (((e * b) + d) - d) >= (ry - d) by XREAL_1: 9;

        then ((b * e) / e) >= ((ry - d) / e) by A2, XREAL_1: 72;

        then

         A15: b >= ((ry - d) / e) by A2, XCMPLX_1: 89;

        ((e * a) + d) <= ry by A6, A14, XXREAL_1: 1;

        then (((e * a) + d) - d) <= (ry - d) by XREAL_1: 9;

        then ((a * e) / e) <= ((ry - d) / e) by A2, XREAL_1: 72;

        then a <= ((ry - d) / e) by A2, XCMPLX_1: 89;

        then

         A16: ((ry - d) / e) in [.a, b.] by A15, XXREAL_1: 1;

        

        then (f2 . ((ry - d) / e)) = ((e * ((ry - d) / e)) + d) by A3, A11

        .= ((ry - d) + d) by A2, XCMPLX_1: 87

        .= ry;

        hence thesis by A3, A13, A16, FUNCT_1: 3;

      end;

      then

       A17: ( rng f2) = ( [#] T) by XBOOLE_0:def 10;

      then

      reconsider f3 = f1 as Function of S, R^1 by A6, A13, FUNCT_2: 2, TOPMETR: 17;

      for x1,x2 be object st x1 in ( dom f2) & x2 in ( dom f2) & (f2 . x1) = (f2 . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A18: x1 in ( dom f2) and

         A19: x2 in ( dom f2) and

         A20: (f2 . x1) = (f2 . x2);

        reconsider r2 = x2 as Real by A19;

        reconsider r1 = x1 as Real by A18;

        (f2 . x1) = ((e * r1) + d) by A11, A18;

        

        then (((e * r1) + d) - d) = (((e * r2) + d) - d) by A11, A19, A20

        .= (e * r2);

        then ((r1 * e) / e) = r2 by A2, XCMPLX_1: 89;

        hence thesis by A2, XCMPLX_1: 89;

      end;

      then

       A21: ( dom f2) = ( [#] S) & f2 is one-to-one by FUNCT_1:def 4, FUNCT_2:def 1;

      

       A22: for x be object st x in the carrier of R^1 holds ex y be object st y in the carrier of R^1 & P1[x, y]

      proof

        let x be object;

        assume x in the carrier of R^1 ;

        then

        reconsider rx = x as Real;

        reconsider ry = ((e * rx) + d) as Element of REAL by XREAL_0:def 1;

        for r be Real st r = x holds ry = ((e * r) + d);

        hence thesis by TOPMETR: 17;

      end;

      ex f4 be Function of the carrier of R^1 , the carrier of R^1 st for x be object st x in the carrier of R^1 holds P1[x, (f4 . x)] from FUNCT_2:sch 1( A22);

      then

      consider f4 be Function of the carrier of R^1 , the carrier of R^1 such that

       A23: for x be object st x in the carrier of R^1 holds P1[x, (f4 . x)];

      reconsider f5 = f4 as Function of R^1 , R^1 ;

      

       A24: for x be Real holds (f5 . x) = ((e * x) + d) by A23, TOPMETR: 17, XREAL_0:def 1;

      

       A25: (( dom f5) /\ B) = ( REAL /\ B) by FUNCT_2:def 1, TOPMETR: 17

      .= B by TOPMETR: 17, XBOOLE_1: 28;

      

       A26: for x be object st x in ( dom f3) holds (f3 . x) = (f5 . x)

      proof

        let x be object;

        assume

         A27: x in ( dom f3);

        then

        reconsider rx = x as Element of REAL by A3, A13;

        (f4 . x) = ((e * rx) + d) by A23, TOPMETR: 17;

        hence thesis by A11, A27;

      end;

      ( dom f3) = B by FUNCT_2:def 1;

      then f3 = (f5 | B) by A25, A26, FUNCT_1: 46;

      then

       A28: f3 is continuous by A24, A4, TOPMETR: 7, TOPMETR: 21;

      

       A29: S is compact by A1, HEINE: 4;

      ( R^1 | C) = T by A5, A6, TOPMETR: 19, XREAL_1: 7;

      then f2 is being_homeomorphism by A21, A17, A28, A29, A7, COMPTS_1: 17, TOPMETR: 6;

      hence thesis by A12;

    end;

    theorem :: JGRAPH_5:37

    

     Th37: for a,b,d,e be Real st a <= b & e < 0 holds ex f be Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st f is being_homeomorphism & for r be Real st r in [.a, b.] holds (f . r) = ((e * r) + d)

    proof

      let a,b,d,e be Real;

      assume that

       A1: a <= b and

       A2: e < 0 ;

      set S = ( Closed-Interval-TSpace (a,b));

      defpred P[ object, object] means (for r be Real st $1 = r holds $2 = ((e * r) + d));

      set X = the carrier of ( Closed-Interval-TSpace (a,b));

      

       A3: X = [.a, b.] by A1, TOPMETR: 18;

      then

      reconsider B = the carrier of S as Subset of R^1 by TOPMETR: 17;

      

       A4: ( R^1 | B) = S by A1, A3, TOPMETR: 19;

      set T = ( Closed-Interval-TSpace (((e * b) + d),((e * a) + d)));

      set Y = the carrier of ( Closed-Interval-TSpace (((e * b) + d),((e * a) + d)));

      

       A5: (e * a) >= (e * b) by A1, A2, XREAL_1: 65;

      then

       A6: Y = [.((e * b) + d), ((e * a) + d).] by TOPMETR: 18, XREAL_1: 7;

      then

      reconsider C = the carrier of T as Subset of R^1 by TOPMETR: 17;

      defpred P1[ object, object] means for r be Real st r = $1 holds $2 = ((e * r) + d);

      T = ( TopSpaceMetr ( Closed-Interval-MSpace (((e * b) + d),((e * a) + d)))) by TOPMETR:def 7;

      then

       A7: T is T_2 by PCOMPS_1: 34;

      

       A8: for x be object st x in X holds ex y be object st y in Y & P[x, y]

      proof

        let x be object;

        assume

         A9: x in X;

        then

        reconsider r1 = x as Real;

        set y1 = ((e * r1) + d);

        r1 <= b by A3, A9, XXREAL_1: 1;

        then (e * r1) >= (e * b) by A2, XREAL_1: 65;

        then

         A10: y1 >= ((e * b) + d) by XREAL_1: 7;

        a <= r1 by A3, A9, XXREAL_1: 1;

        then (e * a) >= (e * r1) by A2, XREAL_1: 65;

        then ((e * a) + d) >= y1 by XREAL_1: 7;

        then (for r be Real st x = r holds y1 = ((e * r) + d)) & y1 in Y by A6, A10, XXREAL_1: 1;

        hence thesis;

      end;

      ex f be Function of X, Y st for x be object st x in X holds P[x, (f . x)] from FUNCT_2:sch 1( A8);

      then

      consider f1 be Function of X, Y such that

       A11: for x be object st x in X holds P[x, (f1 . x)];

      reconsider f2 = f1 as Function of ( Closed-Interval-TSpace (a,b)), ( Closed-Interval-TSpace (((e * b) + d),((e * a) + d)));

      

       A12: for r be Real st r in [.a, b.] holds (f2 . r) = ((e * r) + d) by A3, A11;

      

       A13: ( dom f2) = the carrier of S by FUNCT_2:def 1;

      ( [#] T) c= ( rng f2)

      proof

        let y be object;

        assume

         A14: y in ( [#] T);

        then

        reconsider ry = y as Real;

        ry <= ((e * a) + d) by A6, A14, XXREAL_1: 1;

        then (((e * a) + d) - d) >= (ry - d) by XREAL_1: 9;

        then ((a * e) / e) <= ((ry - d) / e) by A2, XREAL_1: 73;

        then

         A15: a <= ((ry - d) / e) by A2, XCMPLX_1: 89;

        ((e * b) + d) <= ry by A6, A14, XXREAL_1: 1;

        then (((e * b) + d) - d) <= (ry - d) by XREAL_1: 9;

        then ((b * e) / e) >= ((ry - d) / e) by A2, XREAL_1: 73;

        then b >= ((ry - d) / e) by A2, XCMPLX_1: 89;

        then

         A16: ((ry - d) / e) in [.a, b.] by A15, XXREAL_1: 1;

        

        then (f2 . ((ry - d) / e)) = ((e * ((ry - d) / e)) + d) by A3, A11

        .= ((ry - d) + d) by A2, XCMPLX_1: 87

        .= ry;

        hence thesis by A3, A13, A16, FUNCT_1: 3;

      end;

      then

       A17: ( rng f2) = ( [#] T) by XBOOLE_0:def 10;

      then

      reconsider f3 = f1 as Function of S, R^1 by A6, A13, FUNCT_2: 2, TOPMETR: 17;

      for x1,x2 be object st x1 in ( dom f2) & x2 in ( dom f2) & (f2 . x1) = (f2 . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A18: x1 in ( dom f2) and

         A19: x2 in ( dom f2) and

         A20: (f2 . x1) = (f2 . x2);

        reconsider r2 = x2 as Real by A19;

        reconsider r1 = x1 as Real by A18;

        (f2 . x1) = ((e * r1) + d) by A11, A18;

        

        then (((e * r1) + d) - d) = (((e * r2) + d) - d) by A11, A19, A20

        .= (e * r2);

        then ((r1 * e) / e) = r2 by A2, XCMPLX_1: 89;

        hence thesis by A2, XCMPLX_1: 89;

      end;

      then

       A21: ( dom f2) = ( [#] S) & f2 is one-to-one by FUNCT_1:def 4, FUNCT_2:def 1;

      

       A22: for x be object st x in the carrier of R^1 holds ex y be object st y in the carrier of R^1 & P1[x, y]

      proof

        let x be object;

        assume x in the carrier of R^1 ;

        then

        reconsider rx = x as Real;

        reconsider ry = ((e * rx) + d) as Element of REAL by XREAL_0:def 1;

        for r be Real st r = x holds ry = ((e * r) + d);

        hence thesis by TOPMETR: 17;

      end;

      ex f4 be Function of the carrier of R^1 , the carrier of R^1 st for x be object st x in the carrier of R^1 holds P1[x, (f4 . x)] from FUNCT_2:sch 1( A22);

      then

      consider f4 be Function of the carrier of R^1 , the carrier of R^1 such that

       A23: for x be object st x in the carrier of R^1 holds P1[x, (f4 . x)];

      reconsider f5 = f4 as Function of R^1 , R^1 ;

      

       A24: for x be Real holds (f5 . x) = ((e * x) + d) by XREAL_0:def 1, TOPMETR: 17, A23;

      

       A25: (( dom f5) /\ B) = ( REAL /\ B) by FUNCT_2:def 1, TOPMETR: 17

      .= B by TOPMETR: 17, XBOOLE_1: 28;

      

       A26: for x be object st x in ( dom f3) holds (f3 . x) = (f5 . x)

      proof

        let x be object;

        assume

         A27: x in ( dom f3);

        then

        reconsider rx = x as Element of REAL by A3, A13;

        (f4 . x) = ((e * rx) + d) by A23, TOPMETR: 17;

        hence thesis by A11, A27;

      end;

      ( dom f3) = B by FUNCT_2:def 1;

      then f3 = (f5 | B) by A25, A26, FUNCT_1: 46;

      then

       A28: f3 is continuous by A24, A4, TOPMETR: 7, TOPMETR: 21;

      

       A29: S is compact by A1, HEINE: 4;

      ( R^1 | C) = T by A5, A6, TOPMETR: 19, XREAL_1: 7;

      then f2 is being_homeomorphism by A21, A17, A28, A29, A7, COMPTS_1: 17, TOPMETR: 6;

      hence thesis by A12;

    end;

    theorem :: JGRAPH_5:38

    

     Th38: ex f be Function of I[01] , ( Closed-Interval-TSpace (( - 1),1)) st f is being_homeomorphism & (for r be Real st r in [. 0 , 1.] holds (f . r) = ((( - 2) * r) + 1)) & (f . 0 ) = 1 & (f . 1) = ( - 1)

    proof

      consider f be Function of I[01] , ( Closed-Interval-TSpace (((( - 2) * 1) + 1),((( - 2) * 0 ) + 1))) such that

       A1: f is being_homeomorphism and

       A2: for r be Real st r in [. 0 , 1.] holds (f . r) = ((( - 2) * r) + 1) by Th37, TOPMETR: 20;

      1 in [. 0 , 1.] by XXREAL_1: 1;

      then

       A3: (f . 1) = ( - 1) by A2;

      (f . 0 ) = ((( - 2) * 0 ) + 1) by A2, Lm1;

      hence thesis by A1, A2, A3;

    end;

    theorem :: JGRAPH_5:39

    

     Th39: ex f be Function of I[01] , ( Closed-Interval-TSpace (( - 1),1)) st f is being_homeomorphism & (for r be Real st r in [. 0 , 1.] holds (f . r) = ((2 * r) - 1)) & (f . 0 ) = ( - 1) & (f . 1) = 1

    proof

      consider f be Function of I[01] , ( Closed-Interval-TSpace (((2 * 0 ) + ( - 1)),((2 * 1) + ( - 1)))) such that

       A1: f is being_homeomorphism and

       A2: for r be Real st r in [. 0 , 1.] holds (f . r) = ((2 * r) + ( - 1)) by Th36, TOPMETR: 20;

      

       A3: for r be Real st r in [. 0 , 1.] holds (f . r) = ((2 * r) - 1)

      proof

        let r be Real;

        assume r in [. 0 , 1.];

        

        hence (f . r) = ((2 * r) + ( - 1)) by A2

        .= ((2 * r) - 1);

      end;

      1 in [. 0 , 1.] by XXREAL_1: 1;

      

      then

       A4: (f . 1) = ((2 * 1) - 1) by A3

      .= 1;

      (f . 0 ) = ((2 * 0 ) - 1) by A3, Lm1

      .= ( - 1);

      hence thesis by A1, A3, A4;

    end;

     Lm5:

    now

      reconsider B = [.( - 1), 1.] as non empty Subset of R^1 by TOPMETR: 17, XXREAL_1: 1;

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

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

      set K0 = ( Lower_Arc P);

      reconsider g2 = (g | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

      

       A1: for p be Point of (( TOP-REAL 2) | K0) holds (g2 . p) = ( proj1 . p)

      proof

        let p be Point of (( TOP-REAL 2) | K0);

        p in the carrier of (( TOP-REAL 2) | K0);

        then p in K0 by PRE_TOPC: 8;

        hence thesis by FUNCT_1: 49;

      end;

      assume

       A2: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

       A3: K0 c= P by Th33;

      

       A4: ( dom g2) = the carrier of (( TOP-REAL 2) | K0) by FUNCT_2:def 1;

      then

       A5: ( dom g2) = K0 by PRE_TOPC: 8;

      ( rng g2) c= the carrier of ( Closed-Interval-TSpace (( - 1),1))

      proof

        let x be object;

        assume x in ( rng g2);

        then

        consider z be object such that

         A6: z in ( dom g2) and

         A7: x = (g2 . z) by FUNCT_1:def 3;

        z in P by A5, A3, A6;

        then

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

         A8: p = z and

         A9: |.p.| = 1 by A2;

        (1 ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by A9, JGRAPH_3: 1;

        then (1 - ((p `1 ) ^2 )) >= 0 by XREAL_1: 63;

        then ( - (1 - ((p `1 ) ^2 ))) <= 0 ;

        then (((p `1 ) ^2 ) - 1) <= 0 ;

        then

         A10: ( - 1) <= (p `1 ) & (p `1 ) <= 1 by SQUARE_1: 43;

        x = ( proj1 . p) by A1, A6, A7, A8

        .= (p `1 ) by PSCOMP_1:def 5;

        then x in [.( - 1), 1.] by A10, XXREAL_1: 1;

        hence thesis by TOPMETR: 18;

      end;

      then

      reconsider g3 = g2 as Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A4, FUNCT_2: 2;

      ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) by FUNCT_2:def 1;

      then

       A11: ( dom g3) = K0 by PRE_TOPC:def 5;

      

       A12: for x,y be object st x in ( dom g3) & y in ( dom g3) & (g3 . x) = (g3 . y) holds x = y

      proof

        let x,y be object;

        assume that

         A13: x in ( dom g3) and

         A14: y in ( dom g3) and

         A15: (g3 . x) = (g3 . y);

        reconsider p2 = y as Point of ( TOP-REAL 2) by A11, A14;

        

         A16: (g3 . y) = ( proj1 . p2) by A1, A14

        .= (p2 `1 ) by PSCOMP_1:def 5;

        reconsider p1 = x as Point of ( TOP-REAL 2) by A11, A13;

        

         A17: (g3 . x) = ( proj1 . p1) by A1, A13

        .= (p1 `1 ) by PSCOMP_1:def 5;

        p2 in P by A3, A11, A14;

        then ex p22 be Point of ( TOP-REAL 2) st p2 = p22 & |.p22.| = 1 by A2;

        then

         A18: (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

        p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A2, A11, A14, Th35;

        then

         A19: ex p44 be Point of ( TOP-REAL 2) st p2 = p44 & p44 in P & (p44 `2 ) <= 0 ;

        p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A2, A11, A13, Th35;

        then

         A20: ex p33 be Point of ( TOP-REAL 2) st p1 = p33 & p33 in P & (p33 `2 ) <= 0 ;

        p1 in P by A3, A11, A13;

        then ex p11 be Point of ( TOP-REAL 2) st p1 = p11 & |.p11.| = 1 by A2;

        then (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

        then (( - (p1 `2 )) ^2 ) = ((p2 `2 ) ^2 ) by A15, A17, A16, A18;

        then ( - (p1 `2 )) = ( sqrt (( - (p2 `2 )) ^2 )) by A20, SQUARE_1: 22;

        then ( - (p1 `2 )) = ( - (p2 `2 )) by A19, SQUARE_1: 22;

        

        then p1 = |[(p2 `1 ), (p2 `2 )]| by A15, A17, A16, EUCLID: 53

        .= p2 by EUCLID: 53;

        hence thesis;

      end;

      

       A21: ( [#] ( Closed-Interval-TSpace (( - 1),1))) c= ( rng g3)

      proof

        let x be object;

        assume x in ( [#] ( Closed-Interval-TSpace (( - 1),1)));

        then

         A22: x in [.( - 1), 1.] by TOPMETR: 18;

        then

        reconsider r = x as Real;

        ( - 1) <= r & r <= 1 by A22, XXREAL_1: 1;

        then (1 ^2 ) >= (r ^2 ) by SQUARE_1: 49;

        then

         A23: (1 - (r ^2 )) >= 0 by XREAL_1: 48;

        set q = |[r, ( - ( sqrt (1 - (r ^2 ))))]|;

        

         A24: (q `2 ) = ( - ( sqrt (1 - (r ^2 )))) by EUCLID: 52;

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

        .= ( sqrt ((r ^2 ) + ((q `2 ) ^2 ))) by EUCLID: 52

        .= ( sqrt ((r ^2 ) + (( - ( sqrt (1 - (r ^2 )))) ^2 ))) by EUCLID: 52

        .= ( sqrt ((r ^2 ) + (( sqrt (1 - (r ^2 ))) ^2 )));

        

        then |.q.| = ( sqrt ((r ^2 ) + (1 - (r ^2 )))) by A23, SQUARE_1:def 2

        .= 1 by SQUARE_1: 18;

        then

         A25: q in P by A2;

         0 <= ( sqrt (1 - (r ^2 ))) by A23, SQUARE_1:def 2;

        then q in { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A25, A24;

        then

         A26: q in ( dom g3) by A2, A11, Th35;

        

        then (g3 . q) = ( proj1 . q) by A1

        .= (q `1 ) by PSCOMP_1:def 5

        .= r by EUCLID: 52;

        hence thesis by A26, FUNCT_1:def 3;

      end;

      

       A27: ( Closed-Interval-TSpace (( - 1),1)) = ( R^1 | B) by TOPMETR: 19;

      g2 is continuous by A1, JGRAPH_2: 29;

      hence ( proj1 | K0) is continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) & ( proj1 | K0) is one-to-one & ( rng ( proj1 | K0)) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A21, A27, A12, FUNCT_1:def 4, JGRAPH_1: 45, XBOOLE_0:def 10;

    end;

     Lm6:

    now

      reconsider B = [.( - 1), 1.] as non empty Subset of R^1 by TOPMETR: 17, XXREAL_1: 1;

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

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

      set K0 = ( Upper_Arc P);

      reconsider g2 = (g | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

      

       A1: for p be Point of (( TOP-REAL 2) | K0) holds (g2 . p) = ( proj1 . p)

      proof

        let p be Point of (( TOP-REAL 2) | K0);

        p in the carrier of (( TOP-REAL 2) | K0);

        then p in K0 by PRE_TOPC: 8;

        hence thesis by FUNCT_1: 49;

      end;

      assume

       A2: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

       A3: K0 c= P by Th33;

      

       A4: ( dom g2) = the carrier of (( TOP-REAL 2) | K0) by FUNCT_2:def 1;

      then

       A5: ( dom g2) = K0 by PRE_TOPC: 8;

      ( rng g2) c= the carrier of ( Closed-Interval-TSpace (( - 1),1))

      proof

        let x be object;

        assume x in ( rng g2);

        then

        consider z be object such that

         A6: z in ( dom g2) and

         A7: x = (g2 . z) by FUNCT_1:def 3;

        z in P by A5, A3, A6;

        then

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

         A8: p = z and

         A9: |.p.| = 1 by A2;

        (1 ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by A9, JGRAPH_3: 1;

        then (1 - ((p `1 ) ^2 )) >= 0 by XREAL_1: 63;

        then ( - (1 - ((p `1 ) ^2 ))) <= 0 ;

        then (((p `1 ) ^2 ) - 1) <= 0 ;

        then

         A10: ( - 1) <= (p `1 ) & (p `1 ) <= 1 by SQUARE_1: 43;

        x = ( proj1 . p) by A1, A6, A7, A8

        .= (p `1 ) by PSCOMP_1:def 5;

        then x in [.( - 1), 1.] by A10, XXREAL_1: 1;

        hence thesis by TOPMETR: 18;

      end;

      then

      reconsider g3 = g2 as Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A4, FUNCT_2: 2;

      ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) by FUNCT_2:def 1;

      then

       A11: ( dom g3) = K0 by PRE_TOPC:def 5;

      

       A12: for x,y be object st x in ( dom g3) & y in ( dom g3) & (g3 . x) = (g3 . y) holds x = y

      proof

        let x,y be object;

        assume that

         A13: x in ( dom g3) and

         A14: y in ( dom g3) and

         A15: (g3 . x) = (g3 . y);

        reconsider p2 = y as Point of ( TOP-REAL 2) by A11, A14;

        

         A16: (g3 . y) = ( proj1 . p2) by A1, A14

        .= (p2 `1 ) by PSCOMP_1:def 5;

        reconsider p1 = x as Point of ( TOP-REAL 2) by A11, A13;

        

         A17: (g3 . x) = ( proj1 . p1) by A1, A13

        .= (p1 `1 ) by PSCOMP_1:def 5;

        p2 in P by A3, A11, A14;

        then ex p22 be Point of ( TOP-REAL 2) st p2 = p22 & |.p22.| = 1 by A2;

        then

         A18: (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

        p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A2, A11, A14, Th34;

        then

         A19: ex p44 be Point of ( TOP-REAL 2) st p2 = p44 & p44 in P & (p44 `2 ) >= 0 ;

        p1 in P by A3, A11, A13;

        then ex p11 be Point of ( TOP-REAL 2) st p1 = p11 & |.p11.| = 1 by A2;

        then

         A20: (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

        p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A2, A11, A13, Th34;

        then ex p33 be Point of ( TOP-REAL 2) st p1 = p33 & p33 in P & (p33 `2 ) >= 0 ;

        then (p1 `2 ) = ( sqrt ((p2 `2 ) ^2 )) by A15, A17, A16, A18, A20, SQUARE_1: 22;

        then (p1 `2 ) = (p2 `2 ) by A19, SQUARE_1: 22;

        

        then p1 = |[(p2 `1 ), (p2 `2 )]| by A15, A17, A16, EUCLID: 53

        .= p2 by EUCLID: 53;

        hence thesis;

      end;

      

       A21: ( [#] ( Closed-Interval-TSpace (( - 1),1))) c= ( rng g3)

      proof

        let x be object;

        assume x in ( [#] ( Closed-Interval-TSpace (( - 1),1)));

        then

         A22: x in [.( - 1), 1.] by TOPMETR: 18;

        then

        reconsider r = x as Real;

        ( - 1) <= r & r <= 1 by A22, XXREAL_1: 1;

        then (1 ^2 ) >= (r ^2 ) by SQUARE_1: 49;

        then

         A23: (1 - (r ^2 )) >= 0 by XREAL_1: 48;

        set q = |[r, ( sqrt (1 - (r ^2 )))]|;

        

         A24: (q `2 ) = ( sqrt (1 - (r ^2 ))) by EUCLID: 52;

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

        .= ( sqrt ((r ^2 ) + ((q `2 ) ^2 ))) by EUCLID: 52

        .= ( sqrt ((r ^2 ) + (( sqrt (1 - (r ^2 ))) ^2 ))) by EUCLID: 52;

        

        then |.q.| = ( sqrt ((r ^2 ) + (1 - (r ^2 )))) by A23, SQUARE_1:def 2

        .= 1 by SQUARE_1: 18;

        then

         A25: q in P by A2;

         0 <= ( sqrt (1 - (r ^2 ))) by A23, SQUARE_1:def 2;

        then q in { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A25, A24;

        then

         A26: q in ( dom g3) by A2, A11, Th34;

        

        then (g3 . q) = ( proj1 . q) by A1

        .= (q `1 ) by PSCOMP_1:def 5

        .= r by EUCLID: 52;

        hence thesis by A26, FUNCT_1:def 3;

      end;

      

       A27: ( Closed-Interval-TSpace (( - 1),1)) = ( R^1 | B) by TOPMETR: 19;

      g2 is continuous by A1, JGRAPH_2: 29;

      hence ( proj1 | K0) is continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) & ( proj1 | K0) is one-to-one & ( rng ( proj1 | K0)) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A21, A27, A12, FUNCT_1:def 4, JGRAPH_1: 45, XBOOLE_0:def 10;

    end;

    theorem :: JGRAPH_5:40

    

     Th40: for P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds ex f be Function of ( Closed-Interval-TSpace (( - 1),1)), (( TOP-REAL 2) | ( Lower_Arc P)) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) holds (f . (q `1 )) = q) & (f . ( - 1)) = ( W-min P) & (f . 1) = ( E-max P)

    proof

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

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

      set P4 = ( Lower_Arc P);

      set K0 = ( Lower_Arc P);

      reconsider g2 = (g | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

      

       A1: for p be Point of (( TOP-REAL 2) | K0) holds (g2 . p) = ( proj1 . p)

      proof

        let p be Point of (( TOP-REAL 2) | K0);

        p in the carrier of (( TOP-REAL 2) | K0);

        then p in K0 by PRE_TOPC: 8;

        hence thesis by FUNCT_1: 49;

      end;

      assume

       A2: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

      reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by Lm5;

      

       A3: ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A2, Lm5;

      

       A4: P is being_simple_closed_curve by A2, JGRAPH_3: 26;

      then

       A5: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A6: ( E-max P) in ( Lower_Arc P) by A5, XBOOLE_0:def 4;

      ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

      then

       A7: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

      

       A8: g3 is one-to-one by A2, Lm5;

      

       A9: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) by FUNCT_2:def 1;

      then

       A10: ( dom g3) = K0 by PRE_TOPC:def 5;

      

       A11: g3 is onto by A3, FUNCT_2:def 3;

      

       A12: for q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) holds ((g3 /" ) . (q `1 )) = q

      proof

        reconsider g4 = g3 as Function;

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

        

         A13: q in ( dom g4) implies q = ((g4 " ) . (g4 . q)) & q = (((g4 " ) * g4) . q) by A8, FUNCT_1: 34;

        assume

         A14: q in ( Lower_Arc P);

        

        then (g3 . q) = ( proj1 . q) by A1, A10

        .= (q `1 ) by PSCOMP_1:def 5;

        hence thesis by A11, A9, A8, A14, A13, PRE_TOPC:def 5, TOPS_2:def 4;

      end;

      ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A15: ( W-min P) in ( Lower_Arc P) by A5, XBOOLE_0:def 4;

      

       A16: ( E-max P) = |[1, 0 ]| by A2, Th30;

      

       A17: ( W-min P) = |[( - 1), 0 ]| by A2, Th29;

      ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A4, JORDAN6:def 9;

      then K0 is non empty compact by JORDAN5A: 1;

      then

       A18: (g3 /" ) is being_homeomorphism by A3, A8, A7, COMPTS_1: 17, TOPS_2: 56;

      

       A19: ((g3 /" ) . 1) = ((g3 /" ) . ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ( E-max P) by A6, A12, A16;

      ((g3 /" ) . ( - 1)) = ((g3 /" ) . ( |[( - 1), 0 ]| `1 )) by EUCLID: 52

      .= ( W-min P) by A15, A12, A17;

      hence thesis by A18, A12, A19;

    end;

    theorem :: JGRAPH_5:41

    

     Th41: for P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds ex f be Function of ( Closed-Interval-TSpace (( - 1),1)), (( TOP-REAL 2) | ( Upper_Arc P)) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) holds (f . (q `1 )) = q) & (f . ( - 1)) = ( W-min P) & (f . 1) = ( E-max P)

    proof

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

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

      set P4 = ( Lower_Arc P);

      set K0 = ( Upper_Arc P);

      reconsider g2 = (g | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

      

       A1: for p be Point of (( TOP-REAL 2) | K0) holds (g2 . p) = ( proj1 . p)

      proof

        let p be Point of (( TOP-REAL 2) | K0);

        p in the carrier of (( TOP-REAL 2) | K0);

        then p in K0 by PRE_TOPC: 8;

        hence thesis by FUNCT_1: 49;

      end;

      assume

       A2: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

      reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by Lm6;

      

       A3: ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A2, Lm6;

      

       A4: P is being_simple_closed_curve by A2, JGRAPH_3: 26;

      then

       A5: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A6: ( E-max P) in ( Upper_Arc P) by A5, XBOOLE_0:def 4;

      ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

      then

       A7: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

      

       A8: g3 is one-to-one by A2, Lm6;

      

       A9: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) by FUNCT_2:def 1;

      then

       A10: ( dom g3) = K0 by PRE_TOPC:def 5;

      

       A11: g3 is onto by A3, FUNCT_2:def 3;

      

       A12: for q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) holds ((g3 /" ) . (q `1 )) = q

      proof

        reconsider g4 = g3 as Function;

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

        

         A13: q in ( dom g4) implies q = ((g4 " ) . (g4 . q)) & q = (((g4 " ) * g4) . q) by A8, FUNCT_1: 34;

        assume

         A14: q in ( Upper_Arc P);

        

        then (g3 . q) = ( proj1 . q) by A1, A10

        .= (q `1 ) by PSCOMP_1:def 5;

        hence thesis by A11, A9, A8, A14, A13, PRE_TOPC:def 5, TOPS_2:def 4;

      end;

      ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

      then

       A15: ( W-min P) in ( Upper_Arc P) by A5, XBOOLE_0:def 4;

      

       A16: ( E-max P) = |[1, 0 ]| by A2, Th30;

      

       A17: ( W-min P) = |[( - 1), 0 ]| by A2, Th29;

      ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A4, JORDAN6:def 8;

      then K0 is non empty compact by JORDAN5A: 1;

      then

       A18: (g3 /" ) is being_homeomorphism by A3, A8, A7, COMPTS_1: 17, TOPS_2: 56;

      

       A19: ((g3 /" ) . 1) = ((g3 /" ) . ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ( E-max P) by A6, A12, A16;

      ((g3 /" ) . ( - 1)) = ((g3 /" ) . ( |[( - 1), 0 ]| `1 )) by EUCLID: 52

      .= ( W-min P) by A15, A12, A17;

      hence thesis by A18, A12, A19;

    end;

    theorem :: JGRAPH_5:42

    

     Th42: for P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds ex f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc P)) st f is being_homeomorphism & (for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) > (q2 `1 )) & (f . 0 ) = ( E-max P) & (f . 1) = ( W-min P)

    proof

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

      reconsider T = (( TOP-REAL 2) | ( Lower_Arc P)) as non empty TopSpace;

      consider g be Function of I[01] , ( Closed-Interval-TSpace (( - 1),1)) such that

       A1: g is being_homeomorphism and

       A2: for r be Real st r in [. 0 , 1.] holds (g . r) = ((( - 2) * r) + 1) and

       A3: (g . 0 ) = 1 and

       A4: (g . 1) = ( - 1) by Th38;

      assume

       A5: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

      consider f1 be Function of ( Closed-Interval-TSpace (( - 1),1)), (( TOP-REAL 2) | ( Lower_Arc P)) such that

       A6: f1 is being_homeomorphism and

       A7: for q be Point of ( TOP-REAL 2) st q in ( Lower_Arc P) holds (f1 . (q `1 )) = q and

       A8: (f1 . ( - 1)) = ( W-min P) and

       A9: (f1 . 1) = ( E-max P) by Th40;

      reconsider h = (f1 * g) as Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc P));

      

       A10: ( dom h) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

      then 0 in ( dom h) by XXREAL_1: 1;

      then

       A11: (h . 0 ) = ( E-max P) by A9, A3, FUNCT_1: 12;

      

       A12: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (h . r1) = q1 & (h . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) > (q2 `1 )

      proof

        let q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real;

        assume that

         A13: (h . r1) = q1 and

         A14: (h . r2) = q2 and

         A15: r1 in [. 0 , 1.] and

         A16: r2 in [. 0 , 1.];

         A17:

        now

          set s1 = ((( - 2) * r2) + 1), s2 = ((( - 2) * r1) + 1);

          set p1 = |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]|, p2 = |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]|;

          

           A18: ( |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| `2 ) = ( - ( sqrt (1 - (s1 ^2 )))) by EUCLID: 52;

          r2 <= 1 by A16, XXREAL_1: 1;

          then (( - 2) * r2) >= (( - 2) * 1) by XREAL_1: 65;

          then ((( - 2) * r2) + 1) >= ((( - 2) * 1) + 1) by XREAL_1: 7;

          then

           A19: ( - 1) <= s1;

          r2 >= 0 by A16, XXREAL_1: 1;

          then ((( - 2) * r2) + 1) <= ((( - 2) * 0 ) + 1) by XREAL_1: 7;

          then (s1 ^2 ) <= (1 ^2 ) by A19, SQUARE_1: 49;

          then

           A20: (1 - (s1 ^2 )) >= 0 by XREAL_1: 48;

          then

           A21: ( sqrt (1 - (s1 ^2 ))) >= 0 by SQUARE_1:def 2;

           |.p1.| = ( sqrt (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ((s1 ^2 ) + (( sqrt (1 - (s1 ^2 ))) ^2 ))) by A18, EUCLID: 52

          .= ( sqrt ((s1 ^2 ) + (1 - (s1 ^2 )))) by A20, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p1 in P by A5;

          then p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A18, A21;

          then

           A22: ( |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| `1 ) = s1 & |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| in ( Lower_Arc P) by A5, Th35, EUCLID: 52;

          (g . r2) = ((( - 2) * r2) + 1) & ( dom h) = [. 0 , 1.] by A2, A16, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r2) = (f1 . s1) by A16, FUNCT_1: 12

          .= p1 by A7, A22;

          then

           A23: (q2 `1 ) = s1 by A14, EUCLID: 52;

          

           A24: ( |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| `1 ) = s2 by EUCLID: 52;

          r1 <= 1 by A15, XXREAL_1: 1;

          then (( - 2) * r1) >= (( - 2) * 1) by XREAL_1: 65;

          then ((( - 2) * r1) + 1) >= ((( - 2) * 1) + 1) by XREAL_1: 7;

          then

           A25: ( - 1) <= s2;

          r1 >= 0 by A15, XXREAL_1: 1;

          then ((( - 2) * r1) + 1) <= ((( - 2) * 0 ) + 1) by XREAL_1: 7;

          then (s2 ^2 ) <= (1 ^2 ) by A25, SQUARE_1: 49;

          then

           A26: (1 - (s2 ^2 )) >= 0 by XREAL_1: 48;

          then

           A27: ( sqrt (1 - (s2 ^2 ))) >= 0 by SQUARE_1:def 2;

          assume r2 < r1;

          then

           A28: (( - 2) * r2) > (( - 2) * r1) by XREAL_1: 69;

          

           A29: ( |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| `2 ) = ( - ( sqrt (1 - (s2 ^2 )))) by EUCLID: 52;

           |.p2.| = ( sqrt (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ((s2 ^2 ) + (( sqrt (1 - (s2 ^2 ))) ^2 ))) by A29, EUCLID: 52

          .= ( sqrt ((s2 ^2 ) + (1 - (s2 ^2 )))) by A26, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p2 in P by A5;

          then p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A29, A27;

          then

           A30: |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| in ( Lower_Arc P) by A5, Th35;

          (g . r1) = ((( - 2) * r1) + 1) & ( dom h) = [. 0 , 1.] by A2, A15, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r1) = (f1 . s2) by A15, FUNCT_1: 12

          .= p2 by A7, A24, A30;

          hence (q2 `1 ) > (q1 `1 ) by A13, A28, A23, A24, XREAL_1: 8;

        end;

         A31:

        now

          assume

           A32: (q1 `1 ) > (q2 `1 );

          now

            assume

             A33: r1 >= r2;

            now

              per cases by A33, XXREAL_0: 1;

                case r1 > r2;

                hence contradiction by A17, A32;

              end;

                case r1 = r2;

                hence contradiction by A13, A14, A32;

              end;

            end;

            hence contradiction;

          end;

          hence r1 < r2;

        end;

        now

          assume r1 < r2;

          then (( - 2) * r1) > (( - 2) * r2) by XREAL_1: 69;

          then

           A34: ((( - 2) * r1) + 1) > ((( - 2) * r2) + 1) by XREAL_1: 8;

          set s1 = ((( - 2) * r1) + 1), s2 = ((( - 2) * r2) + 1);

          set p1 = |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]|, p2 = |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]|;

          

           A35: ( |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| `2 ) = ( - ( sqrt (1 - (s1 ^2 )))) by EUCLID: 52;

          r1 <= 1 by A15, XXREAL_1: 1;

          then (( - 2) * r1) >= (( - 2) * 1) by XREAL_1: 65;

          then ((( - 2) * r1) + 1) >= ((( - 2) * 1) + 1) by XREAL_1: 7;

          then

           A36: ( - 1) <= s1;

          r1 >= 0 by A15, XXREAL_1: 1;

          then ((( - 2) * r1) + 1) <= ((( - 2) * 0 ) + 1) by XREAL_1: 7;

          then (s1 ^2 ) <= (1 ^2 ) by A36, SQUARE_1: 49;

          then

           A37: (1 - (s1 ^2 )) >= 0 by XREAL_1: 48;

          then

           A38: ( sqrt (1 - (s1 ^2 ))) >= 0 by SQUARE_1:def 2;

           |.p1.| = ( sqrt (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ((s1 ^2 ) + (( sqrt (1 - (s1 ^2 ))) ^2 ))) by A35, EUCLID: 52

          .= ( sqrt ((s1 ^2 ) + (1 - (s1 ^2 )))) by A37, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p1 in P by A5;

          then p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A35, A38;

          then

           A39: ( |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| `1 ) = s1 & |[s1, ( - ( sqrt (1 - (s1 ^2 ))))]| in ( Lower_Arc P) by A5, Th35, EUCLID: 52;

          (g . r1) = ((( - 2) * r1) + 1) & ( dom h) = [. 0 , 1.] by A2, A15, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r1) = (f1 . s1) by A15, FUNCT_1: 12

          .= p1 by A7, A39;

          then

           A40: (q1 `1 ) = s1 by A13, EUCLID: 52;

          

           A41: ( |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| `2 ) = ( - ( sqrt (1 - (s2 ^2 )))) by EUCLID: 52;

          r2 <= 1 by A16, XXREAL_1: 1;

          then (( - 2) * r2) >= (( - 2) * 1) by XREAL_1: 65;

          then ((( - 2) * r2) + 1) >= ((( - 2) * 1) + 1) by XREAL_1: 7;

          then

           A42: ( - 1) <= s2;

          r2 >= 0 by A16, XXREAL_1: 1;

          then ((( - 2) * r2) + 1) <= ((( - 2) * 0 ) + 1) by XREAL_1: 7;

          then (s2 ^2 ) <= (1 ^2 ) by A42, SQUARE_1: 49;

          then

           A43: (1 - (s2 ^2 )) >= 0 by XREAL_1: 48;

          then

           A44: ( sqrt (1 - (s2 ^2 ))) >= 0 by SQUARE_1:def 2;

           |.p2.| = ( sqrt (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 ))) by JGRAPH_3: 1

          .= ( sqrt ((s2 ^2 ) + (( sqrt (1 - (s2 ^2 ))) ^2 ))) by A41, EUCLID: 52

          .= ( sqrt ((s2 ^2 ) + (1 - (s2 ^2 )))) by A43, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p2 in P by A5;

          then p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) <= 0 } by A41, A44;

          then

           A45: ( |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| `1 ) = s2 & |[s2, ( - ( sqrt (1 - (s2 ^2 ))))]| in ( Lower_Arc P) by A5, Th35, EUCLID: 52;

          (g . r2) = ((( - 2) * r2) + 1) & ( dom h) = [. 0 , 1.] by A2, A16, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r2) = (f1 . s2) by A16, FUNCT_1: 12

          .= p2 by A7, A45;

          hence (q1 `1 ) > (q2 `1 ) by A14, A34, A40, EUCLID: 52;

        end;

        hence thesis by A31;

      end;

      1 in ( dom h) by A10, XXREAL_1: 1;

      then

       A46: (h . 1) = ( W-min P) by A8, A4, FUNCT_1: 12;

      reconsider f2 = f1 as Function of ( Closed-Interval-TSpace (( - 1),1)), T;

      (f2 * g) is being_homeomorphism by A6, A1, TOPS_2: 57;

      hence thesis by A12, A11, A46;

    end;

    theorem :: JGRAPH_5:43

    

     Th43: for P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds ex f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc P)) st f is being_homeomorphism & (for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) < (q2 `1 )) & (f . 0 ) = ( W-min P) & (f . 1) = ( E-max P)

    proof

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

      reconsider T = (( TOP-REAL 2) | ( Upper_Arc P)) as non empty TopSpace;

      consider g be Function of I[01] , ( Closed-Interval-TSpace (( - 1),1)) such that

       A1: g is being_homeomorphism and

       A2: for r be Real st r in [. 0 , 1.] holds (g . r) = ((2 * r) - 1) and

       A3: (g . 0 ) = ( - 1) and

       A4: (g . 1) = 1 by Th39;

      assume

       A5: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      then

      consider f1 be Function of ( Closed-Interval-TSpace (( - 1),1)), (( TOP-REAL 2) | ( Upper_Arc P)) such that

       A6: f1 is being_homeomorphism and

       A7: for q be Point of ( TOP-REAL 2) st q in ( Upper_Arc P) holds (f1 . (q `1 )) = q and

       A8: (f1 . ( - 1)) = ( W-min P) and

       A9: (f1 . 1) = ( E-max P) by Th41;

      reconsider h = (f1 * g) as Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc P));

      

       A10: ( dom h) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

      then 0 in ( dom h) by XXREAL_1: 1;

      then

       A11: (h . 0 ) = ( W-min P) by A8, A3, FUNCT_1: 12;

      

       A12: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (h . r1) = q1 & (h . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) < (q2 `1 )

      proof

        let q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real;

        assume that

         A13: (h . r1) = q1 and

         A14: (h . r2) = q2 and

         A15: r1 in [. 0 , 1.] and

         A16: r2 in [. 0 , 1.];

         A17:

        now

          r2 <= 1 by A16, XXREAL_1: 1;

          then (2 * r2) <= (2 * 1) by XREAL_1: 64;

          then

           A18: ((2 * r2) - 1) <= ((2 * 1) - 1) by XREAL_1: 9;

          r1 >= 0 by A15, XXREAL_1: 1;

          then

           A19: ((2 * r1) - 1) >= ((2 * 0 ) - 1) by XREAL_1: 9;

          set s1 = ((2 * r1) - 1), s2 = ((2 * r2) - 1);

          set p1 = |[s1, ( sqrt (1 - (s1 ^2 )))]|, p2 = |[s2, ( sqrt (1 - (s2 ^2 )))]|;

          

           A20: ( |[s1, ( sqrt (1 - (s1 ^2 )))]| `1 ) = s1 by EUCLID: 52;

          r2 >= 0 by A16, XXREAL_1: 1;

          then

           A21: ((2 * r2) - 1) >= ((2 * 0 ) - 1) by XREAL_1: 9;

          ((2 * 0 ) - 1) = ( - 1);

          then (s2 ^2 ) <= (1 ^2 ) by A18, A21, SQUARE_1: 49;

          then

           A22: (1 - (s2 ^2 )) >= 0 by XREAL_1: 48;

          then

           A23: ( sqrt (1 - (s2 ^2 ))) >= 0 by SQUARE_1:def 2;

          r1 <= 1 by A15, XXREAL_1: 1;

          then (2 * r1) <= (2 * 1) by XREAL_1: 64;

          then

           A24: ((2 * r1) - 1) <= ((2 * 1) - 1) by XREAL_1: 9;

          assume r1 > r2;

          then

           A25: (2 * r1) > (2 * r2) by XREAL_1: 68;

          ((2 * 0 ) - 1) = ( - 1);

          then (s1 ^2 ) <= (1 ^2 ) by A24, A19, SQUARE_1: 49;

          then

           A26: (1 - (s1 ^2 )) >= 0 by XREAL_1: 48;

          then

           A27: ( sqrt (1 - (s1 ^2 ))) >= 0 by SQUARE_1:def 2;

          

           A28: ( |[s1, ( sqrt (1 - (s1 ^2 )))]| `2 ) = ( sqrt (1 - (s1 ^2 ))) by EUCLID: 52;

          

          then |.p1.| = ( sqrt ((s1 ^2 ) + (( sqrt (1 - (s1 ^2 ))) ^2 ))) by A20, JGRAPH_3: 1

          .= ( sqrt ((s1 ^2 ) + (1 - (s1 ^2 )))) by A26, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p1 in P by A5;

          then p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A28, A27;

          then

           A29: |[s1, ( sqrt (1 - (s1 ^2 )))]| in ( Upper_Arc P) by A5, Th34;

          (g . r1) = ((2 * r1) - 1) & ( dom h) = [. 0 , 1.] by A2, A15, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r1) = (f1 . s1) by A15, FUNCT_1: 12

          .= p1 by A7, A20, A29;

          then

           A30: (q1 `1 ) = s1 by A13, EUCLID: 52;

          

           A31: ( |[s2, ( sqrt (1 - (s2 ^2 )))]| `1 ) = s2 by EUCLID: 52;

          

           A32: ( |[s2, ( sqrt (1 - (s2 ^2 )))]| `2 ) = ( sqrt (1 - (s2 ^2 ))) by EUCLID: 52;

          

          then |.p2.| = ( sqrt ((s2 ^2 ) + (( sqrt (1 - (s2 ^2 ))) ^2 ))) by A31, JGRAPH_3: 1

          .= ( sqrt ((s2 ^2 ) + (1 - (s2 ^2 )))) by A22, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p2 in P by A5;

          then p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A32, A23;

          then

           A33: |[s2, ( sqrt (1 - (s2 ^2 )))]| in ( Upper_Arc P) by A5, Th34;

          (g . r2) = ((2 * r2) - 1) & ( dom h) = [. 0 , 1.] by A2, A16, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r2) = (f1 . s2) by A16, FUNCT_1: 12

          .= p2 by A7, A31, A33;

          hence (q1 `1 ) > (q2 `1 ) by A14, A25, A30, A31, XREAL_1: 14;

        end;

         A34:

        now

          assume

           A35: (q1 `1 ) < (q2 `1 );

          now

            assume

             A36: r1 >= r2;

            now

              per cases by A36, XXREAL_0: 1;

                case r1 > r2;

                hence contradiction by A17, A35;

              end;

                case r1 = r2;

                hence contradiction by A13, A14, A35;

              end;

            end;

            hence contradiction;

          end;

          hence r1 < r2;

        end;

        now

          assume r2 > r1;

          then

           A37: (2 * r2) > (2 * r1) by XREAL_1: 68;

          set s1 = ((2 * r2) - 1), s2 = ((2 * r1) - 1);

          set p1 = |[s1, ( sqrt (1 - (s1 ^2 )))]|, p2 = |[s2, ( sqrt (1 - (s2 ^2 )))]|;

          

           A38: ( |[s1, ( sqrt (1 - (s1 ^2 )))]| `1 ) = s1 by EUCLID: 52;

          r2 >= 0 by A16, XXREAL_1: 1;

          then ((2 * r2) - 1) >= ((2 * 0 ) - 1) by XREAL_1: 9;

          then

           A39: ( - 1) <= s1;

          r2 <= 1 by A16, XXREAL_1: 1;

          then (2 * r2) <= (2 * 1) by XREAL_1: 64;

          then ((2 * r2) - 1) <= ((2 * 1) - 1) by XREAL_1: 9;

          then (s1 ^2 ) <= (1 ^2 ) by A39, SQUARE_1: 49;

          then

           A40: (1 - (s1 ^2 )) >= 0 by XREAL_1: 48;

          then

           A41: ( sqrt (1 - (s1 ^2 ))) >= 0 by SQUARE_1:def 2;

          

           A42: ( |[s1, ( sqrt (1 - (s1 ^2 )))]| `2 ) = ( sqrt (1 - (s1 ^2 ))) by EUCLID: 52;

          

          then |.p1.| = ( sqrt ((s1 ^2 ) + (( sqrt (1 - (s1 ^2 ))) ^2 ))) by A38, JGRAPH_3: 1

          .= ( sqrt ((s1 ^2 ) + (1 - (s1 ^2 )))) by A40, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p1 in P by A5;

          then p1 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A42, A41;

          then

           A43: |[s1, ( sqrt (1 - (s1 ^2 )))]| in ( Upper_Arc P) by A5, Th34;

          (g . r2) = ((2 * r2) - 1) & ( dom h) = [. 0 , 1.] by A2, A16, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r2) = (f1 . s1) by A16, FUNCT_1: 12

          .= p1 by A7, A38, A43;

          then

           A44: (q2 `1 ) = s1 by A14, EUCLID: 52;

          

           A45: ( |[s2, ( sqrt (1 - (s2 ^2 )))]| `1 ) = s2 by EUCLID: 52;

          r1 >= 0 by A15, XXREAL_1: 1;

          then ((2 * r1) - 1) >= ((2 * 0 ) - 1) by XREAL_1: 9;

          then

           A46: ( - 1) <= s2;

          r1 <= 1 by A15, XXREAL_1: 1;

          then (2 * r1) <= (2 * 1) by XREAL_1: 64;

          then ((2 * r1) - 1) <= ((2 * 1) - 1) by XREAL_1: 9;

          then (s2 ^2 ) <= (1 ^2 ) by A46, SQUARE_1: 49;

          then

           A47: (1 - (s2 ^2 )) >= 0 by XREAL_1: 48;

          then

           A48: ( sqrt (1 - (s2 ^2 ))) >= 0 by SQUARE_1:def 2;

          

           A49: ( |[s2, ( sqrt (1 - (s2 ^2 )))]| `2 ) = ( sqrt (1 - (s2 ^2 ))) by EUCLID: 52;

          

          then |.p2.| = ( sqrt ((s2 ^2 ) + (( sqrt (1 - (s2 ^2 ))) ^2 ))) by A45, JGRAPH_3: 1

          .= ( sqrt ((s2 ^2 ) + (1 - (s2 ^2 )))) by A47, SQUARE_1:def 2

          .= 1 by SQUARE_1: 18;

          then p2 in P by A5;

          then p2 in { p3 where p3 be Point of ( TOP-REAL 2) : p3 in P & (p3 `2 ) >= 0 } by A49, A48;

          then

           A50: |[s2, ( sqrt (1 - (s2 ^2 )))]| in ( Upper_Arc P) by A5, Th34;

          (g . r1) = ((2 * r1) - 1) & ( dom h) = [. 0 , 1.] by A2, A15, BORSUK_1: 40, FUNCT_2:def 1;

          

          then (h . r1) = (f1 . s2) by A15, FUNCT_1: 12

          .= p2 by A7, A45, A50;

          hence (q2 `1 ) > (q1 `1 ) by A13, A37, A44, A45, XREAL_1: 14;

        end;

        hence thesis by A34;

      end;

      1 in ( dom h) by A10, XXREAL_1: 1;

      then

       A51: (h . 1) = ( E-max P) by A9, A4, FUNCT_1: 12;

      reconsider f2 = f1 as Function of ( Closed-Interval-TSpace (( - 1),1)), T;

      (f2 * g) is being_homeomorphism by A6, A1, TOPS_2: 57;

      hence thesis by A12, A11, A51;

    end;

    theorem :: JGRAPH_5:44

    

     Th44: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p2 in ( Upper_Arc P) & LE (p1,p2,P) holds p1 in ( Upper_Arc P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p2 in ( Upper_Arc P) and

       A3: LE (p1,p2,P);

      set P4b = ( Lower_Arc P);

      

       A4: p1 in ( Upper_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P) or p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P) & LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P)) or p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P) & LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A3;

      

       A5: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A6: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by JORDAN6:def 9;

      

       A7: (( Upper_Arc P) /\ P4b) = {( W-min P), ( E-max P)} by A5, JORDAN6:def 9;

      then ( E-max P) in (( Upper_Arc P) /\ P4b) by TARSKI:def 2;

      then

       A8: ( E-max P) in ( Upper_Arc P) by XBOOLE_0:def 4;

      now

        assume

         A9: not p1 in ( Upper_Arc P);

        then p2 in (( Upper_Arc P) /\ P4b) by A2, A4, XBOOLE_0:def 4;

        then

         A10: p2 = ( E-max P) by A7, A4, A9, TARSKI:def 2;

        then LE (p2,p1,( Lower_Arc P),( E-max P),( W-min P)) by A6, A4, A9, JORDAN5C: 10;

        hence contradiction by A6, A8, A4, A9, A10, JORDAN5C: 12;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:45

    

     Th45: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & p1 <> p2 & (p1 `1 ) < 0 & (p1 `2 ) < 0 & (p2 `2 ) < 0 holds (p1 `1 ) > (p2 `1 ) & (p1 `2 ) < (p2 `2 )

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: p1 <> p2 and

       A4: (p1 `1 ) < 0 and

       A5: (p1 `2 ) < 0 and

       A6: (p2 `2 ) < 0 ;

      consider f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc P)) such that

       A7: f is being_homeomorphism and

       A8: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) > (q2 `1 ) and

       A9: (f . 0 ) = ( E-max P) & (f . 1) = ( W-min P) by A1, Th42;

      

       A10: ( rng f) = ( [#] (( TOP-REAL 2) | ( Lower_Arc P))) by A7, TOPS_2:def 5

      .= ( Lower_Arc P) by PRE_TOPC:def 5;

      

       A11: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

       A12:

      now

        assume p1 in ( Upper_Arc P);

        then ex p be Point of ( TOP-REAL 2) st p1 = p & p in P & (p `2 ) >= 0 by A11;

        hence contradiction by A5;

      end;

      then

       A13: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A2;

      p2 in ( Lower_Arc P) by A2, A12;

      then

      consider x2 be object such that

       A14: x2 in ( dom f) and

       A15: p2 = (f . x2) by A10, FUNCT_1:def 3;

      

       A16: ( dom f) = ( [#] I[01] ) by A7, TOPS_2:def 5

      .= [. 0 , 1.] by BORSUK_1: 40;

      reconsider r22 = x2 as Real by A14;

      

       A17: 0 <= r22 & r22 <= 1 by A14, A16, XXREAL_1: 1;

      p1 in ( Lower_Arc P) by A2, A12;

      then

      consider x1 be object such that

       A18: x1 in ( dom f) and

       A19: p1 = (f . x1) by A10, FUNCT_1:def 3;

      reconsider r11 = x1 as Real by A18;

      r11 <= 1 by A18, A16, XXREAL_1: 1;

      then

       A20: r11 <= r22 by A13, A7, A9, A19, A15, A17, JORDAN5C:def 3;

      

       A21: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then p1 in P by A2, JORDAN7: 5;

      then ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1;

      then (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

      then ((1 ^2 ) - ((p1 `1 ) ^2 )) = (( - (p1 `2 )) ^2 );

      then ( - (p1 `2 )) = ( sqrt ((1 ^2 ) - (( - (p1 `1 )) ^2 ))) by A5, SQUARE_1: 22;

      then

       A22: (p1 `2 ) = ( - ( sqrt ((1 ^2 ) - (( - (p1 `1 )) ^2 ))));

      p2 in P by A2, A21, JORDAN7: 5;

      then ex p4 be Point of ( TOP-REAL 2) st p4 = p2 & |.p4.| = 1 by A1;

      then

       A23: (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

      then ((1 ^2 ) - ((p2 `1 ) ^2 )) = (( - (p2 `2 )) ^2 );

      then ( - (p2 `2 )) = ( sqrt ((1 ^2 ) - (( - (p2 `1 )) ^2 ))) by A6, SQUARE_1: 22;

      then

       A24: (p2 `2 ) = ( - ( sqrt ((1 ^2 ) - (( - (p2 `1 )) ^2 ))));

      

       A25: r11 < r22 iff (p1 `1 ) > (p2 `1 ) by A8, A18, A19, A14, A15, A16;

      then ( - (p1 `1 )) < ( - (p2 `1 )) by A3, A19, A15, A20, XREAL_1: 24, XXREAL_0: 1;

      then (( - (p1 `1 )) ^2 ) < (( - (p2 `1 )) ^2 ) by A4, SQUARE_1: 16;

      then ((1 ^2 ) - (( - (p1 `1 )) ^2 )) > ((1 ^2 ) - (( - (p2 `1 )) ^2 )) by XREAL_1: 15;

      then ( sqrt ((1 ^2 ) - (( - (p1 `1 )) ^2 ))) > ( sqrt ((1 ^2 ) - (( - (p2 `1 )) ^2 ))) by A23, SQUARE_1: 27, XREAL_1: 63;

      hence thesis by A19, A15, A25, A20, A22, A24, XREAL_1: 24, XXREAL_0: 1;

    end;

    theorem :: JGRAPH_5:46

    

     Th46: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & p1 <> p2 & (p2 `1 ) < 0 & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 holds (p1 `1 ) < (p2 `1 ) & (p1 `2 ) < (p2 `2 )

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: p1 <> p2 and

       A4: (p2 `1 ) < 0 and

       A5: (p1 `2 ) >= 0 and

       A6: (p2 `2 ) >= 0 ;

      set P4 = ( Lower_Arc P);

      

       A7: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A8: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      

       A9: p1 in P by A2, A7, JORDAN7: 5;

       A10:

      now

        assume p2 = ( W-min P);

        then LE (p2,p1,P) by A7, A9, JORDAN7: 3;

        hence contradiction by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 57;

      end;

      

       A11: p2 in P by A2, A7, JORDAN7: 5;

      then ex p4 be Point of ( TOP-REAL 2) st p4 = p2 & |.p4.| = 1 by A1;

      then (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

      then

       A12: (p2 `2 ) = ( sqrt ((1 ^2 ) - (( - (p2 `1 )) ^2 ))) by A6, SQUARE_1: 22;

      

       A13: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

       A14:

      now

        assume

         A15: p2 in ( Lower_Arc P);

        p2 in ( Upper_Arc P) by A6, A11, A13;

        then p2 in {( W-min P), ( E-max P)} by A8, A15, XBOOLE_0:def 4;

        then

         A16: p2 = ( W-min P) or p2 = ( E-max P) by TARSKI:def 2;

        ( E-max P) = |[1, 0 ]| by A1, Th30;

        hence contradiction by A4, A10, A16, EUCLID: 52;

      end;

      then

       A17: LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P)) by A2;

      

       A18: ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A9;

      then (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

      then

       A19: (p1 `2 ) = ( sqrt ((1 ^2 ) - (( - (p1 `1 )) ^2 ))) by A5, SQUARE_1: 22;

      (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A18, JGRAPH_3: 1;

      then

       A20: ((1 ^2 ) - (( - (p1 `1 )) ^2 )) >= 0 by XREAL_1: 63;

      consider f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc P)) such that

       A21: f is being_homeomorphism and

       A22: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) < (q2 `1 ) and

       A23: (f . 0 ) = ( W-min P) & (f . 1) = ( E-max P) by A1, Th43;

      

       A24: ( rng f) = ( [#] (( TOP-REAL 2) | ( Upper_Arc P))) by A21, TOPS_2:def 5

      .= ( Upper_Arc P) by PRE_TOPC:def 5;

      p2 in ( Upper_Arc P) by A2, A14;

      then

      consider x2 be object such that

       A25: x2 in ( dom f) and

       A26: p2 = (f . x2) by A24, FUNCT_1:def 3;

      

       A27: ( dom f) = ( [#] I[01] ) by A21, TOPS_2:def 5

      .= [. 0 , 1.] by BORSUK_1: 40;

      reconsider r22 = x2 as Real by A25;

      

       A28: 0 <= r22 & r22 <= 1 by A25, A27, XXREAL_1: 1;

      p1 in ( Upper_Arc P) by A2, A14;

      then

      consider x1 be object such that

       A29: x1 in ( dom f) and

       A30: p1 = (f . x1) by A24, FUNCT_1:def 3;

      reconsider r11 = x1 as Real by A29;

      r11 <= 1 by A29, A27, XXREAL_1: 1;

      then

       A31: r11 <= r22 by A17, A21, A23, A30, A26, A28, JORDAN5C:def 3;

      

       A32: r11 < r22 iff (p1 `1 ) < (p2 `1 ) by A22, A29, A30, A25, A26, A27;

      then ( - (p1 `1 )) > ( - (p2 `1 )) by A3, A30, A26, A31, XREAL_1: 24, XXREAL_0: 1;

      then (( - (p1 `1 )) ^2 ) > (( - (p2 `1 )) ^2 ) by A4, SQUARE_1: 16;

      then ((1 ^2 ) - (( - (p1 `1 )) ^2 )) < ((1 ^2 ) - (( - (p2 `1 )) ^2 )) by XREAL_1: 15;

      hence thesis by A30, A26, A32, A31, A19, A12, A20, SQUARE_1: 27, XXREAL_0: 1;

    end;

    theorem :: JGRAPH_5:47

    

     Th47: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & p1 <> p2 & (p2 `2 ) >= 0 holds (p1 `1 ) < (p2 `1 )

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: p1 <> p2 and

       A4: (p2 `2 ) >= 0 ;

      

       A5: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A6: p1 in P by A2, JORDAN7: 5;

      set P4 = ( Lower_Arc P);

      

       A7: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      

       A8: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by A5, JORDAN6:def 9;

      

       A9: p2 in P by A2, A5, JORDAN7: 5;

       A10:

      now

         A11:

        now

          assume p2 = ( W-min P);

          then LE (p2,p1,P) by A5, A6, JORDAN7: 3;

          hence contradiction by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 57;

        end;

        assume

         A12: p2 in ( Lower_Arc P);

        p2 in ( Upper_Arc P) by A4, A9, A7;

        then p2 in {( W-min P), ( E-max P)} by A8, A12, XBOOLE_0:def 4;

        then p2 = ( W-min P) or p2 = ( E-max P) by TARSKI:def 2;

        then

         A13: p2 = |[1, 0 ]| by A1, A11, Th30;

        then

         A14: (p2 `1 ) = 1 by EUCLID: 52;

        

         A15: ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & |.p8.| = 1 by A1, A6;

         A16:

        now

          assume

           A17: (p1 `1 ) = 1;

          (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A15, JGRAPH_3: 1;

          then (p1 `2 ) = 0 by A17, XCMPLX_1: 6;

          hence contradiction by A3, A13, A17, EUCLID: 53;

        end;

        (p1 `1 ) <= 1 by A15, Th1;

        hence thesis by A14, A16, XXREAL_0: 1;

      end;

      now

        assume p2 = ( W-min P);

        then LE (p2,p1,P) by A5, A6, JORDAN7: 3;

        hence contradiction by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 57;

      end;

      then

       A18: p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P) & not p2 = ( W-min P) & LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P)) or (p1 `1 ) < (p2 `1 ) by A2, A10;

      consider f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc P)) such that

       A19: f is being_homeomorphism and

       A20: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) < (q2 `1 ) and

       A21: (f . 0 ) = ( W-min P) & (f . 1) = ( E-max P) by A1, Th43;

      

       A22: ( rng f) = ( [#] (( TOP-REAL 2) | ( Upper_Arc P))) by A19, TOPS_2:def 5

      .= ( Upper_Arc P) by PRE_TOPC:def 5;

      now

        per cases ;

          case

           A23: not (p1 `1 ) < (p2 `1 );

          then

          consider x1 be object such that

           A24: x1 in ( dom f) and

           A25: p1 = (f . x1) by A18, A22, FUNCT_1:def 3;

          consider x2 be object such that

           A26: x2 in ( dom f) and

           A27: p2 = (f . x2) by A18, A22, A23, FUNCT_1:def 3;

          

           A28: ( dom f) = ( [#] I[01] ) by A19, TOPS_2:def 5

          .= [. 0 , 1.] by BORSUK_1: 40;

          reconsider r22 = x2 as Real by A26;

          

           A29: 0 <= r22 & r22 <= 1 by A26, A28, XXREAL_1: 1;

          reconsider r11 = x1 as Real by A24;

          

           A30: r11 < r22 iff (p1 `1 ) < (p2 `1 ) by A20, A24, A25, A26, A27, A28;

          r11 <= 1 by A24, A28, XXREAL_1: 1;

          then r11 <= r22 or (p1 `1 ) < (p2 `1 ) by A18, A19, A21, A25, A27, A29, JORDAN5C:def 3;

          hence thesis by A3, A25, A27, A30, XXREAL_0: 1;

        end;

          case (p1 `1 ) < (p2 `1 );

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:48

    

     Th48: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & p1 <> p2 & (p1 `2 ) <= 0 & p1 <> ( W-min P) holds (p1 `1 ) > (p2 `1 )

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: p1 <> p2 and

       A4: (p1 `2 ) <= 0 and

       A5: p1 <> ( W-min P);

      

       A6: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A7: p2 in P by A2, JORDAN7: 5;

      set P4 = ( Lower_Arc P);

      

       A8: ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A1, Th35;

      

       A9: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by A6, JORDAN6:def 9;

      

       A10: p1 in P by A2, A6, JORDAN7: 5;

      now

        assume

         A11: p1 in ( Upper_Arc P);

        p1 in ( Lower_Arc P) by A4, A10, A8;

        then p1 in {( W-min P), ( E-max P)} by A9, A11, XBOOLE_0:def 4;

        then p1 = ( W-min P) or p1 = ( E-max P) by TARSKI:def 2;

        then

         A12: p1 = |[1, 0 ]| by A1, A5, Th30;

        then

         A13: (p1 `1 ) = 1 by EUCLID: 52;

        

         A14: ex p9 be Point of ( TOP-REAL 2) st p9 = p2 & |.p9.| = 1 by A1, A7;

         A15:

        now

          assume

           A16: (p2 `1 ) = 1;

          (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A14, JGRAPH_3: 1;

          then (p2 `2 ) = 0 by A16, XCMPLX_1: 6;

          hence contradiction by A3, A12, A16, EUCLID: 53;

        end;

        (p2 `1 ) <= 1 by A14, Th1;

        hence thesis by A13, A15, XXREAL_0: 1;

      end;

      then

       A17: p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P) & LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) or (p1 `1 ) > (p2 `1 ) by A2;

      consider f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc P)) such that

       A18: f is being_homeomorphism and

       A19: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) > (q2 `1 ) and

       A20: (f . 0 ) = ( E-max P) & (f . 1) = ( W-min P) by A1, Th42;

      

       A21: ( rng f) = ( [#] (( TOP-REAL 2) | ( Lower_Arc P))) by A18, TOPS_2:def 5

      .= ( Lower_Arc P) by PRE_TOPC:def 5;

      now

        per cases ;

          case

           A22: not (p1 `1 ) > (p2 `1 );

          then

          consider x1 be object such that

           A23: x1 in ( dom f) and

           A24: p1 = (f . x1) by A17, A21, FUNCT_1:def 3;

          consider x2 be object such that

           A25: x2 in ( dom f) and

           A26: p2 = (f . x2) by A17, A21, A22, FUNCT_1:def 3;

          

           A27: ( dom f) = ( [#] I[01] ) by A18, TOPS_2:def 5

          .= [. 0 , 1.] by BORSUK_1: 40;

          reconsider r22 = x2 as Real by A25;

          

           A28: 0 <= r22 & r22 <= 1 by A25, A27, XXREAL_1: 1;

          reconsider r11 = x1 as Real by A23;

          

           A29: r11 < r22 iff (p1 `1 ) > (p2 `1 ) by A19, A23, A24, A25, A26, A27;

          r11 <= 1 by A23, A27, XXREAL_1: 1;

          then r11 <= r22 or (p1 `1 ) > (p2 `1 ) by A17, A18, A20, A24, A26, A28, JORDAN5C:def 3;

          hence thesis by A3, A24, A26, A29, XXREAL_0: 1;

        end;

          case (p1 `1 ) > (p2 `1 );

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:49

    

     Th49: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & ((p2 `2 ) >= 0 or (p2 `1 ) >= 0 ) & LE (p1,p2,P) holds (p1 `2 ) >= 0 or (p1 `1 ) >= 0

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: (p2 `2 ) >= 0 or (p2 `1 ) >= 0 and

       A3: LE (p1,p2,P);

      

       A4: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      

       A5: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A6: p2 in P by A3, JORDAN7: 5;

      

       A7: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A5, JORDAN6:def 9;

      per cases by A2;

        suppose (p2 `2 ) >= 0 ;

        then p2 in ( Upper_Arc P) by A6, A4;

        then p1 in ( Upper_Arc P) by A1, A3, Th44;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & p8 in P & (p8 `2 ) >= 0 by A4;

        hence thesis;

      end;

        suppose

         A8: (p2 `2 ) < 0 & (p2 `1 ) >= 0 ;

        then not ex p8 be Point of ( TOP-REAL 2) st p8 = p2 & p8 in P & (p8 `2 ) >= 0 ;

        then

         A9: not p2 in ( Upper_Arc P) by A4;

        now

          per cases by A3, A9;

            case p1 in ( Upper_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P);

            then ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & p8 in P & (p8 `2 ) >= 0 by A4;

            hence thesis;

          end;

            case

             A10: p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P) & LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P));

            now

              assume

               A11: p1 = ( W-min P);

              then LE (p2,p1,( Lower_Arc P),( E-max P),( W-min P)) by A7, A10, JORDAN5C: 10;

              hence contradiction by A7, A10, A11, JORDAN5C: 12;

            end;

            hence thesis by A1, A3, A8, Th48;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JGRAPH_5:50

    

     Th50: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & p1 <> p2 & (p1 `1 ) >= 0 & (p2 `1 ) >= 0 holds (p1 `2 ) > (p2 `2 )

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: p1 <> p2 and

       A4: (p1 `1 ) >= 0 and

       A5: (p2 `1 ) >= 0 ;

      

       A6: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      

       A7: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A8: p2 in P by A2, JORDAN7: 5;

      then

       A9: ex p3 be Point of ( TOP-REAL 2) st p3 = p2 & |.p3.| = 1 by A1;

      ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      then

       A10: (( W-min P) `2 ) = 0 by EUCLID: 52;

      

       A11: ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A1, Th35;

      

       A12: p1 in P by A2, A7, JORDAN7: 5;

      then

       A13: ex p4 be Point of ( TOP-REAL 2) st p4 = p1 & |.p4.| = 1 by A1;

      now

        per cases ;

          case

           A14: (p1 `2 ) >= 0 & (p2 `2 ) >= 0 ;

          then (p1 `1 ) < (p2 `1 ) by A1, A2, A3, Th47;

          then ((p1 `1 ) ^2 ) < ((p2 `1 ) ^2 ) by A4, SQUARE_1: 16;

          then

           A15: ((1 ^2 ) - ((p1 `1 ) ^2 )) > ((1 ^2 ) - ((p2 `1 ) ^2 )) by XREAL_1: 15;

          (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A13, JGRAPH_3: 1;

          then

           A16: (p1 `2 ) = ( sqrt ((1 ^2 ) - ((p1 `1 ) ^2 ))) by A14, SQUARE_1: 22;

          

           A17: (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

          then (p2 `2 ) = ( sqrt ((1 ^2 ) - ((p2 `1 ) ^2 ))) by A14, SQUARE_1: 22;

          hence thesis by A15, A16, A17, SQUARE_1: 27, XREAL_1: 63;

        end;

          case (p1 `2 ) >= 0 & (p2 `2 ) < 0 ;

          hence thesis;

        end;

          case

           A18: (p1 `2 ) < 0 & (p2 `2 ) >= 0 ;

          then p1 in ( Lower_Arc P) & p2 in ( Upper_Arc P) by A12, A8, A6, A11;

          then LE (p2,p1,P) by A10, A18;

          hence contradiction by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 57;

        end;

          case

           A19: (p1 `2 ) < 0 & (p2 `2 ) < 0 ;

          ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A12;

          then

           A20: (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by JGRAPH_3: 1;

          then ((1 ^2 ) - ((p1 `1 ) ^2 )) = (( - (p1 `2 )) ^2 );

          then

           A21: ( - (p1 `2 )) = ( sqrt ((1 ^2 ) - ((p1 `1 ) ^2 ))) by A19, SQUARE_1: 22;

           not ex p be Point of ( TOP-REAL 2) st p = p1 & p in P & (p `2 ) >= 0 by A19;

          then

           A22: not p1 in ( Upper_Arc P) by A6;

          then

           A23: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A2;

          ex p4 be Point of ( TOP-REAL 2) st p4 = p2 & |.p4.| = 1 by A1, A8;

          then (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by JGRAPH_3: 1;

          then ((1 ^2 ) - ((p2 `1 ) ^2 )) = (( - (p2 `2 )) ^2 );

          then

           A24: ( - (p2 `2 )) = ( sqrt ((1 ^2 ) - ((p2 `1 ) ^2 ))) by A19, SQUARE_1: 22;

          consider f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc P)) such that

           A25: f is being_homeomorphism and

           A26: for q1,q2 be Point of ( TOP-REAL 2), r1,r2 be Real st (f . r1) = q1 & (f . r2) = q2 & r1 in [. 0 , 1.] & r2 in [. 0 , 1.] holds r1 < r2 iff (q1 `1 ) > (q2 `1 ) and

           A27: (f . 0 ) = ( E-max P) & (f . 1) = ( W-min P) by A1, Th42;

          

           A28: ( rng f) = ( [#] (( TOP-REAL 2) | ( Lower_Arc P))) by A25, TOPS_2:def 5

          .= ( Lower_Arc P) by PRE_TOPC:def 5;

          p2 in ( Lower_Arc P) by A2, A22;

          then

          consider x2 be object such that

           A29: x2 in ( dom f) and

           A30: p2 = (f . x2) by A28, FUNCT_1:def 3;

          

           A31: ( dom f) = ( [#] I[01] ) by A25, TOPS_2:def 5

          .= [. 0 , 1.] by BORSUK_1: 40;

          reconsider r22 = x2 as Real by A29;

          

           A32: 0 <= r22 & r22 <= 1 by A29, A31, XXREAL_1: 1;

          p1 in ( Lower_Arc P) by A2, A22;

          then

          consider x1 be object such that

           A33: x1 in ( dom f) and

           A34: p1 = (f . x1) by A28, FUNCT_1:def 3;

          reconsider r11 = x1 as Real by A33;

          

           A35: r11 < r22 iff (p1 `1 ) > (p2 `1 ) by A26, A33, A34, A29, A30, A31;

          r11 <= 1 by A33, A31, XXREAL_1: 1;

          then r11 <= r22 by A23, A25, A27, A34, A30, A32, JORDAN5C:def 3;

          then ((p1 `1 ) ^2 ) > ((p2 `1 ) ^2 ) by A3, A5, A34, A30, A35, SQUARE_1: 16, XXREAL_0: 1;

          then ((1 ^2 ) - ((p1 `1 ) ^2 )) < ((1 ^2 ) - ((p2 `1 ) ^2 )) by XREAL_1: 15;

          then ( sqrt ((1 ^2 ) - ((p1 `1 ) ^2 ))) < ( sqrt ((1 ^2 ) - ((p2 `1 ) ^2 ))) by A20, SQUARE_1: 27, XREAL_1: 63;

          hence thesis by A21, A24, XREAL_1: 24;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:51

    

     Th51: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `1 ) < 0 & (p2 `1 ) < 0 & (p1 `2 ) < 0 & (p2 `2 ) < 0 & ((p1 `1 ) >= (p2 `1 ) or (p1 `2 ) <= (p2 `2 )) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `1 ) < 0 and

       A5: (p2 `1 ) < 0 and

       A6: (p1 `2 ) < 0 and

       A7: (p2 `2 ) < 0 and

       A8: (p1 `1 ) >= (p2 `1 ) or (p1 `2 ) <= (p2 `2 );

      

       A9: ex p3 be Point of ( TOP-REAL 2) st p3 = p2 & |.p3.| = 1 by A1, A3;

      set P4 = ( Lower_Arc P);

      

       A10: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A11: (( Upper_Arc P) \/ P4) = P by JORDAN6:def 9;

      

       A12: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

       A13:

      now

        assume not p1 in ( Lower_Arc P);

        then p1 in ( Upper_Arc P) by A2, A11, XBOOLE_0:def 3;

        then ex p be Point of ( TOP-REAL 2) st p1 = p & p in P & (p `2 ) >= 0 by A12;

        hence contradiction by A6;

      end;

       A14:

      now

        assume not p2 in ( Lower_Arc P);

        then p2 in ( Upper_Arc P) by A3, A11, XBOOLE_0:def 3;

        then ex p be Point of ( TOP-REAL 2) st p2 = p & p in P & (p `2 ) >= 0 by A12;

        hence contradiction by A7;

      end;

      

       A15: ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A2;

       A16:

      now

        assume (p1 `2 ) <= (p2 `2 );

        then ( - (p1 `2 )) >= ( - (p2 `2 )) by XREAL_1: 24;

        then (( - (p1 `2 )) ^2 ) >= (( - (p2 `2 )) ^2 ) by A7, SQUARE_1: 15;

        then

         A17: ((1 ^2 ) - (( - (p1 `2 )) ^2 )) <= ((1 ^2 ) - (( - (p2 `2 )) ^2 )) by XREAL_1: 13;

        

         A18: (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A15, JGRAPH_3: 1;

        then ((1 ^2 ) - (( - (p1 `2 )) ^2 )) >= 0 by XREAL_1: 63;

        then

         A19: ( sqrt ((1 ^2 ) - (( - (p1 `2 )) ^2 ))) <= ( sqrt ((1 ^2 ) - (( - (p2 `2 )) ^2 ))) by A17, SQUARE_1: 26;

        (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

        then ((1 ^2 ) - (( - (p2 `2 )) ^2 )) = (( - (p2 `1 )) ^2 );

        then

         A20: ( - (p2 `1 )) = ( sqrt ((1 ^2 ) - (( - (p2 `2 )) ^2 ))) by A5, SQUARE_1: 22;

        ((1 ^2 ) - (( - (p1 `2 )) ^2 )) = (( - (p1 `1 )) ^2 ) by A18;

        then ( - (p1 `1 )) = ( sqrt ((1 ^2 ) - (( - (p1 `2 )) ^2 ))) by A4, SQUARE_1: 22;

        hence (p1 `1 ) >= (p2 `1 ) by A20, A19, XREAL_1: 24;

      end;

      

       A21: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by A10, JORDAN6:def 9;

      

       A22: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A10, JORDAN6:def 9;

      

       A23: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      for g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max P) & (g . 1) = ( W-min P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

      proof

        ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

        then

         A24: ( W-min P) in ( Lower_Arc P) by A21, XBOOLE_0:def 4;

        set K0 = ( Lower_Arc P);

        reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

        ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

        then

         A25: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

        reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm5;

        let g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real;

        assume that

         A26: g is being_homeomorphism and (g . 0 ) = ( E-max P) and

         A27: (g . 1) = ( W-min P) and

         A28: (g . s1) = p1 and

         A29: 0 <= s1 & s1 <= 1 and

         A30: (g . s2) = p2 and

         A31: 0 <= s2 & s2 <= 1;

        

         A32: s2 in [. 0 , 1.] by A31, XXREAL_1: 1;

        reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

        

         A33: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm5, FUNCT_2:def 1;

        g3 is one-to-one & K0 is non empty compact by A1, A22, Lm5, JORDAN5A: 1;

        then g3 is being_homeomorphism by A33, A25, COMPTS_1: 17;

        then

         A34: h is being_homeomorphism by A26, TOPMETR: 20, TOPS_2: 57;

        

         A35: ( dom g) = ( [#] I[01] ) by A26, TOPS_2:def 5

        .= [. 0 , 1.] by BORSUK_1: 40;

        then

         A36: 1 in ( dom g) by XXREAL_1: 1;

        

         A37: ( - 1) = ( |[( - 1), 0 ]| `1 ) by EUCLID: 52

        .= ( proj1 . |[( - 1), 0 ]|) by PSCOMP_1:def 5

        .= (g3 . (g . 1)) by A23, A27, A24, FUNCT_1: 49

        .= (h . 1) by A36, FUNCT_1: 13;

        

         A38: s1 in [. 0 , 1.] by A29, XXREAL_1: 1;

        

         A39: (p2 `1 ) = ( proj1 . p2) by PSCOMP_1:def 5

        .= (g3 . (g . s2)) by A14, A30, FUNCT_1: 49

        .= (h . s2) by A35, A32, FUNCT_1: 13;

        (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

        .= (g3 . (g . s1)) by A13, A28, FUNCT_1: 49

        .= (h . s1) by A35, A38, FUNCT_1: 13;

        hence thesis by A8, A16, A34, A38, A32, A37, A39, Th9;

      end;

      then

       A40: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A13, A14, JORDAN5C:def 3;

      now

        assume

         A41: p2 = ( W-min P);

        ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

        hence contradiction by A7, A41, EUCLID: 52;

      end;

      hence thesis by A13, A14, A40;

    end;

    theorem :: JGRAPH_5:52

    for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `1 ) > 0 & (p2 `1 ) > 0 & (p1 `2 ) < 0 & (p2 `2 ) < 0 & ((p1 `1 ) >= (p2 `1 ) or (p1 `2 ) >= (p2 `2 )) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `1 ) > 0 and

       A5: (p2 `1 ) > 0 and

       A6: (p1 `2 ) < 0 and

       A7: (p2 `2 ) < 0 and

       A8: (p1 `1 ) >= (p2 `1 ) or (p1 `2 ) >= (p2 `2 );

      

       A9: ex p3 be Point of ( TOP-REAL 2) st p3 = p2 & |.p3.| = 1 by A1, A3;

      set P4 = ( Lower_Arc P);

      

       A10: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A11: (( Upper_Arc P) \/ P4) = P by JORDAN6:def 9;

      

       A12: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

       A13:

      now

        assume not p1 in ( Lower_Arc P);

        then p1 in ( Upper_Arc P) by A2, A11, XBOOLE_0:def 3;

        then ex p be Point of ( TOP-REAL 2) st p1 = p & p in P & (p `2 ) >= 0 by A12;

        hence contradiction by A6;

      end;

       A14:

      now

        assume not p2 in ( Lower_Arc P);

        then p2 in ( Upper_Arc P) by A3, A11, XBOOLE_0:def 3;

        then ex p be Point of ( TOP-REAL 2) st p2 = p & p in P & (p `2 ) >= 0 by A12;

        hence contradiction by A7;

      end;

      

       A15: ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A2;

       A16:

      now

        assume (p1 `2 ) >= (p2 `2 );

        then ( - (p1 `2 )) <= ( - (p2 `2 )) by XREAL_1: 24;

        then (( - (p1 `2 )) ^2 ) <= (( - (p2 `2 )) ^2 ) by A6, SQUARE_1: 15;

        then

         A17: ((1 ^2 ) - (( - (p1 `2 )) ^2 )) >= ((1 ^2 ) - (( - (p2 `2 )) ^2 )) by XREAL_1: 13;

        (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

        then

         A18: (p2 `1 ) = ( sqrt ((1 ^2 ) - (( - (p2 `2 )) ^2 ))) & ((1 ^2 ) - (( - (p2 `2 )) ^2 )) >= 0 by A5, SQUARE_1: 22;

        (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A15, JGRAPH_3: 1;

        then (p1 `1 ) = ( sqrt ((1 ^2 ) - (( - (p1 `2 )) ^2 ))) by A4, SQUARE_1: 22;

        hence (p1 `1 ) >= (p2 `1 ) by A17, A18, SQUARE_1: 26;

      end;

      

       A19: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by A10, JORDAN6:def 9;

      

       A20: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A10, JORDAN6:def 9;

      

       A21: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      for g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max P) & (g . 1) = ( W-min P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

      proof

        ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

        then

         A22: ( W-min P) in ( Lower_Arc P) by A19, XBOOLE_0:def 4;

        set K0 = ( Lower_Arc P);

        reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

        ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

        then

         A23: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

        reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm5;

        let g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real;

        assume that

         A24: g is being_homeomorphism and (g . 0 ) = ( E-max P) and

         A25: (g . 1) = ( W-min P) and

         A26: (g . s1) = p1 and

         A27: 0 <= s1 & s1 <= 1 and

         A28: (g . s2) = p2 and

         A29: 0 <= s2 & s2 <= 1;

        

         A30: s2 in [. 0 , 1.] by A29, XXREAL_1: 1;

        reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

        

         A31: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm5, FUNCT_2:def 1;

        g3 is one-to-one & K0 is non empty compact by A1, A20, Lm5, JORDAN5A: 1;

        then g3 is being_homeomorphism by A31, A23, COMPTS_1: 17;

        then

         A32: h is being_homeomorphism by A24, TOPMETR: 20, TOPS_2: 57;

        

         A33: ( dom g) = ( [#] I[01] ) by A24, TOPS_2:def 5

        .= [. 0 , 1.] by BORSUK_1: 40;

        then

         A34: 1 in ( dom g) by XXREAL_1: 1;

        

         A35: ( - 1) = ( |[( - 1), 0 ]| `1 ) by EUCLID: 52

        .= ( proj1 . |[( - 1), 0 ]|) by PSCOMP_1:def 5

        .= (g3 . (g . 1)) by A21, A25, A22, FUNCT_1: 49

        .= (h . 1) by A34, FUNCT_1: 13;

        

         A36: s1 in [. 0 , 1.] by A27, XXREAL_1: 1;

        

         A37: (p2 `1 ) = ( proj1 . p2) by PSCOMP_1:def 5

        .= (g3 . p2) by A14, FUNCT_1: 49

        .= (h . s2) by A28, A33, A30, FUNCT_1: 13;

        (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

        .= (g3 . (g . s1)) by A13, A26, FUNCT_1: 49

        .= (h . s1) by A33, A36, FUNCT_1: 13;

        hence thesis by A8, A16, A32, A36, A30, A35, A37, Th9;

      end;

      then

       A38: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A13, A14, JORDAN5C:def 3;

      now

        assume

         A39: p2 = ( W-min P);

        ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

        hence contradiction by A5, A39, EUCLID: 52;

      end;

      hence thesis by A13, A14, A38;

    end;

    theorem :: JGRAPH_5:53

    

     Th53: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `1 ) < 0 & (p2 `1 ) < 0 & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 & ((p1 `1 ) <= (p2 `1 ) or (p1 `2 ) <= (p2 `2 )) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `1 ) < 0 and

       A5: (p2 `1 ) < 0 and

       A6: (p1 `2 ) >= 0 and

       A7: (p2 `2 ) >= 0 and

       A8: (p1 `1 ) <= (p2 `1 ) or (p1 `2 ) <= (p2 `2 );

      

       A9: ex p3 be Point of ( TOP-REAL 2) st p3 = p2 & |.p3.| = 1 by A1, A3;

      set P4b = ( Upper_Arc P);

      set P4 = ( Lower_Arc P);

      

       A10: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A11: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      

       A12: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      then

       A13: p1 in ( Upper_Arc P) by A2, A6;

      

       A14: p2 in ( Upper_Arc P) by A3, A7, A12;

      

       A15: ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A2;

       A16:

      now

        assume (p1 `2 ) <= (p2 `2 );

        then ((p1 `2 ) ^2 ) <= ((p2 `2 ) ^2 ) by A6, SQUARE_1: 15;

        then

         A17: ((1 ^2 ) - ((p1 `2 ) ^2 )) >= ((1 ^2 ) - ((p2 `2 ) ^2 )) by XREAL_1: 13;

        

         A18: (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

        then ((1 ^2 ) - ((p2 `2 ) ^2 )) >= 0 by XREAL_1: 63;

        then

         A19: ( sqrt ((1 ^2 ) - ((p1 `2 ) ^2 ))) >= ( sqrt ((1 ^2 ) - ((p2 `2 ) ^2 ))) by A17, SQUARE_1: 26;

        (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A15, JGRAPH_3: 1;

        then ((1 ^2 ) - ((p1 `2 ) ^2 )) = (( - (p1 `1 )) ^2 );

        then

         A20: ( - (p1 `1 )) = ( sqrt ((1 ^2 ) - ((p1 `2 ) ^2 ))) by A4, SQUARE_1: 22;

        ((1 ^2 ) - ((p2 `2 ) ^2 )) = (( - (p2 `1 )) ^2 ) by A18;

        then ( - (p2 `1 )) = ( sqrt ((1 ^2 ) - ((p2 `2 ) ^2 ))) by A5, SQUARE_1: 22;

        hence (p1 `1 ) <= (p2 `1 ) by A20, A19, XREAL_1: 24;

      end;

      

       A21: ( E-max P) = |[1, 0 ]| by A1, Th30;

      

       A22: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A10, JORDAN6:def 8;

      for g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min P) & (g . 1) = ( E-max P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

      proof

        ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

        then

         A23: ( E-max P) in ( Upper_Arc P) by A11, XBOOLE_0:def 4;

        set K0 = ( Upper_Arc P);

        reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

        ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

        then

         A24: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

        reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm6;

        let g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real;

        assume that

         A25: g is being_homeomorphism and (g . 0 ) = ( W-min P) and

         A26: (g . 1) = ( E-max P) and

         A27: (g . s1) = p1 and

         A28: 0 <= s1 & s1 <= 1 and

         A29: (g . s2) = p2 and

         A30: 0 <= s2 & s2 <= 1;

        

         A31: s2 in [. 0 , 1.] by A30, XXREAL_1: 1;

        reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

        

         A32: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm6, FUNCT_2:def 1;

        g3 is one-to-one & K0 is non empty compact by A1, A22, Lm6, JORDAN5A: 1;

        then g3 is being_homeomorphism by A32, A24, COMPTS_1: 17;

        then

         A33: h is being_homeomorphism by A25, TOPMETR: 20, TOPS_2: 57;

        

         A34: ( dom g) = ( [#] I[01] ) by A25, TOPS_2:def 5

        .= [. 0 , 1.] by BORSUK_1: 40;

        then

         A35: 1 in ( dom g) by XXREAL_1: 1;

        

         A36: 1 = ( |[1, 0 ]| `1 ) by EUCLID: 52

        .= (g0 . |[1, 0 ]|) by PSCOMP_1:def 5

        .= (g3 . |[1, 0 ]|) by A21, A23, FUNCT_1: 49

        .= (h . 1) by A21, A26, A35, FUNCT_1: 13;

        

         A37: s1 in [. 0 , 1.] by A28, XXREAL_1: 1;

        

         A38: (p2 `1 ) = (g0 . p2) by PSCOMP_1:def 5

        .= (g3 . p2) by A14, FUNCT_1: 49

        .= (h . s2) by A29, A34, A31, FUNCT_1: 13;

        (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

        .= (g3 . (g . s1)) by A13, A27, FUNCT_1: 49

        .= (h . s1) by A34, A37, FUNCT_1: 13;

        hence thesis by A8, A16, A33, A37, A31, A36, A38, Th8;

      end;

      then

       A39: LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P)) by A13, A14, JORDAN5C:def 3;

      p1 in ( Upper_Arc P) by A2, A6, A12;

      hence thesis by A14, A39;

    end;

    theorem :: JGRAPH_5:54

    

     Th54: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 & (p1 `1 ) <= (p2 `1 ) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `2 ) >= 0 and

       A5: (p2 `2 ) >= 0 and

       A6: (p1 `1 ) <= (p2 `1 );

      

       A7: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      then

       A8: p1 in ( Upper_Arc P) by A2, A4;

      

       A9: p2 in ( Upper_Arc P) by A3, A5, A7;

      set P4b = ( Upper_Arc P);

      set P4 = ( Lower_Arc P);

      

       A10: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A11: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      

       A12: ( E-max P) = |[1, 0 ]| by A1, Th30;

      

       A13: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A10, JORDAN6:def 8;

      for g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min P) & (g . 1) = ( E-max P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

      proof

        ( E-max P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

        then

         A14: ( E-max P) in ( Upper_Arc P) by A11, XBOOLE_0:def 4;

        set K0 = ( Upper_Arc P);

        reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

        ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

        then

         A15: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

        reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm6;

        let g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real;

        assume that

         A16: g is being_homeomorphism and (g . 0 ) = ( W-min P) and

         A17: (g . 1) = ( E-max P) and

         A18: (g . s1) = p1 and

         A19: 0 <= s1 & s1 <= 1 and

         A20: (g . s2) = p2 and

         A21: 0 <= s2 & s2 <= 1;

        

         A22: s2 in [. 0 , 1.] by A21, XXREAL_1: 1;

        reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

        

         A23: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm6, FUNCT_2:def 1;

        g3 is one-to-one & K0 is non empty compact by A1, A13, Lm6, JORDAN5A: 1;

        then g3 is being_homeomorphism by A23, A15, COMPTS_1: 17;

        then

         A24: h is being_homeomorphism by A16, TOPMETR: 20, TOPS_2: 57;

        

         A25: ( dom g) = ( [#] I[01] ) by A16, TOPS_2:def 5

        .= [. 0 , 1.] by BORSUK_1: 40;

        then

         A26: 1 in ( dom g) by XXREAL_1: 1;

        

         A27: 1 = ( |[1, 0 ]| `1 ) by EUCLID: 52

        .= (g0 . |[1, 0 ]|) by PSCOMP_1:def 5

        .= (g3 . |[1, 0 ]|) by A12, A14, FUNCT_1: 49

        .= (h . 1) by A12, A17, A26, FUNCT_1: 13;

        

         A28: s1 in [. 0 , 1.] by A19, XXREAL_1: 1;

        

         A29: (p2 `1 ) = (g0 . p2) by PSCOMP_1:def 5

        .= (g3 . p2) by A9, FUNCT_1: 49

        .= (h . s2) by A20, A25, A22, FUNCT_1: 13;

        (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

        .= (g3 . p1) by A8, FUNCT_1: 49

        .= (h . s1) by A18, A25, A28, FUNCT_1: 13;

        hence thesis by A6, A24, A28, A22, A27, A29, Th8;

      end;

      then

       A30: LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P)) by A8, A9, JORDAN5C:def 3;

      p1 in ( Upper_Arc P) by A2, A4, A7;

      hence thesis by A9, A30;

    end;

    theorem :: JGRAPH_5:55

    

     Th55: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `1 ) >= 0 & (p2 `1 ) >= 0 & (p1 `2 ) >= (p2 `2 ) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `1 ) >= 0 and

       A5: (p2 `1 ) >= 0 and

       A6: (p1 `2 ) >= (p2 `2 );

      

       A7: ex p3 be Point of ( TOP-REAL 2) st p3 = p1 & |.p3.| = 1 by A1, A2;

      

       A8: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      

       A9: ex p3 be Point of ( TOP-REAL 2) st p3 = p2 & |.p3.| = 1 by A1, A3;

      

       A10: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A1, Th34;

      set P4b = ( Lower_Arc P);

      

       A11: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A12: (( Upper_Arc P) /\ P4b) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      

       A13: (( Upper_Arc P) \/ P4b) = P by A11, JORDAN6:def 9;

      

       A14: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A11, JORDAN6:def 9;

      now

        per cases ;

          case

           A15: p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P);

          (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A7, JGRAPH_3: 1;

          then

           A16: (p1 `1 ) = ( sqrt ((1 ^2 ) - ((p1 `2 ) ^2 ))) & ((1 ^2 ) - ((p1 `2 ) ^2 )) >= 0 by A4, SQUARE_1: 22;

          (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

          then

           A17: (p2 `1 ) = ( sqrt ((1 ^2 ) - ((p2 `2 ) ^2 ))) by A5, SQUARE_1: 22;

          

           A18: ex p22 be Point of ( TOP-REAL 2) st p2 = p22 & p22 in P & (p22 `2 ) >= 0 by A10, A15;

          then ((p1 `2 ) ^2 ) >= ((p2 `2 ) ^2 ) by A6, SQUARE_1: 15;

          then ((1 ^2 ) - ((p1 `2 ) ^2 )) <= ((1 ^2 ) - ((p2 `2 ) ^2 )) by XREAL_1: 13;

          hence thesis by A1, A2, A6, A18, A17, A16, Th54, SQUARE_1: 26;

        end;

          case

           A19: p1 in ( Upper_Arc P) & not p2 in ( Upper_Arc P);

           A20:

          now

            assume

             A21: p2 = ( W-min P);

            ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

            then (p2 `2 ) = 0 by A21, EUCLID: 52;

            hence contradiction by A3, A10, A19;

          end;

          p2 in ( Lower_Arc P) by A3, A13, A19, XBOOLE_0:def 3;

          hence thesis by A19, A20;

        end;

          case

           A22: not p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P);

          then ex p9 be Point of ( TOP-REAL 2) st p2 = p9 & p9 in P & (p9 `2 ) >= 0 by A10;

          hence contradiction by A2, A6, A10, A22;

        end;

          case

           A23: not p1 in ( Upper_Arc P) & not p2 in ( Upper_Arc P);

          

           A24: ( - (p1 `2 )) <= ( - (p2 `2 )) by A6, XREAL_1: 24;

          (p1 `2 ) < 0 by A2, A10, A23;

          then (( - (p1 `2 )) ^2 ) <= (( - (p2 `2 )) ^2 ) by A24, SQUARE_1: 15;

          then

           A25: ((1 ^2 ) - (( - (p1 `2 )) ^2 )) >= ((1 ^2 ) - (( - (p2 `2 )) ^2 )) by XREAL_1: 13;

          (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A9, JGRAPH_3: 1;

          then

           A26: (p2 `1 ) = ( sqrt ((1 ^2 ) - (( - (p2 `2 )) ^2 ))) & ((1 ^2 ) - (( - (p2 `2 )) ^2 )) >= 0 by A5, SQUARE_1: 22;

          

           A27: p2 in ( Lower_Arc P) by A3, A13, A23, XBOOLE_0:def 3;

           A28:

          now

            assume

             A29: p2 = ( W-min P);

            ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

            then (p2 `2 ) = 0 by A29, EUCLID: 52;

            hence contradiction by A3, A10, A23;

          end;

          

           A30: p1 in ( Lower_Arc P) by A2, A13, A23, XBOOLE_0:def 3;

          (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A7, JGRAPH_3: 1;

          then (p1 `1 ) = ( sqrt ((1 ^2 ) - (( - (p1 `2 )) ^2 ))) by A4, SQUARE_1: 22;

          then

           A31: (p1 `1 ) >= (p2 `1 ) by A25, A26, SQUARE_1: 26;

          for g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max P) & (g . 1) = ( W-min P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

          proof

            ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

            then

             A32: ( W-min P) in ( Lower_Arc P) by A12, XBOOLE_0:def 4;

            set K0 = ( Lower_Arc P);

            reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

            reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

            ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

            then

             A33: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

            reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm5;

            let g be Function of I[01] , (( TOP-REAL 2) | P4b), s1,s2 be Real;

            assume that

             A34: g is being_homeomorphism and (g . 0 ) = ( E-max P) and

             A35: (g . 1) = ( W-min P) and

             A36: (g . s1) = p1 and

             A37: 0 <= s1 & s1 <= 1 and

             A38: (g . s2) = p2 and

             A39: 0 <= s2 & s2 <= 1;

            

             A40: s2 in [. 0 , 1.] by A39, XXREAL_1: 1;

            reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

            

             A41: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm5, FUNCT_2:def 1;

            g3 is one-to-one & K0 is non empty compact by A1, A14, Lm5, JORDAN5A: 1;

            then g3 is being_homeomorphism by A41, A33, COMPTS_1: 17;

            then

             A42: h is being_homeomorphism by A34, TOPMETR: 20, TOPS_2: 57;

            

             A43: ( dom g) = ( [#] I[01] ) by A34, TOPS_2:def 5

            .= [. 0 , 1.] by BORSUK_1: 40;

            then

             A44: 1 in ( dom g) by XXREAL_1: 1;

            

             A45: ( - 1) = ( |[( - 1), 0 ]| `1 ) by EUCLID: 52

            .= ( proj1 . |[( - 1), 0 ]|) by PSCOMP_1:def 5

            .= (g3 . |[( - 1), 0 ]|) by A8, A32, FUNCT_1: 49

            .= (h . 1) by A8, A35, A44, FUNCT_1: 13;

            

             A46: s1 in [. 0 , 1.] by A37, XXREAL_1: 1;

            

             A47: (p2 `1 ) = (g0 . p2) by PSCOMP_1:def 5

            .= (g3 . p2) by A27, FUNCT_1: 49

            .= (h . s2) by A38, A43, A40, FUNCT_1: 13;

            (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

            .= (g3 . p1) by A30, FUNCT_1: 49

            .= (h . s1) by A36, A43, A46, FUNCT_1: 13;

            hence thesis by A31, A42, A46, A40, A45, A47, Th9;

          end;

          then

           A48: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A30, A27, JORDAN5C:def 3;

          p1 in ( Lower_Arc P) by A2, A13, A23, XBOOLE_0:def 3;

          hence thesis by A27, A28, A48;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:56

    

     Th56: for p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & (p1 `2 ) <= 0 & (p2 `2 ) <= 0 & p2 <> ( W-min P) & (p1 `1 ) >= (p2 `1 ) holds LE (p1,p2,P)

    proof

      let p1,p2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p1 in P and

       A3: p2 in P and

       A4: (p1 `2 ) <= 0 and

       A5: (p2 `2 ) <= 0 and

       A6: p2 <> ( W-min P) and

       A7: (p1 `1 ) >= (p2 `1 );

      

       A8: ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A1, Th35;

      then

       A9: p1 in ( Lower_Arc P) by A2, A4;

      set P4 = ( Lower_Arc P);

      

       A10: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A11: (( Upper_Arc P) /\ P4) = {( W-min P), ( E-max P)} by JORDAN6:def 9;

      

       A12: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      

       A13: p2 in ( Lower_Arc P) by A3, A5, A8;

      

       A14: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A10, JORDAN6:def 9;

      for g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max P) & (g . 1) = ( W-min P) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

      proof

        ( W-min P) in {( W-min P), ( E-max P)} by TARSKI:def 2;

        then

         A15: ( W-min P) in ( Lower_Arc P) by A11, XBOOLE_0:def 4;

        set K0 = ( Lower_Arc P);

        reconsider g0 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider g2 = (g0 | K0) as Function of (( TOP-REAL 2) | K0), R^1 by PRE_TOPC: 9;

        ( Closed-Interval-TSpace (( - 1),1)) = ( TopSpaceMetr ( Closed-Interval-MSpace (( - 1),1))) by TOPMETR:def 7;

        then

         A16: ( Closed-Interval-TSpace (( - 1),1)) is T_2 by PCOMPS_1: 34;

        reconsider g3 = g2 as continuous Function of (( TOP-REAL 2) | K0), ( Closed-Interval-TSpace (( - 1),1)) by A1, Lm5;

        let g be Function of I[01] , (( TOP-REAL 2) | P4), s1,s2 be Real;

        assume that

         A17: g is being_homeomorphism and (g . 0 ) = ( E-max P) and

         A18: (g . 1) = ( W-min P) and

         A19: (g . s1) = p1 and

         A20: 0 <= s1 & s1 <= 1 and

         A21: (g . s2) = p2 and

         A22: 0 <= s2 & s2 <= 1;

        

         A23: s2 in [. 0 , 1.] by A22, XXREAL_1: 1;

        reconsider h = (g3 * g) as Function of ( Closed-Interval-TSpace ( 0 ,1)), ( Closed-Interval-TSpace (( - 1),1)) by TOPMETR: 20;

        

         A24: ( dom g3) = ( [#] (( TOP-REAL 2) | K0)) & ( rng g3) = ( [#] ( Closed-Interval-TSpace (( - 1),1))) by A1, Lm5, FUNCT_2:def 1;

        g3 is one-to-one & K0 is non empty compact by A1, A14, Lm5, JORDAN5A: 1;

        then g3 is being_homeomorphism by A24, A16, COMPTS_1: 17;

        then

         A25: h is being_homeomorphism by A17, TOPMETR: 20, TOPS_2: 57;

        

         A26: ( dom g) = ( [#] I[01] ) by A17, TOPS_2:def 5

        .= [. 0 , 1.] by BORSUK_1: 40;

        then

         A27: 1 in ( dom g) by XXREAL_1: 1;

        

         A28: ( - 1) = ( |[( - 1), 0 ]| `1 ) by EUCLID: 52

        .= ( proj1 . |[( - 1), 0 ]|) by PSCOMP_1:def 5

        .= (g3 . |[( - 1), 0 ]|) by A12, A15, FUNCT_1: 49

        .= (h . 1) by A12, A18, A27, FUNCT_1: 13;

        

         A29: s1 in [. 0 , 1.] by A20, XXREAL_1: 1;

        

         A30: (p2 `1 ) = (g0 . p2) by PSCOMP_1:def 5

        .= (g3 . p2) by A13, FUNCT_1: 49

        .= (h . s2) by A21, A26, A23, FUNCT_1: 13;

        (p1 `1 ) = (g0 . p1) by PSCOMP_1:def 5

        .= (g3 . p1) by A9, FUNCT_1: 49

        .= (h . s1) by A19, A26, A29, FUNCT_1: 13;

        hence thesis by A7, A25, A29, A23, A28, A30, Th9;

      end;

      then

       A31: LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) by A9, A13, JORDAN5C:def 3;

      p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P) by A2, A3, A4, A5, A8;

      hence thesis by A6, A31;

    end;

    theorem :: JGRAPH_5:57

    

     Th57: for cn be Real, q be Point of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & (q `2 ) <= 0 holds for p be Point of ( TOP-REAL 2) st p = ((cn -FanMorphS ) . q) holds (p `2 ) <= 0

    proof

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

      assume that

       A1: ( - 1) < cn and

       A2: cn < 1 and

       A3: (q `2 ) <= 0 ;

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

      assume

       A4: p = ((cn -FanMorphS ) . q);

      per cases by A3;

        suppose

         A5: (q `2 ) < 0 ;

        now

          per cases ;

            case ((q `1 ) / |.q.|) < cn;

            hence thesis by A1, A4, A5, JGRAPH_4: 138;

          end;

            case ((q `1 ) / |.q.|) >= cn;

            hence thesis by A2, A4, A5, JGRAPH_4: 137;

          end;

        end;

        hence thesis;

      end;

        suppose (q `2 ) = 0 ;

        hence thesis by A4, JGRAPH_4: 113;

      end;

    end;

    theorem :: JGRAPH_5:58

    

     Th58: for cn be Real, p1,p2,q1,q2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st ( - 1) < cn & cn < 1 & P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & q1 = ((cn -FanMorphS ) . p1) & q2 = ((cn -FanMorphS ) . p2) holds LE (q1,q2,P)

    proof

      let cn be Real, p1,p2,q1,q2 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: ( - 1) < cn & cn < 1 and

       A2: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A3: LE (p1,p2,P) and

       A4: q1 = ((cn -FanMorphS ) . p1) and

       A5: q2 = ((cn -FanMorphS ) . p2);

      

       A6: P is being_simple_closed_curve by A2, JGRAPH_3: 26;

      ( W-min P) = |[( - 1), 0 ]| by A2, Th29;

      then

       A7: (( W-min P) `2 ) = 0 by EUCLID: 52;

      then

       A8: ((cn -FanMorphS ) . ( W-min P)) = ( W-min P) by JGRAPH_4: 113;

      p2 in the carrier of ( TOP-REAL 2);

      then

       A9: p2 in ( dom (cn -FanMorphS )) by FUNCT_2:def 1;

      ( W-min P) in the carrier of ( TOP-REAL 2);

      then

       A10: ( W-min P) in ( dom (cn -FanMorphS )) by FUNCT_2:def 1;

      

       A11: ( Lower_Arc P) c= P by A2, Th33;

      

       A12: (cn -FanMorphS ) is one-to-one by A1, JGRAPH_4: 133;

      

       A13: ( Upper_Arc P) c= P by A2, Th33;

       A14:

      now

        per cases by A3;

          case p1 in ( Upper_Arc P);

          hence p1 in P by A13;

        end;

          case p1 in ( Lower_Arc P);

          hence p1 in P by A11;

        end;

      end;

       A15:

      now

        assume

         A16: q2 = ( W-min P);

        then p2 = ( W-min P) by A5, A8, A10, A9, A12, FUNCT_1:def 4;

        then LE (p2,p1,P) by A6, A14, JORDAN7: 3;

        then

         A17: q1 = q2 by A2, A3, A4, A5, JGRAPH_3: 26, JORDAN6: 57;

        ( W-min P) in ( Lower_Arc P) by A6, JORDAN7: 1;

        then LE (q1,q2,P) by A2, A11, A16, A17, JGRAPH_3: 26, JORDAN6: 56;

        hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

      end;

      

       A18: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A2, Th34;

      

       A19: ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A2, Th35;

      per cases by A3;

        suppose

         A20: p1 in ( Upper_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P);

        

         A21: |.q2.| = |.p2.| by A5, JGRAPH_4: 128;

        

         A22: ex p9 be Point of ( TOP-REAL 2) st p9 = p2 & p9 in P & (p9 `2 ) <= 0 by A19, A20;

        then ex p10 be Point of ( TOP-REAL 2) st p10 = p2 & |.p10.| = 1 by A2;

        then

         A23: q2 in P by A2, A21;

        

         A24: ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & p8 in P & (p8 `2 ) >= 0 by A18, A20;

        (q2 `2 ) <= 0 by A1, A5, A22, Th57;

        hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P)) by A4, A19, A15, A20, A24, A23, JGRAPH_4: 113;

      end;

        suppose

         A25: p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P) & LE (p1,p2,( Upper_Arc P),( W-min P),( E-max P));

        then ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & p8 in P & (p8 `2 ) >= 0 by A18;

        then

         A26: p1 = ((cn -FanMorphS ) . p1) by JGRAPH_4: 113;

        ex p9 be Point of ( TOP-REAL 2) st p9 = p2 & p9 in P & (p9 `2 ) >= 0 by A18, A25;

        hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P)) by A4, A5, A25, A26, JGRAPH_4: 113;

      end;

        suppose

         A27: p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P) & not p2 = ( W-min P) & LE (p1,p2,( Lower_Arc P),( E-max P),( W-min P)) & not p1 in ( Upper_Arc P);

        then

         A28: ex p8 be Point of ( TOP-REAL 2) st p8 = p1 & p8 in P & (p8 `2 ) <= 0 by A19;

        then

         A29: ex p10 be Point of ( TOP-REAL 2) st p10 = p1 & |.p10.| = 1 by A2;

        

         A30: ex p9 be Point of ( TOP-REAL 2) st p9 = p2 & p9 in P & (p9 `2 ) <= 0 by A19, A27;

        then

         A31: ex p11 be Point of ( TOP-REAL 2) st p11 = p2 & |.p11.| = 1 by A2;

        

         A32: (q2 `2 ) <= 0 by A1, A5, A30, Th57;

        

         A33: |.q2.| = |.p2.| by A5, JGRAPH_4: 128;

        then

         A34: q2 in P by A2, A31;

        

         A35: (q1 `2 ) <= 0 by A1, A4, A28, Th57;

        

         A36: |.q1.| = |.p1.| by A4, JGRAPH_4: 128;

        then

         A37: q1 in P by A2, A29;

        now

          per cases ;

            case

             A38: p1 = ( W-min P);

            then p1 = ((cn -FanMorphS ) . p1) by A7, JGRAPH_4: 113;

            then LE (q1,q2,P) by A4, A6, A34, A38, JORDAN7: 3;

            hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

          end;

            case

             A39: p1 <> ( W-min P);

            now

              per cases by A2, A3, A28, A39, Th48;

                case

                 A40: (p1 `1 ) = (p2 `1 );

                

                 A41: p2 = |[(p2 `1 ), (p2 `2 )]| by EUCLID: 53;

                 A42:

                now

                  assume

                   A43: (p1 `2 ) = ( - (p2 `2 ));

                  then (p2 `2 ) = 0 by A28, A30, XREAL_1: 58;

                  hence p1 = p2 by A40, A41, A43, EUCLID: 53;

                end;

                (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) = (1 ^2 ) by A29, JGRAPH_3: 1

                .= (((p1 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A31, A40, JGRAPH_3: 1;

                then

                 A44: (p1 `2 ) = (p2 `2 ) or (p1 `2 ) = ( - (p2 `2 )) by SQUARE_1: 40;

                p1 = |[(p1 `1 ), (p1 `2 )]| by EUCLID: 53;

                then LE (q1,q2,P) by A2, A4, A5, A34, A40, A44, A41, A42, JGRAPH_3: 26, JORDAN6: 56;

                hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

              end;

                case (p1 `1 ) > (p2 `1 );

                then ((p1 `1 ) / |.p1.|) > ((p2 `1 ) / |.p2.|) by A29, A31;

                then

                 A45: ((q1 `1 ) / |.q1.|) >= ((q2 `1 ) / |.q2.|) by A1, A4, A5, A28, A30, A29, A31, Th27;

                q2 <> ( W-min P) by A5, A8, A10, A9, A12, A27, FUNCT_1:def 4;

                then LE (q1,q2,P) by A2, A36, A33, A35, A32, A29, A31, A37, A34, A45, Th56;

                hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

              end;

            end;

            hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

          end;

        end;

        hence q1 in ( Upper_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) or q1 in ( Upper_Arc P) & q2 in ( Upper_Arc P) & LE (q1,q2,( Upper_Arc P),( W-min P),( E-max P)) or q1 in ( Lower_Arc P) & q2 in ( Lower_Arc P) & not q2 = ( W-min P) & LE (q1,q2,( Lower_Arc P),( E-max P),( W-min P));

      end;

    end;

    theorem :: JGRAPH_5:59

    

     Th59: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & (p1 `1 ) < 0 & (p1 `2 ) >= 0 & (p2 `1 ) < 0 & (p2 `2 ) >= 0 & (p3 `1 ) < 0 & (p3 `2 ) >= 0 & (p4 `1 ) < 0 & (p4 `2 ) >= 0 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 & (q4 `2 ) < 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: LE (p2,p3,P) and

       A4: LE (p3,p4,P) and

       A5: (p1 `1 ) < 0 and

       A6: (p1 `2 ) >= 0 and

       A7: (p2 `1 ) < 0 and

       A8: (p2 `2 ) >= 0 and

       A9: (p3 `1 ) < 0 and

       A10: (p3 `2 ) >= 0 and

       A11: (p4 `1 ) < 0 and

       A12: (p4 `2 ) >= 0 ;

      consider r be Real such that

       A13: (p4 `1 ) < r and

       A14: r < 0 by A11, XREAL_1: 5;

      reconsider r1 = r as Real;

      set s = ( sqrt (1 - (r1 ^2 )));

      

       A15: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then p4 in P by A4, JORDAN7: 5;

      then

       A16: ex p be Point of ( TOP-REAL 2) st p = p4 & |.p.| = 1 by A1;

      then ( - 1) <= (p4 `1 ) by Th1;

      then ( - 1) <= r1 by A13, XXREAL_0: 2;

      then (r1 ^2 ) <= (1 ^2 ) by A14, SQUARE_1: 49;

      then

       A17: (1 - (r1 ^2 )) >= 0 by XREAL_1: 48;

      then

       A18: (s ^2 ) = (1 - (r1 ^2 )) by SQUARE_1:def 2;

      then

       A19: ((1 - (s ^2 )) + (s ^2 )) > ( 0 + (s ^2 )) by A14, SQUARE_1: 12, XREAL_1: 8;

      then

       A20: ( - 1) < s by SQUARE_1: 52;

      

       A21: s < 1 by A19, SQUARE_1: 52;

      then

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

       A22: f1 = (s -FanMorphW ) and

       A23: f1 is being_homeomorphism by A20, JGRAPH_4: 41;

      set q11 = (f1 . p1), q22 = (f1 . p2), q33 = (f1 . p3), q44 = (f1 . p4);

      

       A24: s >= 0 by A17, SQUARE_1:def 2;

      p3 in P by A3, A15, JORDAN7: 5;

      then

       A25: ex p33 be Point of ( TOP-REAL 2) st p33 = p3 & |.p33.| = 1 by A1;

      then ((p3 `2 ) / |.p3.|) < ((p4 `2 ) / |.p4.|) or p3 = p4 by A1, A4, A11, A12, A16, Th46;

      then

       A26: ((q33 `2 ) / |.q33.|) < ((q44 `2 ) / |.q44.|) or p3 = p4 by A9, A11, A20, A21, A22, JGRAPH_4: 46;

      ((p4 `1 ) ^2 ) > (r1 ^2 ) by A13, A14, SQUARE_1: 44;

      then

       A27: (1 - ((p4 `1 ) ^2 )) < (1 - (r1 ^2 )) by XREAL_1: 15;

      

       A28: (p3 `1 ) < (p4 `1 ) or p3 = p4 by A1, A4, A9, A10, A12, Th46;

      then ( - (p3 `1 )) >= ( - (p4 `1 )) by XREAL_1: 24;

      then (( - (p3 `1 )) ^2 ) >= (( - (p4 `1 )) ^2 ) by A11, SQUARE_1: 15;

      then (1 - ((p3 `1 ) ^2 )) <= (1 - ((p4 `1 ) ^2 )) by XREAL_1: 10;

      then

       A29: (1 - ((p3 `1 ) ^2 )) < (s ^2 ) by A27, A18, XXREAL_0: 2;

      (p2 `1 ) < (p3 `1 ) or p2 = p3 by A1, A3, A7, A8, A10, Th46;

      then

       A30: (p2 `1 ) <= (p4 `1 ) by A28, XXREAL_0: 2;

      then ( - (p2 `1 )) >= ( - (p4 `1 )) by XREAL_1: 24;

      then (( - (p2 `1 )) ^2 ) >= (( - (p4 `1 )) ^2 ) by A11, SQUARE_1: 15;

      then (1 - ((p2 `1 ) ^2 )) <= (1 - ((p4 `1 ) ^2 )) by XREAL_1: 10;

      then

       A31: (1 - ((p2 `1 ) ^2 )) < (s ^2 ) by A27, A18, XXREAL_0: 2;

      (p1 `1 ) < (p2 `1 ) or p1 = p2 by A1, A2, A5, A6, A8, Th46;

      then (p1 `1 ) <= (p4 `1 ) by A30, XXREAL_0: 2;

      then ( - (p1 `1 )) >= ( - (p4 `1 )) by XREAL_1: 24;

      then (( - (p1 `1 )) ^2 ) >= (( - (p4 `1 )) ^2 ) by A11, SQUARE_1: 15;

      then (1 - ((p1 `1 ) ^2 )) <= (1 - ((p4 `1 ) ^2 )) by XREAL_1: 10;

      then

       A32: (1 - ((p1 `1 ) ^2 )) < (s ^2 ) by A27, A18, XXREAL_0: 2;

      (1 ^2 ) = (((p3 `1 ) ^2 ) + ((p3 `2 ) ^2 )) by A25, JGRAPH_3: 1;

      then

       A33: ((p3 `2 ) / |.p3.|) < s by A25, A24, A29, SQUARE_1: 48;

      then

       A34: (q33 `1 ) < 0 by A9, A20, A22, JGRAPH_4: 43;

      p2 in P by A2, A15, JORDAN7: 5;

      then

       A35: ex p22 be Point of ( TOP-REAL 2) st p22 = p2 & |.p22.| = 1 by A1;

      then

       A36: |.q22.| = 1 by A22, JGRAPH_4: 33;

      then

       A37: q22 in P by A1;

      ((p2 `2 ) / |.p2.|) < ((p3 `2 ) / |.p3.|) or p2 = p3 by A1, A3, A9, A10, A35, A25, Th46;

      then

       A38: ((q22 `2 ) / |.q22.|) < ((q33 `2 ) / |.q33.|) or p2 = p3 by A7, A9, A20, A21, A22, JGRAPH_4: 46;

      

       A39: |.q33.| = 1 by A25, A22, JGRAPH_4: 33;

      then

       A40: q33 in P by A1;

      (1 ^2 ) = (((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) by A35, JGRAPH_3: 1;

      then

       A41: ((p2 `2 ) / |.p2.|) < s by A35, A24, A31, SQUARE_1: 48;

      then

       A42: (q22 `2 ) < 0 by A7, A20, A22, JGRAPH_4: 43;

      

       A43: (q22 `1 ) < 0 by A7, A20, A22, A41, JGRAPH_4: 43;

      (1 ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by A16, JGRAPH_3: 1;

      then ((p4 `2 ) / |.p4.|) < s by A27, A16, A18, A24, SQUARE_1: 48;

      then

       A44: (q44 `1 ) < 0 & (q44 `2 ) < 0 by A11, A20, A22, JGRAPH_4: 43;

      p1 in P by A2, A15, JORDAN7: 5;

      then

       A45: ex p11 be Point of ( TOP-REAL 2) st p11 = p1 & |.p11.| = 1 by A1;

      then ((p1 `2 ) / |.p1.|) < ((p2 `2 ) / |.p2.|) or p1 = p2 by A1, A2, A7, A8, A35, Th46;

      then

       A46: ((q11 `2 ) / |.q11.|) < ((q22 `2 ) / |.q22.|) or p1 = p2 by A5, A7, A20, A21, A22, JGRAPH_4: 46;

      (1 ^2 ) = (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 )) by A45, JGRAPH_3: 1;

      then

       A47: ((p1 `2 ) / |.p1.|) < s by A45, A24, A32, SQUARE_1: 48;

      then

       A48: (q11 `1 ) < 0 by A5, A20, A22, JGRAPH_4: 43;

      

       A49: |.q11.| = 1 by A45, A22, JGRAPH_4: 33;

      then q11 in P by A1;

      then

       A50: LE (q11,q22,P) by A1, A49, A36, A37, A48, A43, A42, A46, Th51;

      

       A51: (q22 `1 ) < 0 & (q22 `2 ) < 0 by A7, A20, A22, A41, JGRAPH_4: 43;

      

       A52: (q11 `1 ) < 0 & (q11 `2 ) < 0 by A5, A20, A22, A47, JGRAPH_4: 43;

      

       A53: for q be Point of ( TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A22, JGRAPH_4: 33;

      (q33 `1 ) < 0 & (q33 `2 ) < 0 by A9, A20, A22, A33, JGRAPH_4: 43;

      then

       A54: LE (q22,q33,P) by A1, A36, A37, A39, A40, A43, A38, Th51;

      

       A55: (q33 `2 ) < 0 by A9, A20, A22, A33, JGRAPH_4: 43;

      

       A56: |.q44.| = 1 by A16, A22, JGRAPH_4: 33;

      then q44 in P by A1;

      then LE (q33,q44,P) by A1, A39, A40, A56, A34, A44, A26, Th51;

      hence thesis by A23, A53, A52, A51, A34, A55, A44, A50, A54;

    end;

    theorem :: JGRAPH_5:60

    

     Th60: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 & (p3 `2 ) >= 0 & (p4 `2 ) > 0 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) >= 0 & (q2 `1 ) < 0 & (q2 `2 ) >= 0 & (q3 `1 ) < 0 & (q3 `2 ) >= 0 & (q4 `1 ) < 0 & (q4 `2 ) >= 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: LE (p2,p3,P) and

       A4: LE (p3,p4,P) and

       A5: (p1 `2 ) >= 0 and

       A6: (p2 `2 ) >= 0 and

       A7: (p3 `2 ) >= 0 and

       A8: (p4 `2 ) > 0 ;

      

       A9: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then p4 in P by A4, JORDAN7: 5;

      then

       A10: ex p be Point of ( TOP-REAL 2) st p = p4 & |.p.| = 1 by A1;

       A11:

      now

        assume (p4 `1 ) = 1;

        then (1 ^2 ) = (1 + ((p4 `2 ) ^2 )) by A10, JGRAPH_3: 1;

        hence contradiction by A8, XCMPLX_1: 6;

      end;

      (p4 `1 ) <= 1 by A10, Th1;

      then (p4 `1 ) < 1 by A11, XXREAL_0: 1;

      then

      consider r be Real such that

       A12: (p4 `1 ) < r and

       A13: r < 1 by XREAL_1: 5;

      reconsider r1 = r as Real;

      ( - 1) <= (p4 `1 ) by A10, Th1;

      then

       A14: ( - 1) < r1 by A12, XXREAL_0: 2;

      then

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

       A15: f1 = (r1 -FanMorphN ) and

       A16: f1 is being_homeomorphism by A13, JGRAPH_4: 74;

      set q11 = (f1 . p1), q22 = (f1 . p2), q33 = (f1 . p3), q44 = (f1 . p4);

      

       A17: for q be Point of ( TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A15, JGRAPH_4: 66;

      

       A18: (p3 `1 ) < (p4 `1 ) or p3 = p4 by A1, A4, A8, Th47;

      then

       A19: (p3 `1 ) < r1 by A12, XXREAL_0: 2;

      p3 in P by A3, A9, JORDAN7: 5;

      then

       A20: ex p33 be Point of ( TOP-REAL 2) st p33 = p3 & |.p33.| = 1 by A1;

      then ((p3 `1 ) / |.p3.|) < ((p4 `1 ) / |.p4.|) or p3 = p4 by A1, A4, A8, A10, Th47;

      then

       A21: ((q33 `1 ) / |.q33.|) < ((q44 `1 ) / |.q44.|) or p3 = p4 by A7, A8, A10, A20, A13, A14, A15, Th21;

      

       A22: ((p3 `1 ) / |.p3.|) < r1 by A20, A12, A18, XXREAL_0: 2;

      then

       A23: (q33 `2 ) >= 0 by A7, A20, A13, A14, A15, Th20;

      

       A24: (p1 `1 ) < (p2 `1 ) or p1 = p2 by A1, A2, A6, Th47;

      ((p4 `1 ) / |.p4.|) < r1 by A10, A12;

      then

       A25: (q44 `1 ) < 0 & (q44 `2 ) > 0 by A8, A14, A15, JGRAPH_4: 76;

      p2 in P by A2, A9, JORDAN7: 5;

      then

       A26: ex p22 be Point of ( TOP-REAL 2) st p22 = p2 & |.p22.| = 1 by A1;

      then

       A27: |.q22.| = 1 by A15, JGRAPH_4: 66;

      then

       A28: q22 in P by A1;

      

       A29: (p2 `1 ) < (p3 `1 ) or p2 = p3 by A1, A3, A7, Th47;

      then

       A30: ((p2 `1 ) / |.p2.|) < r1 by A26, A19, XXREAL_0: 2;

      then

       A31: (q22 `2 ) >= 0 by A6, A26, A13, A14, A15, Th20;

      p1 in P by A2, A9, JORDAN7: 5;

      then

       A32: ex p11 be Point of ( TOP-REAL 2) st p11 = p1 & |.p11.| = 1 by A1;

      then ((p1 `1 ) / |.p1.|) < ((p2 `1 ) / |.p2.|) or p1 = p2 by A1, A2, A6, A26, Th47;

      then

       A33: ((q11 `1 ) / |.q11.|) < ((q22 `1 ) / |.q22.|) or p1 = p2 by A5, A6, A32, A26, A13, A14, A15, Th21;

      (p2 `1 ) < r1 by A29, A19, XXREAL_0: 2;

      then

       A34: ((p1 `1 ) / |.p1.|) < r1 by A32, A24, XXREAL_0: 2;

      then

       A35: (q11 `2 ) >= 0 by A5, A32, A13, A14, A15, Th20;

      

       A36: (q22 `1 ) < 0 by A6, A26, A13, A14, A15, A30, Th20;

      

       A37: |.q11.| = 1 by A32, A15, JGRAPH_4: 66;

      then q11 in P by A1;

      then

       A38: LE (q11,q22,P) by A1, A37, A27, A28, A31, A36, A35, A33, Th53;

      

       A39: |.q33.| = 1 by A20, A15, JGRAPH_4: 66;

      then

       A40: q33 in P by A1;

      

       A41: (q33 `1 ) < 0 by A7, A20, A13, A14, A15, A22, Th20;

      

       A42: (q22 `1 ) < 0 & (q22 `2 ) >= 0 by A6, A26, A13, A14, A15, A30, Th20;

      

       A43: (q11 `1 ) < 0 & (q11 `2 ) >= 0 or (q11 `1 ) < 0 & (q11 `2 ) = 0 by A5, A32, A13, A14, A15, A34, Th20;

      

       A44: |.q44.| = 1 by A10, A15, JGRAPH_4: 66;

      then q44 in P by A1;

      then

       A45: LE (q33,q44,P) by A1, A39, A40, A44, A25, A23, A21, Th53;

      ((p2 `1 ) / |.p2.|) < ((p3 `1 ) / |.p3.|) or p2 = p3 by A1, A3, A7, A26, A20, Th47;

      then ((q22 `1 ) / |.q22.|) < ((q33 `1 ) / |.q33.|) or p2 = p3 by A6, A7, A26, A20, A13, A14, A15, Th21;

      then LE (q22,q33,P) by A1, A27, A28, A39, A40, A31, A23, A41, Th53;

      hence thesis by A16, A17, A25, A43, A42, A38, A23, A41, A45;

    end;

    theorem :: JGRAPH_5:61

    

     Th61: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 & (p3 `2 ) >= 0 & (p4 `2 ) > 0 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 & (q4 `2 ) < 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & (p1 `2 ) >= 0 & (p2 `2 ) >= 0 & (p3 `2 ) >= 0 & (p4 `2 ) > 0 ;

      consider f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) such that

       A3: f is being_homeomorphism and

       A4: for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.| and

       A5: q1 = (f . p1) & q2 = (f . p2) and

       A6: q3 = (f . p3) & q4 = (f . p4) and

       A7: (q1 `1 ) < 0 & (q1 `2 ) >= 0 & (q2 `1 ) < 0 & (q2 `2 ) >= 0 & (q3 `1 ) < 0 & (q3 `2 ) >= 0 & (q4 `1 ) < 0 & (q4 `2 ) >= 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P) by A1, A2, Th60;

      consider f2 be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1b,q2b,q3b,q4b be Point of ( TOP-REAL 2) such that

       A8: f2 is being_homeomorphism and

       A9: for q be Point of ( TOP-REAL 2) holds |.(f2 . q).| = |.q.| and

       A10: q1b = (f2 . q1) & q2b = (f2 . q2) and

       A11: q3b = (f2 . q3) & q4b = (f2 . q4) and

       A12: (q1b `1 ) < 0 & (q1b `2 ) < 0 & (q2b `1 ) < 0 & (q2b `2 ) < 0 & (q3b `1 ) < 0 & (q3b `2 ) < 0 & (q4b `1 ) < 0 & (q4b `2 ) < 0 & LE (q1b,q2b,P) & LE (q2b,q3b,P) & LE (q3b,q4b,P) by A1, A7, Th59;

      reconsider f3 = (f2 * f) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A13: f3 is being_homeomorphism by A3, A8, TOPS_2: 57;

      

       A14: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      then

       A15: (f3 . p3) = q3b & (f3 . p4) = q4b by A6, A11, FUNCT_1: 13;

      

       A16: for q be Point of ( TOP-REAL 2) holds |.(f3 . q).| = |.q.|

      proof

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

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

        then (f3 . q) = (f2 . (f . q)) by FUNCT_1: 13;

        

        hence |.(f3 . q).| = |.(f . q).| by A9

        .= |.q.| by A4;

      end;

      (f3 . p1) = q1b & (f3 . p2) = q2b by A5, A10, A14, FUNCT_1: 13;

      hence thesis by A12, A13, A16, A15;

    end;

    theorem :: JGRAPH_5:62

    

     Th62: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & ((p1 `2 ) >= 0 or (p1 `1 ) >= 0 ) & ((p2 `2 ) >= 0 or (p2 `1 ) >= 0 ) & ((p3 `2 ) >= 0 or (p3 `1 ) >= 0 ) & ((p4 `2 ) > 0 or (p4 `1 ) > 0 ) holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `2 ) >= 0 & (q2 `2 ) >= 0 & (q3 `2 ) >= 0 & (q4 `2 ) > 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: LE (p2,p3,P) and

       A4: LE (p3,p4,P) and

       A5: (p1 `2 ) >= 0 or (p1 `1 ) >= 0 and

       A6: (p2 `2 ) >= 0 or (p2 `1 ) >= 0 and

       A7: (p3 `2 ) >= 0 or (p3 `1 ) >= 0 and

       A8: (p4 `2 ) > 0 or (p4 `1 ) > 0 ;

      

       A9: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A10: p4 in P by A4, JORDAN7: 5;

      then

       A11: ex p44 be Point of ( TOP-REAL 2) st p44 = p4 & |.p44.| = 1 by A1;

      then

       A12: ( - 1) <= (p4 `2 ) by Th1;

      now

        assume

         A13: (p4 `2 ) = ( - 1);

        (1 ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by A11, JGRAPH_3: 1

        .= (((p4 `1 ) ^2 ) + 1) by A13;

        hence contradiction by A8, A13, XCMPLX_1: 6;

      end;

      then (p4 `2 ) > ( - 1) by A12, XXREAL_0: 1;

      then

      consider r be Real such that

       A14: ( - 1) < r and

       A15: r < (p4 `2 ) by XREAL_1: 5;

      reconsider r1 = r as Real;

      (p4 `2 ) <= 1 by A11, Th1;

      then

       A16: r1 < 1 by A15, XXREAL_0: 2;

      then

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

       A17: f1 = (r1 -FanMorphE ) and

       A18: f1 is being_homeomorphism by A14, JGRAPH_4: 105;

      set q11 = (f1 . p1), q22 = (f1 . p2), q33 = (f1 . p3), q44 = (f1 . p4);

      

       A19: |.q44.| = 1 by A11, A17, JGRAPH_4: 97;

      then

       A20: q44 in P by A1;

      

       A21: p1 in P by A2, A9, JORDAN7: 5;

      then

       A22: ex p11 be Point of ( TOP-REAL 2) st p11 = p1 & |.p11.| = 1 by A1;

      then

       A23: |.q11.| = 1 by A17, JGRAPH_4: 97;

      then

       A24: q11 in P by A1;

      

       A25: p3 in P by A3, A9, JORDAN7: 5;

      then

       A26: ex p33 be Point of ( TOP-REAL 2) st p33 = p3 & |.p33.| = 1 by A1;

      then

       A27: |.q33.| = 1 by A17, JGRAPH_4: 97;

      then

       A28: q33 in P by A1;

      

       A29: p2 in P by A2, A9, JORDAN7: 5;

      then

       A30: ex p22 be Point of ( TOP-REAL 2) st p22 = p2 & |.p22.| = 1 by A1;

      then

       A31: |.q22.| = 1 by A17, JGRAPH_4: 97;

      then

       A32: q22 in P by A1;

      now

        per cases ;

          case

           A33: (p4 `2 ) <= 0 ;

          

           A34: ( Upper_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) >= 0 } by A1, Th34;

          

           A35: ((p4 `2 ) / |.p4.|) > r1 by A11, A15;

          then

           A36: (q44 `1 ) > 0 by A8, A15, A17, A33, JGRAPH_4: 106;

           A37:

          now

            set q8 = |[( sqrt (1 - (r1 ^2 ))), r1]|;

            assume

             A38: (q44 `2 ) = 0 ;

            (1 ^2 ) = (((q44 `1 ) ^2 ) + ((q44 `2 ) ^2 )) by A19, JGRAPH_3: 1

            .= ((q44 `1 ) ^2 ) by A38;

            then (q44 `1 ) = ( - 1) or (q44 `1 ) = 1 by SQUARE_1: 41;

            then

             A39: q44 = |[1, 0 ]| by A8, A15, A17, A33, A35, A38, EUCLID: 53, JGRAPH_4: 106;

            set r8 = (f1 . q8);

            (1 ^2 ) > (r1 ^2 ) by A14, A16, SQUARE_1: 50;

            then

             A40: (1 - (r1 ^2 )) > 0 by XREAL_1: 50;

            

             A41: (q8 `1 ) = ( sqrt (1 - (r1 ^2 ))) by EUCLID: 52;

            then

             A42: (q8 `1 ) > 0 by A40, SQUARE_1: 25;

            (q8 `2 ) = r1 by EUCLID: 52;

            then |.q8.| = ( sqrt ((( sqrt (1 - (r1 ^2 ))) ^2 ) + (r1 ^2 ))) by A41, JGRAPH_3: 1;

            

            then

             A43: |.q8.| = ( sqrt ((1 - (r1 ^2 )) + (r1 ^2 ))) by A40, SQUARE_1:def 2

            .= 1 by SQUARE_1: 18;

            then

             A44: ((q8 `2 ) / |.q8.|) = r1 by EUCLID: 52;

            then

             A45: (r8 `2 ) = 0 by A17, A42, JGRAPH_4: 111;

             |.r8.| = 1 by A17, A43, JGRAPH_4: 97;

            

            then (1 ^2 ) = (((r8 `1 ) ^2 ) + ((r8 `2 ) ^2 )) by JGRAPH_3: 1

            .= ((r8 `1 ) ^2 ) by A45;

            then (r8 `1 ) = ( - 1) or (r8 `1 ) = 1 by SQUARE_1: 41;

            then

             A46: (f1 . |[( sqrt (1 - (r1 ^2 ))), r1]|) = |[1, 0 ]| by A17, A44, A42, A45, EUCLID: 53, JGRAPH_4: 111;

            f1 is one-to-one & ( dom f1) = the carrier of ( TOP-REAL 2) by A14, A16, A17, FUNCT_2:def 1, JGRAPH_4: 102;

            then p4 = |[( sqrt (1 - (r1 ^2 ))), r1]| by A39, A46, FUNCT_1:def 4;

            hence contradiction by A15, EUCLID: 52;

          end;

          

           A47: (q44 `2 ) >= 0 by A8, A15, A17, A33, A35, JGRAPH_4: 106;

          

           A48: ( Lower_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) <= 0 } by A1, Th35;

           A49:

          now

            per cases ;

              case

               A50: (p3 `1 ) <= 0 ;

              then

               A51: q33 = p3 by A17, JGRAPH_4: 82;

               A52:

              now

                per cases by A50;

                  case

                   A53: (p3 `1 ) = 0 ;

                   A54:

                  now

                    assume (q33 `2 ) = ( - 1);

                    then ( - 1) >= (p4 `2 ) by A1, A4, A7, A8, A33, A51, Th50;

                    hence contradiction by A14, A15, XXREAL_0: 2;

                  end;

                  (1 ^2 ) = (( 0 ^2 ) + ((q33 `2 ) ^2 )) by A26, A51, A53, JGRAPH_3: 1

                  .= ((q33 `2 ) ^2 );

                  hence (q33 `2 ) >= 0 by A54, SQUARE_1: 41;

                end;

                  case (p3 `1 ) < 0 ;

                  hence (q33 `2 ) >= 0 by A7, A17, JGRAPH_4: 82;

                end;

              end;

              now

                per cases ;

                  case

                   A55: p2 <> ( W-min P);

                   A56:

                  now

                    

                     A57: p3 in ( Upper_Arc P) by A25, A34, A51, A52;

                    assume

                     A58: (p2 `2 ) < 0 ;

                    then p2 in ( Lower_Arc P) by A29, A48;

                    then LE (p3,p2,P) by A55, A57;

                    hence contradiction by A1, A3, A51, A52, A58, JGRAPH_3: 26, JORDAN6: 57;

                  end;

                  

                   A59: (p2 `1 ) <= (p3 `1 ) by A1, A3, A51, A52, Th47;

                  then

                   A60: q22 = p2 by A17, A50, JGRAPH_4: 82;

                  now

                    per cases ;

                      case

                       A61: p1 <> ( W-min P);

                       A62:

                      now

                        

                         A63: p2 in ( Upper_Arc P) by A29, A34, A56;

                        assume

                         A64: (p1 `2 ) < 0 ;

                        then p1 in ( Lower_Arc P) by A21, A48;

                        then LE (p2,p1,P) by A61, A63;

                        hence contradiction by A1, A2, A56, A64, JGRAPH_3: 26, JORDAN6: 57;

                      end;

                      (p1 `1 ) <= (p2 `1 ) by A1, A2, A56, Th47;

                      hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P) by A1, A2, A3, A17, A28, A20, A36, A47, A37, A51, A52, A56, A59, A60, A62, Th54, JGRAPH_4: 82;

                    end;

                      case

                       A65: p1 = ( W-min P);

                      

                       A66: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

                      then (p1 `1 ) = ( - 1) by A65, EUCLID: 52;

                      then p1 = q11 by A17, JGRAPH_4: 82;

                      hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P) by A1, A2, A3, A25, A17, A20, A36, A47, A37, A51, A52, A56, A59, A65, A66, Th54, EUCLID: 52, JGRAPH_4: 82;

                    end;

                  end;

                  hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P);

                end;

                  case

                   A67: p2 = ( W-min P);

                  ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

                  then

                   A68: (p2 `1 ) = ( - 1) by A67, EUCLID: 52;

                  then p2 = q22 & (p1 `1 ) <= (p2 `1 ) by A1, A2, A6, A17, Th47, JGRAPH_4: 82;

                  hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P) by A1, A2, A3, A5, A6, A14, A15, A17, A28, A20, A33, A36, A47, A37, A51, A52, A68, Th54, JGRAPH_4: 82;

                end;

              end;

              hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P);

            end;

              case

               A69: (p3 `1 ) > 0 ;

               A70:

              now

                per cases ;

                  case

                   A71: p3 <> p4;

                   A72:

                  now

                    

                     A73: LE (p2,p4,P) by A1, A3, A4, JGRAPH_3: 26, JORDAN6: 58;

                    assume that

                     A74: (p2 `1 ) = 0 and

                     A75: (p2 `2 ) = ( - 1);

                    (p2 `2 ) <= (p4 `2 ) by A11, A75, Th1;

                    then LE (p4,p2,P) by A1, A8, A29, A10, A33, A74, Th55;

                    hence contradiction by A1, A8, A74, A75, A73, JGRAPH_3: 26, JORDAN6: 57;

                  end;

                  (p3 `2 ) > (p4 `2 ) by A1, A4, A8, A33, A69, A71, Th50;

                  then

                   A76: ((p3 `2 ) / |.p3.|) >= r1 by A26, A15, XXREAL_0: 2;

                  then

                   A77: (q33 `1 ) > 0 by A16, A17, A69, JGRAPH_4: 106;

                  

                   A78: (q33 `2 ) >= 0 by A16, A17, A69, A76, JGRAPH_4: 106;

                   A79:

                  now

                    assume (p2 `1 ) = 0 ;

                    then (1 ^2 ) = (( 0 ^2 ) + ((p2 `2 ) ^2 )) by A30, JGRAPH_3: 1;

                    hence (p2 `2 ) = 1 or (p2 `2 ) = ( - 1) by SQUARE_1: 40;

                  end;

                   A80:

                  now

                    per cases by A6, A79, A72;

                      case

                       A81: (p2 `1 ) <= 0 & (p2 `2 ) >= 0 ;

                      then q22 = p2 by A17, JGRAPH_4: 82;

                      hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A1, A29, A28, A77, A78, A81, Th54;

                    end;

                      case

                       A82: (p2 `1 ) > 0 ;

                      then

                       A83: (q22 `1 ) > 0 by A14, A16, A17, Th22;

                      now

                        per cases ;

                          case p2 = p3;

                          hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A9, A16, A17, A28, A69, A76, JGRAPH_4: 106, JORDAN6: 56;

                        end;

                          case p2 <> p3;

                          then ((p2 `2 ) / |.p2.|) > ((p3 `2 ) / |.p3.|) by A1, A3, A30, A26, A69, A82, Th50;

                          then ((q22 `2 ) / |.q22.|) > ((q33 `2 ) / |.q33.|) by A30, A26, A14, A16, A17, A69, A82, Th24;

                          hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A1, A16, A17, A31, A32, A27, A28, A69, A76, A77, A83, Th55, JGRAPH_4: 106;

                        end;

                      end;

                      hence (q22 `2 ) >= 0 & LE (q22,q33,P);

                    end;

                  end;

                  ((p3 `2 ) / |.p3.|) > ((p4 `2 ) / |.p4.|) by A1, A4, A8, A11, A26, A33, A69, A71, Th50;

                  then ((q33 `2 ) / |.q33.|) > ((q44 `2 ) / |.q44.|) by A8, A11, A26, A14, A15, A17, A33, A69, Th24;

                  then ((q33 `2 ) ^2 ) > ((q44 `2 ) ^2 ) by A27, A19, A47, SQUARE_1: 16;

                  then

                   A84: ((1 ^2 ) - ((q33 `2 ) ^2 )) < ((1 ^2 ) - ((q44 `2 ) ^2 )) by XREAL_1: 15;

                  (1 ^2 ) = (((q44 `1 ) ^2 ) + ((q44 `2 ) ^2 )) by A19, JGRAPH_3: 1;

                  then

                   A85: (q44 `1 ) = ( sqrt ((1 ^2 ) - ((q44 `2 ) ^2 ))) by A36, SQUARE_1: 22;

                  

                   A86: (1 ^2 ) = (((q33 `1 ) ^2 ) + ((q33 `2 ) ^2 )) by A27, JGRAPH_3: 1;

                  then (q33 `1 ) = ( sqrt ((1 ^2 ) - ((q33 `2 ) ^2 ))) by A77, SQUARE_1: 22;

                  then (q33 `1 ) < (q44 `1 ) by A86, A85, A84, SQUARE_1: 27, XREAL_1: 63;

                  hence (q22 `2 ) >= 0 & LE (q22,q33,P) & (q33 `2 ) >= 0 & LE (q33,q44,P) by A1, A28, A20, A47, A78, A80, Th54;

                end;

                  case

                   A87: p3 = p4;

                   A88:

                  now

                    

                     A89: LE (p2,p4,P) by A1, A3, A4, JGRAPH_3: 26, JORDAN6: 58;

                    assume

                     A90: (p2 `1 ) = 0 & (p2 `2 ) = ( - 1);

                    then LE (p4,p2,P) by A1, A8, A29, A10, A12, A33, Th55;

                    hence contradiction by A1, A8, A90, A89, JGRAPH_3: 26, JORDAN6: 57;

                  end;

                   A91:

                  now

                    assume (p2 `1 ) = 0 ;

                    then (1 ^2 ) = (( 0 ^2 ) + ((p2 `2 ) ^2 )) by A30, JGRAPH_3: 1;

                    hence (p2 `2 ) = 1 or (p2 `2 ) = ( - 1) by SQUARE_1: 40;

                  end;

                  

                   A92: ((p3 `2 ) / |.p3.|) >= r1 by A26, A15, A87;

                  then

                   A93: (q33 `1 ) > 0 by A16, A17, A69, JGRAPH_4: 106;

                  

                   A94: (q33 `2 ) >= 0 by A16, A17, A69, A92, JGRAPH_4: 106;

                  now

                    per cases by A6, A91, A88;

                      case

                       A95: (p2 `1 ) <= 0 & (p2 `2 ) >= 0 ;

                      then q22 = p2 by A17, JGRAPH_4: 82;

                      hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A1, A29, A28, A93, A94, A95, Th54;

                    end;

                      case

                       A96: (p2 `1 ) > 0 ;

                      then

                       A97: (q22 `1 ) > 0 by A14, A16, A17, Th22;

                      now

                        per cases ;

                          case p2 = p3;

                          hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A9, A16, A17, A28, A69, A92, JGRAPH_4: 106, JORDAN6: 56;

                        end;

                          case p2 <> p3;

                          then ((p2 `2 ) / |.p2.|) > ((p3 `2 ) / |.p3.|) by A1, A3, A30, A26, A69, A96, Th50;

                          then ((q22 `2 ) / |.q22.|) > ((q33 `2 ) / |.q33.|) by A30, A26, A14, A16, A17, A69, A96, Th24;

                          hence (q22 `2 ) >= 0 & LE (q22,q33,P) by A1, A16, A17, A31, A32, A27, A28, A69, A92, A93, A97, Th55, JGRAPH_4: 106;

                        end;

                      end;

                      hence (q22 `2 ) >= 0 & LE (q22,q33,P);

                    end;

                  end;

                  hence (q22 `2 ) >= 0 & LE (q22,q33,P) & (q33 `2 ) >= 0 & LE (q33,q44,P) by A1, A28, A36, A47, A87, Th54;

                end;

              end;

               A98:

              now

                 LE (p1,p3,P) by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 58;

                then

                 A99: LE (p1,p4,P) by A1, A4, JGRAPH_3: 26, JORDAN6: 58;

                assume

                 A100: (p1 `1 ) = 0 & (p1 `2 ) = ( - 1);

                then LE (p4,p1,P) by A1, A8, A21, A10, A12, A33, Th55;

                hence contradiction by A1, A8, A100, A99, JGRAPH_3: 26, JORDAN6: 57;

              end;

               A101:

              now

                assume (p2 `1 ) = 0 ;

                then (1 ^2 ) = (( 0 ^2 ) + ((p2 `2 ) ^2 )) by A30, JGRAPH_3: 1;

                hence (p2 `2 ) = 1 or (p2 `2 ) = ( - 1) by SQUARE_1: 40;

              end;

               A102:

              now

                

                 A103: LE (p2,p4,P) by A1, A3, A4, JGRAPH_3: 26, JORDAN6: 58;

                assume that

                 A104: (p2 `1 ) = 0 and

                 A105: (p2 `2 ) = ( - 1);

                (p2 `2 ) <= (p4 `2 ) by A11, A105, Th1;

                then LE (p4,p2,P) by A1, A8, A29, A10, A33, A104, Th55;

                hence contradiction by A1, A8, A104, A105, A103, JGRAPH_3: 26, JORDAN6: 57;

              end;

               A106:

              now

                assume (p1 `1 ) = 0 ;

                then (1 ^2 ) = (( 0 ^2 ) + ((p1 `2 ) ^2 )) by A22, JGRAPH_3: 1;

                hence (p1 `2 ) = 1 or (p1 `2 ) = ( - 1) by SQUARE_1: 40;

              end;

              now

                per cases by A5, A106, A98;

                  case

                   A107: (p1 `1 ) <= 0 & (p1 `2 ) >= 0 ;

                  then

                   A108: p1 = q11 by A17, JGRAPH_4: 82;

                  

                   A109: (q11 `2 ) >= 0 by A17, A107, JGRAPH_4: 82;

                  now

                    per cases by A6, A101, A102;

                      case (p2 `1 ) <= 0 & (p2 `2 ) >= 0 ;

                      hence (q11 `2 ) >= 0 & LE (q11,q22,P) by A2, A17, A107, A108, JGRAPH_4: 82;

                    end;

                      case (p2 `1 ) > 0 ;

                      then (q11 `1 ) < (q22 `1 ) by A14, A16, A17, A107, A108, Th22;

                      hence (q11 `2 ) >= 0 & LE (q11,q22,P) by A1, A24, A32, A70, A109, Th54;

                    end;

                  end;

                  hence (q11 `2 ) >= 0 & LE (q11,q22,P);

                end;

                  case

                   A110: (p1 `1 ) > 0 ;

                  then

                   A111: (q11 `1 ) > 0 by A14, A16, A17, Th22;

                  now

                    per cases by A6, A101, A102;

                      case

                       A112: (p2 `1 ) <= 0 & (p2 `2 ) >= 0 ;

                      now

                        

                         A113: p2 in ( Upper_Arc P) by A29, A34, A112;

                        assume

                         A114: (p1 `2 ) < 0 ;

                        ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

                        then

                         A115: p1 <> ( W-min P) by A114, EUCLID: 52;

                        p1 in ( Lower_Arc P) by A21, A48, A114;

                        then LE (p2,p1,P) by A113, A115;

                        hence contradiction by A1, A2, A110, A112, JGRAPH_3: 26, JORDAN6: 57;

                      end;

                      then LE (p2,p1,P) by A1, A21, A29, A110, A112, Th54;

                      then q11 = q22 by A1, A2, JGRAPH_3: 26, JORDAN6: 57;

                      hence (q11 `2 ) >= 0 & LE (q11,q22,P) by A9, A17, A24, A112, JGRAPH_4: 82, JORDAN6: 56;

                    end;

                      case

                       A116: (p2 `1 ) > 0 ;

                      then

                       A117: (q22 `1 ) > 0 by A14, A16, A17, Th22;

                      now

                        per cases ;

                          case p1 = p2;

                          hence (q11 `2 ) >= 0 & LE (q11,q22,P) by A1, A24, A70, JGRAPH_3: 26, JORDAN6: 56;

                        end;

                          case p1 <> p2;

                          then ((p1 `2 ) / |.p1.|) > ((p2 `2 ) / |.p2.|) by A1, A2, A22, A30, A110, A116, Th50;

                          then ((q11 `2 ) / |.q11.|) > ((q22 `2 ) / |.q22.|) by A22, A30, A14, A16, A17, A110, A116, Th24;

                          hence (q11 `2 ) >= 0 & LE (q11,q22,P) by A1, A23, A24, A31, A32, A70, A111, A117, Th55;

                        end;

                      end;

                      hence (q11 `2 ) >= 0 & LE (q11,q22,P);

                    end;

                  end;

                  hence (q11 `2 ) >= 0 & LE (q11,q22,P);

                end;

              end;

              hence (q11 `2 ) >= 0 & (q22 `2 ) >= 0 & (q33 `2 ) >= 0 & (q44 `2 ) > 0 & LE (q11,q22,P) & LE (q22,q33,P) & LE (q33,q44,P) by A8, A15, A17, A33, A35, A37, A70, JGRAPH_4: 106;

            end;

          end;

          for q be Point of ( TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A17, JGRAPH_4: 97;

          hence thesis by A18, A49;

        end;

          case

           A118: (p4 `2 ) > 0 ;

          

           A119: ( Lower_Arc P) = { p8 where p8 be Point of ( TOP-REAL 2) : p8 in P & (p8 `2 ) <= 0 } by A1, Th35;

           A120:

          now

            assume p4 in ( Lower_Arc P);

            then ex p9 be Point of ( TOP-REAL 2) st p9 = p4 & p9 in P & (p9 `2 ) <= 0 by A119;

            hence contradiction by A118;

          end;

          

           A121: ( Upper_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) >= 0 } by A1, Th34;

          p3 in ( Upper_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) or p3 in ( Upper_Arc P) & p4 in ( Upper_Arc P) & LE (p3,p4,( Upper_Arc P),( W-min P),( E-max P)) or p3 in ( Lower_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) & LE (p3,p4,( Lower_Arc P),( E-max P),( W-min P)) by A4;

          then

           A122: ex p33 be Point of ( TOP-REAL 2) st p33 = p3 & p33 in P & (p33 `2 ) >= 0 by A121, A120;

          set f4 = ( id ( TOP-REAL 2));

          

           A123: (f4 . p3) = p3 & (f4 . p4) = p4;

          

           A124: for q be Point of ( TOP-REAL 2) holds |.(f4 . q).| = |.q.|;

          

           A125: LE (p2,p4,P) by A1, A3, A4, JGRAPH_3: 26, JORDAN6: 58;

          then p2 in ( Upper_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) or p2 in ( Upper_Arc P) & p4 in ( Upper_Arc P) & LE (p2,p4,( Upper_Arc P),( W-min P),( E-max P)) or p2 in ( Lower_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) & LE (p2,p4,( Lower_Arc P),( E-max P),( W-min P));

          then

           A126: ex p22 be Point of ( TOP-REAL 2) st p22 = p2 & p22 in P & (p22 `2 ) >= 0 by A121, A120;

           LE (p1,p4,P) by A1, A2, A125, JGRAPH_3: 26, JORDAN6: 58;

          then p1 in ( Upper_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) or p1 in ( Upper_Arc P) & p4 in ( Upper_Arc P) & LE (p1,p4,( Upper_Arc P),( W-min P),( E-max P)) or p1 in ( Lower_Arc P) & p4 in ( Lower_Arc P) & not p4 = ( W-min P) & LE (p1,p4,( Lower_Arc P),( E-max P),( W-min P));

          then

           A127: ex p11 be Point of ( TOP-REAL 2) st p11 = p1 & p11 in P & (p11 `2 ) >= 0 by A121, A120;

          (f4 . p1) = p1 & (f4 . p2) = p2;

          hence thesis by A2, A3, A4, A118, A122, A126, A127, A123, A124;

        end;

      end;

      hence thesis;

    end;

    theorem :: JGRAPH_5:63

    

     Th63: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & ((p1 `2 ) >= 0 or (p1 `1 ) >= 0 ) & ((p2 `2 ) >= 0 or (p2 `1 ) >= 0 ) & ((p3 `2 ) >= 0 or (p3 `1 ) >= 0 ) & ((p4 `2 ) > 0 or (p4 `1 ) > 0 ) holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 & (q4 `2 ) < 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & ((p1 `2 ) >= 0 or (p1 `1 ) >= 0 ) & ((p2 `2 ) >= 0 or (p2 `1 ) >= 0 ) & ((p3 `2 ) >= 0 or (p3 `1 ) >= 0 ) & ((p4 `2 ) > 0 or (p4 `1 ) > 0 );

      consider f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) such that

       A3: f is being_homeomorphism and

       A4: for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.| and

       A5: q1 = (f . p1) & q2 = (f . p2) and

       A6: q3 = (f . p3) & q4 = (f . p4) and

       A7: (q1 `2 ) >= 0 & (q2 `2 ) >= 0 & (q3 `2 ) >= 0 & (q4 `2 ) > 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P) by A1, A2, Th62;

      consider f2 be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1b,q2b,q3b,q4b be Point of ( TOP-REAL 2) such that

       A8: f2 is being_homeomorphism and

       A9: for q be Point of ( TOP-REAL 2) holds |.(f2 . q).| = |.q.| and

       A10: q1b = (f2 . q1) & q2b = (f2 . q2) and

       A11: q3b = (f2 . q3) & q4b = (f2 . q4) and

       A12: (q1b `1 ) < 0 & (q1b `2 ) < 0 & (q2b `1 ) < 0 & (q2b `2 ) < 0 & (q3b `1 ) < 0 & (q3b `2 ) < 0 & (q4b `1 ) < 0 & (q4b `2 ) < 0 & LE (q1b,q2b,P) & LE (q2b,q3b,P) & LE (q3b,q4b,P) by A1, A7, Th61;

      reconsider f3 = (f2 * f) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A13: f3 is being_homeomorphism by A3, A8, TOPS_2: 57;

      

       A14: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      then

       A15: (f3 . p3) = q3b & (f3 . p4) = q4b by A6, A11, FUNCT_1: 13;

      

       A16: for q be Point of ( TOP-REAL 2) holds |.(f3 . q).| = |.q.|

      proof

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

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

        then (f3 . q) = (f2 . (f . q)) by FUNCT_1: 13;

        

        hence |.(f3 . q).| = |.(f . q).| by A9

        .= |.q.| by A4;

      end;

      (f3 . p1) = q1b & (f3 . p2) = q2b by A5, A10, A14, FUNCT_1: 13;

      hence thesis by A12, A13, A16, A15;

    end;

    theorem :: JGRAPH_5:64

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & p4 = ( W-min P) & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 & (q4 `2 ) < 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: p4 = ( W-min P) and

       A3: LE (p1,p2,P) and

       A4: LE (p2,p3,P) and

       A5: LE (p3,p4,P);

      

       A6: ( Upper_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) >= 0 } by A1, Th34;

      

       A7: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      then

       A8: (( W-min P) `2 ) = 0 by EUCLID: 52;

      

       A9: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then p4 in P by A5, JORDAN7: 5;

      then

       A10: p4 in ( Upper_Arc P) by A2, A6, A8;

      

       A11: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A9, JORDAN6:def 8;

      

       A12: p3 in ( Upper_Arc P) by A1, A5, A10, Th44;

      then LE (p4,p3,( Upper_Arc P),( W-min P),( E-max P)) by A2, A11, JORDAN5C: 10;

      then LE (p4,p3,P) by A10, A12;

      then

       A13: p3 = p4 by A1, A5, JGRAPH_3: 26, JORDAN6: 57;

      

       A14: LE (p2,p4,P) by A1, A4, A5, JGRAPH_3: 26, JORDAN6: 58;

      

       A15: p2 in ( Upper_Arc P) by A1, A4, A12, Th44;

      then LE (p4,p2,( Upper_Arc P),( W-min P),( E-max P)) by A2, A11, JORDAN5C: 10;

      then LE (p4,p2,P) by A10, A15;

      then

       A16: p2 = p4 by A1, A14, JGRAPH_3: 26, JORDAN6: 57;

      

       A17: (( W-min P) `1 ) = ( - 1) by A7, EUCLID: 52;

      

       A18: p1 in ( Upper_Arc P) by A1, A3, A15, Th44;

      then LE (p4,p1,( Upper_Arc P),( W-min P),( E-max P)) by A2, A11, JORDAN5C: 10;

      then LE (p4,p1,P) by A10, A18;

      then p1 = p4 by A1, A3, A16, JGRAPH_3: 26, JORDAN6: 57;

      hence thesis by A1, A2, A3, A17, A8, A13, A16, Th59;

    end;

    theorem :: JGRAPH_5:65

    

     Th65: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & q1 = (f . p1) & q2 = (f . p2) & q3 = (f . p3) & q4 = (f . p4) & (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 & (q4 `2 ) < 0 & LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: LE (p2,p3,P) and

       A4: LE (p3,p4,P);

      

       A5: ( Lower_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) <= 0 } by A1, Th35;

      

       A6: ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

      then

       A7: (( W-min P) `2 ) = 0 by EUCLID: 52;

      

       A8: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then

       A9: p1 in P by A2, JORDAN7: 5;

      

       A10: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A8, JORDAN6:def 8;

      

       A11: ( Upper_Arc P) = { p7 where p7 be Point of ( TOP-REAL 2) : p7 in P & (p7 `2 ) >= 0 } by A1, Th34;

      

       A12: p4 in P by A4, A8, JORDAN7: 5;

      then

       A13: ex p44 be Point of ( TOP-REAL 2) st p44 = p4 & |.p44.| = 1 by A1;

      then

       A14: (p4 `1 ) <= 1 by Th1;

      

       A15: ( - 1) <= (p4 `1 ) by A13, Th1;

      now

        per cases ;

          case

           A16: (p4 `1 ) = ( - 1);

          (1 ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by A13, JGRAPH_3: 1

          .= (((p4 `2 ) ^2 ) + 1) by A16;

          then

           A17: (p4 `2 ) = 0 by XCMPLX_1: 6;

          then

           A18: p4 in ( Upper_Arc P) by A12, A11;

          

           A19: p4 = ( W-min P) by A6, A16, A17, EUCLID: 53;

           A20:

          now

            per cases ;

              case

               A21: p1 in ( Upper_Arc P);

              then LE (p4,p1,( Upper_Arc P),( W-min P),( E-max P)) by A10, A19, JORDAN5C: 10;

              hence LE (p4,p1,P) by A18, A21;

            end;

              case not p1 in ( Upper_Arc P);

              then

               A22: (p1 `2 ) < 0 by A9, A11;

              then p1 in ( Lower_Arc P) by A9, A5;

              hence LE (p4,p1,P) by A7, A18, A22;

            end;

          end;

          then

           A23: LE (p4,p2,P) by A1, A2, JGRAPH_3: 26, JORDAN6: 58;

          then LE (p4,p3,P) by A1, A3, JGRAPH_3: 26, JORDAN6: 58;

          then

           A24: p3 = p4 by A1, A4, JGRAPH_3: 26, JORDAN6: 57;

           LE (p2,p4,P) by A1, A3, A4, JGRAPH_3: 26, JORDAN6: 58;

          then

           A25: p2 = p4 by A1, A23, JGRAPH_3: 26, JORDAN6: 57;

           LE (p1,p3,P) by A1, A2, A3, JGRAPH_3: 26, JORDAN6: 58;

          then LE (p1,p4,P) by A1, A4, JGRAPH_3: 26, JORDAN6: 58;

          then p4 = p1 by A1, A20, JGRAPH_3: 26, JORDAN6: 57;

          hence thesis by A1, A2, A16, A17, A25, A24, Th59;

        end;

          case

           A26: (p4 `1 ) <> ( - 1);

          then (p4 `1 ) > ( - 1) by A15, XXREAL_0: 1;

          then

          consider r be Real such that

           A27: ( - 1) < r and

           A28: r < (p4 `1 ) by XREAL_1: 5;

          reconsider r1 = r as Real;

          

           A29: r1 < 1 by A14, A28, XXREAL_0: 2;

          then

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

           A30: f1 = (r1 -FanMorphS ) and

           A31: f1 is being_homeomorphism by A27, JGRAPH_4: 136;

          set q11 = (f1 . p1), q22 = (f1 . p2), q33 = (f1 . p3), q44 = (f1 . p4);

          now

            per cases ;

              case

               A32: (p4 `1 ) > 0 or (p4 `2 ) >= 0 ;

               A33:

              now

                assume that

                 A34: (p4 `2 ) = 0 and

                 A35: (p4 `1 ) <= 0 ;

                (1 ^2 ) = (((p4 `1 ) ^2 ) + ((p4 `2 ) ^2 )) by A13, JGRAPH_3: 1

                .= ((p4 `1 ) ^2 ) by A34;

                hence contradiction by A26, A35, SQUARE_1: 40;

              end;

              

               A36: (p3 `1 ) >= 0 or (p3 `2 ) >= 0 by A1, A4, A32, Th49;

              then

               A37: (p2 `1 ) >= 0 or (p2 `2 ) >= 0 by A1, A3, Th49;

              then (p1 `1 ) >= 0 or (p1 `2 ) >= 0 by A1, A2, Th49;

              hence thesis by A1, A2, A3, A4, A32, A36, A37, A33, Th63;

            end;

              case

               A38: (p4 `1 ) <= 0 & (p4 `2 ) < 0 ;

              ((p4 `1 ) / |.p4.|) > r1 by A13, A28;

              then

               A39: (q44 `1 ) > 0 by A27, A28, A30, A38, Th26;

              

               A40: LE (q33,q44,P) by A1, A4, A27, A29, A30, Th58;

              ( W-min P) = |[( - 1), 0 ]| by A1, Th29;

              then

               A41: (( W-min P) `2 ) = 0 by EUCLID: 52;

               A42:

              now

                per cases ;

                  case (q33 `2 ) >= 0 ;

                  hence (q33 `2 ) >= 0 or (q33 `1 ) >= 0 ;

                end;

                  case (q33 `2 ) < 0 ;

                  thus (q33 `2 ) >= 0 or (q33 `1 ) >= 0 by A1, A39, A40, A41, Th48;

                end;

              end;

              

               A43: LE (q22,q33,P) by A1, A3, A27, A29, A30, Th58;

               A44:

              now

                per cases ;

                  case (q22 `2 ) >= 0 ;

                  hence (q22 `2 ) >= 0 or (q22 `1 ) >= 0 ;

                end;

                  case (q22 `2 ) < 0 ;

                  thus (q22 `2 ) >= 0 or (q22 `1 ) >= 0 by A1, A8, A39, A40, A43, A41, Th48, JORDAN6: 58;

                end;

              end;

              

               A45: LE (q11,q22,P) by A1, A2, A27, A29, A30, Th58;

              

               A46: LE (q22,q44,P) by A1, A40, A43, JGRAPH_3: 26, JORDAN6: 58;

              now

                per cases ;

                  case (q11 `2 ) >= 0 ;

                  hence (q11 `2 ) >= 0 or (q11 `1 ) >= 0 ;

                end;

                  case (q11 `2 ) < 0 ;

                  thus (q11 `2 ) >= 0 or (q11 `1 ) >= 0 by A1, A8, A39, A46, A45, A41, Th48, JORDAN6: 58;

                end;

              end;

              then

              consider f2 be Function of ( TOP-REAL 2), ( TOP-REAL 2), q81,q82,q83,q84 be Point of ( TOP-REAL 2) such that

               A47: f2 is being_homeomorphism and

               A48: for q be Point of ( TOP-REAL 2) holds |.(f2 . q).| = |.q.| and

               A49: q81 = (f2 . q11) & q82 = (f2 . q22) and

               A50: q83 = (f2 . q33) & q84 = (f2 . q44) and

               A51: (q81 `1 ) < 0 & (q81 `2 ) < 0 & (q82 `1 ) < 0 & (q82 `2 ) < 0 & (q83 `1 ) < 0 & (q83 `2 ) < 0 & (q84 `1 ) < 0 & (q84 `2 ) < 0 & LE (q81,q82,P) & LE (q82,q83,P) & LE (q83,q84,P) by A1, A39, A40, A43, A45, A42, A44, Th63;

              reconsider f3 = (f2 * f1) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

              

               A52: ( dom f1) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              then

               A53: (f3 . p1) = q81 & (f3 . p2) = q82 by A49, FUNCT_1: 13;

              

               A54: for q be Point of ( TOP-REAL 2) holds |.(f3 . q).| = |.q.|

              proof

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

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

                then (f3 . q) = (f2 . (f1 . q)) by FUNCT_1: 13;

                

                hence |.(f3 . q).| = |.(f1 . q).| by A48

                .= |.q.| by A30, JGRAPH_4: 128;

              end;

              

               A55: (f3 . p3) = q83 & (f3 . p4) = q84 by A50, A52, FUNCT_1: 13;

              f3 is being_homeomorphism by A31, A47, TOPS_2: 57;

              hence thesis by A51, A54, A53, A55;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    begin

    theorem :: JGRAPH_5:66

    

     Th66: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & p1 <> p2 & p2 <> p3 & p3 <> p4 & (p1 `1 ) < 0 & (p2 `1 ) < 0 & (p3 `1 ) < 0 & (p4 `1 ) < 0 & (p1 `2 ) < 0 & (p2 `2 ) < 0 & (p3 `2 ) < 0 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & |[( - 1), 0 ]| = (f . p1) & |[ 0 , 1]| = (f . p2) & |[1, 0 ]| = (f . p3) & |[ 0 , ( - 1)]| = (f . p4)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) and

       A3: LE (p2,p3,P) and

       A4: LE (p3,p4,P) and

       A5: p1 <> p2 and

       A6: p2 <> p3 and

       A7: p3 <> p4 and

       A8: (p1 `1 ) < 0 and

       A9: (p2 `1 ) < 0 and

       A10: (p3 `1 ) < 0 and

       A11: (p4 `1 ) < 0 and

       A12: (p1 `2 ) < 0 and

       A13: (p2 `2 ) < 0 and

       A14: (p3 `2 ) < 0 ;

      set q2 = (((p1 `2 ) -FanMorphW ) . p2);

      set q3 = (((p1 `2 ) -FanMorphW ) . p3);

      

       A15: (p1 `2 ) < (p2 `2 ) by A1, A2, A5, A8, A12, Th45;

      set q1 = (((p1 `2 ) -FanMorphW ) . p1);

      

       A16: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

      then p1 in P by A2, JORDAN7: 5;

      then

       A17: ex p11 be Point of ( TOP-REAL 2) st p11 = p1 & |.p11.| = 1 by A1;

      then

       A18: ((p1 `2 ) / |.p1.|) = (p1 `2 );

      then

       A19: (q1 `2 ) = 0 by A8, JGRAPH_4: 47;

      

       A20: |.q1.| = 1 by A17, JGRAPH_4: 33;

      then

       A21: ((q1 `2 ) / |.q1.|) = (q1 `2 );

      p2 in P by A2, A16, JORDAN7: 5;

      then

       A22: ex p22 be Point of ( TOP-REAL 2) st p22 = p2 & |.p22.| = 1 by A1;

      then

       A23: ((p2 `2 ) / |.p2.|) = (p2 `2 );

      then

       A24: (q2 `1 ) < 0 by A9, A12, A15, JGRAPH_4: 42;

      

       A25: |.q2.| = 1 by A22, JGRAPH_4: 33;

      then

       A26: ((q2 `2 ) / |.q2.|) = (q2 `2 );

      then

       A27: (q1 `2 ) < (q2 `2 ) by A8, A9, A12, A15, A18, A23, A21, JGRAPH_4: 44;

      then

       A28: (q2 `1 ) < 1 by A25, A19, Th2;

      p3 in P by A3, A16, JORDAN7: 5;

      then

       A29: ex p33 be Point of ( TOP-REAL 2) st p33 = p3 & |.p33.| = 1 by A1;

      then

       A30: |.q3.| = 1 by JGRAPH_4: 33;

      then

       A31: ((q3 `2 ) / |.q3.|) = (q3 `2 );

      set r3 = (((q2 `1 ) -FanMorphN ) . q3);

      set r2 = (((q2 `1 ) -FanMorphN ) . q2);

      

       A32: ((q3 `1 ) / |.q3.|) = (q3 `1 ) by A30;

      

       A33: (p2 `2 ) < (p3 `2 ) by A1, A3, A6, A9, A13, Th45;

      then

       A34: ((p3 `2 ) / |.p3.|) > (p1 `2 ) by A15, A29, XXREAL_0: 2;

      then

       A35: (q3 `1 ) < 0 by A10, A12, JGRAPH_4: 42;

      

       A36: (1 ^2 ) = (((q3 `1 ) ^2 ) + ((q3 `2 ) ^2 )) by A30, JGRAPH_3: 1;

      

       A37: (1 ^2 ) = (((q2 `1 ) ^2 ) + ((q2 `2 ) ^2 )) by A25, JGRAPH_3: 1;

      ((p3 `2 ) / |.p3.|) > (p2 `2 ) by A1, A3, A6, A9, A13, A29, Th45;

      then

       A38: (q2 `2 ) < (q3 `2 ) by A9, A10, A12, A15, A23, A26, A31, A34, JGRAPH_4: 44;

      then ((q3 `2 ) ^2 ) > ((q2 `2 ) ^2 ) by A19, A27, SQUARE_1: 16;

      then (( - (q2 `1 )) ^2 ) > ((q3 `1 ) ^2 ) by A37, A36, XREAL_1: 8;

      then

       A39: ( - ( - (q2 `1 ))) < (q3 `1 ) by A24, SQUARE_1: 48;

      

       A40: 0 < (q3 `2 ) by A8, A10, A12, A18, A21, A31, A19, A34, JGRAPH_4: 44;

      then

       A41: (r3 `2 ) > 0 by A39, A28, A32, JGRAPH_4: 75;

      

       A42: |.r3.| = 1 by A30, JGRAPH_4: 66;

      then

       A43: ((r3 `1 ) / |.r3.|) = (r3 `1 );

      

       A44: ( - 1) < (p1 `2 ) by A8, A12, A17, Th2;

      then

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

       A45: f1 = ((p1 `2 ) -FanMorphW ) and

       A46: f1 is being_homeomorphism by A12, JGRAPH_4: 41;

      

       A47: ( - 1) < (q2 `1 ) by A12, A44, A25, A19, A27, Th2;

      then

      consider f2 be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

       A48: f2 = ((q2 `1 ) -FanMorphN ) and

       A49: f2 is being_homeomorphism by A28, JGRAPH_4: 74;

      

       A50: ((q2 `1 ) / |.q2.|) = (q2 `1 ) by A25;

      then

       A51: (r2 `1 ) = 0 by A19, A27, JGRAPH_4: 80;

      

       A52: |.r2.| = 1 by A25, JGRAPH_4: 66;

      then

       A53: ((r2 `1 ) / |.r2.|) = (r2 `1 );

      then

       A54: (r2 `1 ) < (r3 `1 ) by A19, A27, A38, A39, A47, A28, A50, A32, A43, JGRAPH_4: 79;

      then

       A55: ( - 1) < (r3 `2 ) by A12, A44, A42, A51, Th2;

      (q1 `2 ) < (q2 `2 ) by A8, A9, A12, A15, A18, A23, A21, A26, JGRAPH_4: 44;

      then

       A56: (r2 `1 ) < (r3 `1 ) by A19, A40, A39, A47, A28, A50, A32, A53, A43, JGRAPH_4: 79;

      set q4 = (((p1 `2 ) -FanMorphW ) . p4);

      p4 in P by A4, A16, JORDAN7: 5;

      then

       A57: ex p44 be Point of ( TOP-REAL 2) st p44 = p4 & |.p44.| = 1 by A1;

      then

       A58: |.q4.| = 1 by JGRAPH_4: 33;

      then

       A59: ((q4 `2 ) / |.q4.|) = (q4 `2 );

      (p3 `2 ) < (p4 `2 ) by A1, A4, A7, A10, A14, Th45;

      then ((p4 `2 ) / |.p4.|) > (p2 `2 ) by A33, A57, XXREAL_0: 2;

      then

       A60: ((p4 `2 ) / |.p4.|) > (p1 `2 ) by A15, XXREAL_0: 2;

      ((p4 `2 ) / |.p4.|) > (p3 `2 ) by A1, A4, A7, A10, A14, A57, Th45;

      then (q3 `2 ) < (q4 `2 ) by A10, A11, A12, A29, A31, A59, A34, A60, JGRAPH_4: 44;

      then

       A61: ((q4 `2 ) ^2 ) > ((q3 `2 ) ^2 ) by A19, A27, A38, SQUARE_1: 16;

      (1 ^2 ) = (((q4 `1 ) ^2 ) + ((q4 `2 ) ^2 )) by A58, JGRAPH_3: 1;

      then (( - (q3 `1 )) ^2 ) > ((q4 `1 ) ^2 ) by A36, A61, XREAL_1: 8;

      then ( - ( - (q3 `1 ))) < (q4 `1 ) by A35, SQUARE_1: 48;

      then

       A62: ((q4 `1 ) / |.q4.|) > (q3 `1 ) by A58;

      set r4 = (((q2 `1 ) -FanMorphN ) . q4);

      

       A63: (1 ^2 ) = (((r3 `1 ) ^2 ) + ((r3 `2 ) ^2 )) by A42, JGRAPH_3: 1;

      

       A64: |.r4.| = 1 by A58, JGRAPH_4: 66;

      then

       A65: ((r4 `1 ) / |.r4.|) = (r4 `1 );

      set r1 = (((q2 `1 ) -FanMorphN ) . q1);

      ( |.q1.| ^2 ) = (((q1 `1 ) ^2 ) + ((q1 `2 ) ^2 )) by JGRAPH_3: 1;

      then

       A66: (q1 `1 ) = ( - 1) or (q1 `1 ) = 1 by A20, A19, SQUARE_1: 40;

      then

       A67: (r1 `1 ) = ( - 1) by A8, A18, A19, JGRAPH_4: 47, JGRAPH_4: 49;

      

       A68: (1 ^2 ) = (((r4 `1 ) ^2 ) + ((r4 `2 ) ^2 )) by A64, JGRAPH_3: 1;

       0 < (q4 `2 ) by A8, A11, A12, A18, A21, A59, A19, A60, JGRAPH_4: 44;

      then

       A69: (r3 `1 ) < (r4 `1 ) by A40, A47, A28, A32, A43, A65, A62, JGRAPH_4: 79;

      then ((r4 `1 ) ^2 ) > ((r3 `1 ) ^2 ) by A51, A56, SQUARE_1: 16;

      then ((((r3 `2 ) ^2 ) - ((r4 `2 ) ^2 )) + ((r4 `2 ) ^2 )) > ( 0 + ((r4 `2 ) ^2 )) by A63, A68, XREAL_1: 8;

      then

       A70: (r3 `2 ) > (r4 `2 ) by A41, SQUARE_1: 48;

      set s4 = (((r3 `2 ) -FanMorphE ) . r4);

      set s1 = (((r3 `2 ) -FanMorphE ) . r1);

      (r1 `2 ) = 0 by A19, JGRAPH_4: 49;

      then

       A71: (s1 `2 ) = 0 by A67, JGRAPH_4: 82;

      set t4 = (((s4 `1 ) -FanMorphS ) . s4);

      set s3 = (((r3 `2 ) -FanMorphE ) . r3);

      set s2 = (((r3 `2 ) -FanMorphE ) . r2);

      

       A72: ( |.s3.| ^2 ) = (((s3 `1 ) ^2 ) + ((s3 `2 ) ^2 )) by JGRAPH_3: 1;

      

       A73: ((r3 `2 ) / |.r3.|) = (r3 `2 ) by A42;

      then

       A74: (s3 `2 ) = 0 by A51, A56, JGRAPH_4: 111;

      ( |.r2.| ^2 ) = (((r2 `1 ) ^2 ) + ((r2 `2 ) ^2 )) by JGRAPH_3: 1;

      then

       A75: (r2 `2 ) = ( - 1) or (r2 `2 ) = 1 by A52, A51, SQUARE_1: 40;

      then r2 = |[ 0 , 1]| by A19, A27, A50, A51, EUCLID: 53, JGRAPH_4: 80;

      then

       A76: s2 = |[ 0 , 1]| by A51, JGRAPH_4: 82;

      (s2 `2 ) = 1 by A19, A27, A50, A51, A75, JGRAPH_4: 80, JGRAPH_4: 82;

      then

       A77: (((s4 `1 ) -FanMorphS ) . s2) = |[ 0 , 1]| by A76, JGRAPH_4: 113;

      

       A78: (r3 `2 ) < 1 by A42, A51, A54, Th2;

      then

      consider f3 be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

       A79: f3 = ((r3 `2 ) -FanMorphE ) and

       A80: f3 is being_homeomorphism by A55, JGRAPH_4: 105;

      

       A81: ( dom (f2 * f1)) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A82: ((r4 `2 ) / |.r4.|) = (r4 `2 ) by A64;

      then

       A83: ((s3 `2 ) / |.s3.|) > ((s4 `2 ) / |.s4.|) by A51, A56, A69, A70, A55, A78, A73, JGRAPH_4: 110;

      

       A84: |.s4.| = 1 by A64, JGRAPH_4: 97;

      then

       A85: ((s4 `1 ) / |.s4.|) = (s4 `1 );

      then

       A86: (t4 `1 ) = 0 by A84, A74, A83, JGRAPH_4: 142;

      (s4 `2 ) < 0 by A51, A56, A69, A70, A55, A82, JGRAPH_4: 107;

      then

       A87: (s4 `1 ) < 1 by A84, Th2;

      ( - 1) < (s4 `1 ) by A51, A56, A69, A70, A55, A82, JGRAPH_4: 107;

      then

      consider f4 be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

       A88: f4 = ((s4 `1 ) -FanMorphS ) and

       A89: f4 is being_homeomorphism by A87, JGRAPH_4: 136;

      reconsider g = (f4 * (f3 * (f2 * f1))) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A90: ( dom (f3 * (f2 * f1))) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      (f2 * f1) is being_homeomorphism by A46, A49, TOPS_2: 57;

      then (f3 * (f2 * f1)) is being_homeomorphism by A80, TOPS_2: 57;

      then

       A91: g is being_homeomorphism by A89, TOPS_2: 57;

      

       A92: ( dom g) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

      then

       A93: (g . p2) = (f4 . ((f3 * (f2 * f1)) . p2)) by FUNCT_1: 12

      .= (f4 . (f3 . ((f2 * f1) . p2))) by A90, FUNCT_1: 12

      .= |[ 0 , 1]| by A45, A48, A79, A88, A77, A81, FUNCT_1: 12;

       |.s3.| = 1 by A42, JGRAPH_4: 97;

      then (s3 `1 ) = ( - 1) or (s3 `1 ) = 1 by A74, A72, SQUARE_1: 40;

      then s3 = |[1, 0 ]| by A51, A56, A73, A74, EUCLID: 53, JGRAPH_4: 111;

      then

       A94: (((s4 `1 ) -FanMorphS ) . s3) = |[1, 0 ]| by A74, JGRAPH_4: 113;

      q1 = |[( - 1), 0 ]| by A8, A18, A19, A66, EUCLID: 53, JGRAPH_4: 47;

      then r1 = |[( - 1), 0 ]| by A19, JGRAPH_4: 49;

      then s1 = |[( - 1), 0 ]| by A67, JGRAPH_4: 82;

      then

       A95: (((s4 `1 ) -FanMorphS ) . s1) = |[( - 1), 0 ]| by A71, JGRAPH_4: 113;

      

       A96: ( |.t4.| ^2 ) = (((t4 `1 ) ^2 ) + ((t4 `2 ) ^2 )) by JGRAPH_3: 1;

       |.t4.| = 1 by A84, JGRAPH_4: 128;

      then (t4 `2 ) = ( - 1) or (t4 `2 ) = 1 by A86, A96, SQUARE_1: 40;

      then

       A97: t4 = |[ 0 , ( - 1)]| by A84, A74, A83, A85, A86, EUCLID: 53, JGRAPH_4: 142;

      

       A98: for q be Point of ( TOP-REAL 2) holds |.(g . q).| = |.q.|

      proof

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

        

         A99: |.((f2 * f1) . q).| = |.(f2 . (f1 . q)).| by A81, FUNCT_1: 12

        .= |.(f1 . q).| by A48, JGRAPH_4: 66

        .= |.q.| by A45, JGRAPH_4: 33;

        

         A100: |.((f3 * (f2 * f1)) . q).| = |.(f3 . ((f2 * f1) . q)).| by A90, FUNCT_1: 12

        .= |.q.| by A79, A99, JGRAPH_4: 97;

        

        thus |.(g . q).| = |.(f4 . ((f3 * (f2 * f1)) . q)).| by A92, FUNCT_1: 12

        .= |.q.| by A88, A100, JGRAPH_4: 128;

      end;

      

       A101: (g . p3) = (f4 . ((f3 * (f2 * f1)) . p3)) by A92, FUNCT_1: 12

      .= (f4 . (f3 . ((f2 * f1) . p3))) by A90, FUNCT_1: 12

      .= |[1, 0 ]| by A45, A48, A79, A88, A94, A81, FUNCT_1: 12;

      

       A102: (g . p4) = (f4 . ((f3 * (f2 * f1)) . p4)) by A92, FUNCT_1: 12

      .= (f4 . (f3 . ((f2 * f1) . p4))) by A90, FUNCT_1: 12

      .= |[ 0 , ( - 1)]| by A45, A48, A79, A88, A97, A81, FUNCT_1: 12;

      (g . p1) = (f4 . ((f3 * (f2 * f1)) . p1)) by A92, FUNCT_1: 12

      .= (f4 . (f3 . ((f2 * f1) . p1))) by A90, FUNCT_1: 12

      .= |[( - 1), 0 ]| by A45, A48, A79, A88, A95, A81, FUNCT_1: 12;

      hence thesis by A91, A98, A93, A101, A102;

    end;

    theorem :: JGRAPH_5:67

    

     Th67: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) & p1 <> p2 & p2 <> p3 & p3 <> p4 holds ex f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f is being_homeomorphism & (for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.|) & |[( - 1), 0 ]| = (f . p1) & |[ 0 , 1]| = (f . p2) & |[1, 0 ]| = (f . p3) & |[ 0 , ( - 1)]| = (f . p4)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2);

      assume that

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

       A2: LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) and

       A3: p1 <> p2 & p2 <> p3 and

       A4: p3 <> p4;

      consider f be Function of ( TOP-REAL 2), ( TOP-REAL 2), q1,q2,q3,q4 be Point of ( TOP-REAL 2) such that

       A5: f is being_homeomorphism and

       A6: for q be Point of ( TOP-REAL 2) holds |.(f . q).| = |.q.| and

       A7: q1 = (f . p1) & q2 = (f . p2) and

       A8: q3 = (f . p3) and

       A9: q4 = (f . p4) and

       A10: (q1 `1 ) < 0 & (q1 `2 ) < 0 & (q2 `1 ) < 0 & (q2 `2 ) < 0 & (q3 `1 ) < 0 & (q3 `2 ) < 0 & (q4 `1 ) < 0 and (q4 `2 ) < 0 and

       A11: LE (q1,q2,P) & LE (q2,q3,P) & LE (q3,q4,P) by A1, A2, Th65;

      

       A12: ( dom f) = the carrier of ( TOP-REAL 2) & f is one-to-one by A5, FUNCT_2:def 1, TOPS_2:def 5;

      then

       A13: q3 <> q4 by A4, A8, A9, FUNCT_1:def 4;

      q1 <> q2 & q2 <> q3 by A3, A7, A8, A12, FUNCT_1:def 4;

      then

      consider f2 be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

       A14: f2 is being_homeomorphism and

       A15: for q be Point of ( TOP-REAL 2) holds |.(f2 . q).| = |.q.| and

       A16: |[( - 1), 0 ]| = (f2 . q1) & |[ 0 , 1]| = (f2 . q2) and

       A17: |[1, 0 ]| = (f2 . q3) & |[ 0 , ( - 1)]| = (f2 . q4) by A1, A10, A11, A13, Th66;

      reconsider f3 = (f2 * f) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A18: f3 is being_homeomorphism by A5, A14, TOPS_2: 57;

      

       A19: ( dom f3) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      then

       A20: (f3 . p1) = |[( - 1), 0 ]| & (f3 . p2) = |[ 0 , 1]| by A7, A16, FUNCT_1: 12;

      

       A21: for q be Point of ( TOP-REAL 2) holds |.(f3 . q).| = |.q.|

      proof

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

         |.(f3 . q).| = |.(f2 . (f . q)).| by A19, FUNCT_1: 12

        .= |.(f . q).| by A15

        .= |.q.| by A6;

        hence thesis;

      end;

      (f3 . p3) = |[1, 0 ]| & (f3 . p4) = |[ 0 , ( - 1)]| by A8, A9, A17, A19, FUNCT_1: 12;

      hence thesis by A18, A21, A20;

    end;

    

     Lm7: ( |[( - 1), 0 ]| `1 ) = ( - 1) by EUCLID: 52;

    

     Lm8: ( |[( - 1), 0 ]| `2 ) = 0 by EUCLID: 52;

    

     Lm9: ( |[1, 0 ]| `1 ) = 1 & ( |[1, 0 ]| `2 ) = 0 by EUCLID: 52;

    

     Lm10: ( |[ 0 , ( - 1)]| `1 ) = 0 by EUCLID: 52;

    

     Lm11: ( |[ 0 , ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;

    

     Lm12: ( |[ 0 , 1]| `1 ) = 0 by EUCLID: 52;

    

     Lm13: ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

     Lm14:

    now

      

      thus |. |[( - 1), 0 ]|.| = ( sqrt ((( - 1) ^2 ) + ( 0 ^2 ))) by Lm7, Lm8, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      

      thus |. |[1, 0 ]|.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by Lm9, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      

      thus |. |[ 0 , ( - 1)]|.| = ( sqrt (( 0 ^2 ) + (( - 1) ^2 ))) by Lm10, Lm11, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

      

      thus |. |[ 0 , 1]|.| = ( sqrt (( 0 ^2 ) + (1 ^2 ))) by Lm12, Lm13, JGRAPH_3: 1

      .= 1 by SQUARE_1: 18;

    end;

    

     Lm15: 0 in [. 0 , 1.] by XXREAL_1: 1;

    

     Lm16: 1 in [. 0 , 1.] by XXREAL_1: 1;

    theorem :: JGRAPH_5:68

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P);

      let f,g be Function of I[01] , ( TOP-REAL 2);

      assume

       A2: f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0;

      

       A3: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

       A4: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      per cases ;

        suppose

         A5: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

        now

          per cases by A5;

            case

             A6: p1 = p2;

            p1 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A6, XBOOLE_0: 3;

          end;

            case

             A7: p2 = p3;

            p3 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A7, XBOOLE_0: 3;

          end;

            case

             A8: p3 = p4;

            p3 in ( rng f) & p4 in ( rng g) by A2, A4, A3, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A8, XBOOLE_0: 3;

          end;

        end;

        hence thesis;

      end;

        suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

        then

        consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

         A9: h is being_homeomorphism and

         A10: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

         A11: |[( - 1), 0 ]| = (h . p1) and

         A12: |[ 0 , 1]| = (h . p2) and

         A13: |[1, 0 ]| = (h . p3) and

         A14: |[ 0 , ( - 1)]| = (h . p4) by A1, Th67;

        

         A15: h is one-to-one by A9, TOPS_2:def 5;

        reconsider h1 = h as Function;

        reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

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

        { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

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

        

         A16: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

        

         A17: ( - ( |[ 0 , 1]| `1 )) = 0 by Lm12;

        reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

        

         A18: ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm10;

        

         A19: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;

        then (g2 . 0 ) = |[ 0 , 1]| by A2, A12, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A20: (g2 . O) in KYP by A17, Lm13, Lm14;

        

         A21: ( rng g2) c= C0

        proof

          let y be object;

          assume y in ( rng g2);

          then

          consider x be object such that

           A22: x in ( dom g2) and

           A23: y = (g2 . x) by FUNCT_1:def 3;

          

           A24: (g . x) in ( rng g) by A3, A22, FUNCT_1:def 3;

          then

          reconsider qg = (g . x) as Point of ( TOP-REAL 2);

          (g . x) in C0 by A2, A24;

          then

           A25: ex q5 be Point of ( TOP-REAL 2) st q5 = (g . x) & |.q5.| <= 1 by A2;

          

           A26: |.(h . qg).| = |.qg.| by A10;

          (g2 . x) = (h . (g . x)) by A22, FUNCT_1: 12;

          hence thesis by A2, A23, A25, A26;

        end;

        reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

        

         A27: ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm7;

        

         A28: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;

        then (f2 . 1) = |[1, 0 ]| by A2, A13, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A29: (f2 . I) in KXP by Lm9, Lm14;

        

         A30: ( rng f2) c= C0

        proof

          let y be object;

          assume y in ( rng f2);

          then

          consider x be object such that

           A31: x in ( dom f2) and

           A32: y = (f2 . x) by FUNCT_1:def 3;

          

           A33: (f . x) in ( rng f) by A4, A31, FUNCT_1:def 3;

          then

          reconsider qf = (f . x) as Point of ( TOP-REAL 2);

          (f . x) in C0 by A2, A33;

          then

           A34: ex q5 be Point of ( TOP-REAL 2) st q5 = (f . x) & |.q5.| <= 1 by A2;

          

           A35: |.(h . qf).| = |.qf.| by A10;

          (f2 . x) = (h . (f . x)) by A31, FUNCT_1: 12;

          hence thesis by A2, A32, A34, A35;

        end;

        (g2 . 1) = |[ 0 , ( - 1)]| by A2, A14, A19, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A36: (g2 . I) in KYN by A18, Lm11, Lm14;

        (f2 . 0 ) = |[( - 1), 0 ]| by A2, A11, A28, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A37: (f2 . O) in KXN by A27, Lm8, Lm14;

        f2 is continuous one-to-one & g2 is continuous one-to-one by A2, A9, Th5, Th6;

        then ( rng f2) meets ( rng g2) by A2, A30, A21, A37, A29, A36, A20, Th13;

        then

        consider x2 be object such that

         A38: x2 in ( rng f2) and

         A39: x2 in ( rng g2) by XBOOLE_0: 3;

        consider z3 be object such that

         A40: z3 in ( dom g2) and

         A41: x2 = (g2 . z3) by A39, FUNCT_1:def 3;

        

         A42: (g . z3) in ( rng g) by A3, A40, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A40, A41, FUNCT_1: 12

        .= (g . z3) by A15, A16, A42, FUNCT_1: 34;

        then

         A43: ((h1 " ) . x2) in ( rng g) by A3, A40, FUNCT_1:def 3;

        consider z2 be object such that

         A44: z2 in ( dom f2) and

         A45: x2 = (f2 . z2) by A38, FUNCT_1:def 3;

        

         A46: (f . z2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A44, A45, FUNCT_1: 12

        .= (f . z2) by A15, A16, A46, FUNCT_1: 34;

        then ((h1 " ) . x2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        hence thesis by A43, XBOOLE_0: 3;

      end;

    end;

    theorem :: JGRAPH_5:69

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P);

      let f,g be Function of I[01] , ( TOP-REAL 2);

      assume

       A2: f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0;

      

       A3: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

       A4: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      per cases ;

        suppose

         A5: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

        now

          per cases by A5;

            case

             A6: p1 = p2;

            p1 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A6, XBOOLE_0: 3;

          end;

            case

             A7: p2 = p3;

            p3 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A7, XBOOLE_0: 3;

          end;

            case

             A8: p3 = p4;

            p3 in ( rng f) & p4 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A8, XBOOLE_0: 3;

          end;

        end;

        hence thesis;

      end;

        suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

        then

        consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

         A9: h is being_homeomorphism and

         A10: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

         A11: |[( - 1), 0 ]| = (h . p1) and

         A12: |[ 0 , 1]| = (h . p2) and

         A13: |[1, 0 ]| = (h . p3) and

         A14: |[ 0 , ( - 1)]| = (h . p4) by A1, Th67;

        

         A15: h is one-to-one by A9, TOPS_2:def 5;

        reconsider h1 = h as Function;

        reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

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

        { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

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

        

         A16: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

        

         A17: ( - ( |[ 0 , 1]| `1 )) = 0 by Lm12;

        reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

        

         A18: ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm10;

        

         A19: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;

        then (g2 . 0 ) = |[ 0 , ( - 1)]| by A2, A14, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A20: (g2 . O) in KYN by A18, Lm11, Lm14;

        

         A21: ( rng g2) c= C0

        proof

          let y be object;

          assume y in ( rng g2);

          then

          consider x be object such that

           A22: x in ( dom g2) and

           A23: y = (g2 . x) by FUNCT_1:def 3;

          

           A24: (g . x) in ( rng g) by A3, A22, FUNCT_1:def 3;

          then

          reconsider qg = (g . x) as Point of ( TOP-REAL 2);

          (g . x) in C0 by A2, A24;

          then

           A25: ex q5 be Point of ( TOP-REAL 2) st q5 = (g . x) & |.q5.| <= 1 by A2;

          

           A26: |.(h . qg).| = |.qg.| by A10;

          (g2 . x) = (h . (g . x)) by A22, FUNCT_1: 12;

          hence thesis by A2, A23, A25, A26;

        end;

        reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

        

         A27: ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm7;

        

         A28: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;

        then (f2 . 1) = |[1, 0 ]| by A2, A13, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A29: (f2 . I) in KXP by Lm9, Lm14;

        

         A30: ( rng f2) c= C0

        proof

          let y be object;

          assume y in ( rng f2);

          then

          consider x be object such that

           A31: x in ( dom f2) and

           A32: y = (f2 . x) by FUNCT_1:def 3;

          

           A33: (f . x) in ( rng f) by A4, A31, FUNCT_1:def 3;

          then

          reconsider qf = (f . x) as Point of ( TOP-REAL 2);

          (f . x) in C0 by A2, A33;

          then

           A34: ex q5 be Point of ( TOP-REAL 2) st q5 = (f . x) & |.q5.| <= 1 by A2;

          

           A35: |.(h . qf).| = |.qf.| by A10;

          (f2 . x) = (h . (f . x)) by A31, FUNCT_1: 12;

          hence thesis by A2, A32, A34, A35;

        end;

        (g2 . 1) = |[ 0 , 1]| by A2, A12, A19, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A36: (g2 . I) in KYP by A17, Lm13, Lm14;

        (f2 . 0 ) = |[( - 1), 0 ]| by A2, A11, A28, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A37: (f2 . O) in KXN by A27, Lm8, Lm14;

        f2 is continuous one-to-one & g2 is continuous one-to-one by A2, A9, Th5, Th6;

        then ( rng f2) meets ( rng g2) by A2, A30, A21, A37, A29, A20, A36, JGRAPH_3: 44;

        then

        consider x2 be object such that

         A38: x2 in ( rng f2) and

         A39: x2 in ( rng g2) by XBOOLE_0: 3;

        consider z3 be object such that

         A40: z3 in ( dom g2) and

         A41: x2 = (g2 . z3) by A39, FUNCT_1:def 3;

        

         A42: (g . z3) in ( rng g) by A3, A40, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A40, A41, FUNCT_1: 12

        .= (g . z3) by A15, A16, A42, FUNCT_1: 34;

        then

         A43: ((h1 " ) . x2) in ( rng g) by A3, A40, FUNCT_1:def 3;

        consider z2 be object such that

         A44: z2 in ( dom f2) and

         A45: x2 = (f2 . z2) by A38, FUNCT_1:def 3;

        

         A46: (f . z2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A44, A45, FUNCT_1: 12

        .= (f . z2) by A15, A16, A46, FUNCT_1: 34;

        then ((h1 " ) . x2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        hence thesis by A43, XBOOLE_0: 3;

      end;

    end;

    theorem :: JGRAPH_5:70

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P);

      let f,g be Function of I[01] , ( TOP-REAL 2);

      assume

       A2: f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0;

      

       A3: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

       A4: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      per cases ;

        suppose

         A5: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

        now

          per cases by A5;

            case

             A6: p1 = p2;

            p1 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A6, XBOOLE_0: 3;

          end;

            case

             A7: p2 = p3;

            p3 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A7, XBOOLE_0: 3;

          end;

            case

             A8: p3 = p4;

            p3 in ( rng f) & p4 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A8, XBOOLE_0: 3;

          end;

        end;

        hence thesis;

      end;

        suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

        then

        consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

         A9: h is being_homeomorphism and

         A10: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

         A11: |[( - 1), 0 ]| = (h . p1) and

         A12: |[ 0 , 1]| = (h . p2) and

         A13: |[1, 0 ]| = (h . p3) and

         A14: |[ 0 , ( - 1)]| = (h . p4) by A1, Th67;

        

         A15: h is one-to-one by A9, TOPS_2:def 5;

        reconsider h1 = h as Function;

        reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

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

        { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

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

        

         A16: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

        

         A17: ( - ( |[ 0 , 1]| `1 )) = 0 by Lm12;

        reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

        

         A18: ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm10;

        

         A19: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;

        then (g2 . 0 ) = |[ 0 , ( - 1)]| by A2, A14, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A20: (g2 . O) in KYN by A18, Lm11, Lm14;

        

         A21: ( rng g2) c= C0

        proof

          let y be object;

          assume y in ( rng g2);

          then

          consider x be object such that

           A22: x in ( dom g2) and

           A23: y = (g2 . x) by FUNCT_1:def 3;

          

           A24: (g . x) in ( rng g) by A3, A22, FUNCT_1:def 3;

          then

          reconsider qg = (g . x) as Point of ( TOP-REAL 2);

          (g . x) in C0 by A2, A24;

          then

           A25: ex q5 be Point of ( TOP-REAL 2) st q5 = (g . x) & |.q5.| >= 1 by A2;

          

           A26: |.(h . qg).| = |.qg.| by A10;

          (g2 . x) = (h . (g . x)) by A22, FUNCT_1: 12;

          hence thesis by A2, A23, A25, A26;

        end;

        reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

        

         A27: ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm7;

        

         A28: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;

        then (f2 . 1) = |[1, 0 ]| by A2, A13, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A29: (f2 . I) in KXP by Lm9, Lm14;

        

         A30: ( rng f2) c= C0

        proof

          let y be object;

          assume y in ( rng f2);

          then

          consider x be object such that

           A31: x in ( dom f2) and

           A32: y = (f2 . x) by FUNCT_1:def 3;

          

           A33: (f . x) in ( rng f) by A4, A31, FUNCT_1:def 3;

          then

          reconsider qf = (f . x) as Point of ( TOP-REAL 2);

          (f . x) in C0 by A2, A33;

          then

           A34: ex q5 be Point of ( TOP-REAL 2) st q5 = (f . x) & |.q5.| >= 1 by A2;

          

           A35: |.(h . qf).| = |.qf.| by A10;

          (f2 . x) = (h . (f . x)) by A31, FUNCT_1: 12;

          hence thesis by A2, A32, A34, A35;

        end;

        (g2 . 1) = |[ 0 , 1]| by A2, A12, A19, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A36: (g2 . I) in KYP by A17, Lm13, Lm14;

        (f2 . 0 ) = |[( - 1), 0 ]| by A2, A11, A28, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A37: (f2 . O) in KXN by A27, Lm8, Lm14;

        f2 is continuous one-to-one & g2 is continuous one-to-one by A2, A9, Th5, Th6;

        then ( rng f2) meets ( rng g2) by A2, A30, A21, A37, A29, A20, A36, Th14;

        then

        consider x2 be object such that

         A38: x2 in ( rng f2) and

         A39: x2 in ( rng g2) by XBOOLE_0: 3;

        consider z3 be object such that

         A40: z3 in ( dom g2) and

         A41: x2 = (g2 . z3) by A39, FUNCT_1:def 3;

        

         A42: (g . z3) in ( rng g) by A3, A40, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A40, A41, FUNCT_1: 12

        .= (g . z3) by A15, A16, A42, FUNCT_1: 34;

        then

         A43: ((h1 " ) . x2) in ( rng g) by A3, A40, FUNCT_1:def 3;

        consider z2 be object such that

         A44: z2 in ( dom f2) and

         A45: x2 = (f2 . z2) by A38, FUNCT_1:def 3;

        

         A46: (f . z2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A44, A45, FUNCT_1: 12

        .= (f . z2) by A15, A16, A46, FUNCT_1: 34;

        then ((h1 " ) . x2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        hence thesis by A43, XBOOLE_0: 3;

      end;

    end;

    theorem :: JGRAPH_5:71

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

      assume

       A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P);

      let f,g be Function of I[01] , ( TOP-REAL 2);

      assume

       A2: f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| >= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0;

      

       A3: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

       A4: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      per cases ;

        suppose

         A5: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

        now

          per cases by A5;

            case

             A6: p1 = p2;

            p1 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A6, XBOOLE_0: 3;

          end;

            case

             A7: p2 = p3;

            p3 in ( rng f) & p2 in ( rng g) by A2, A4, A3, Lm15, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A7, XBOOLE_0: 3;

          end;

            case

             A8: p3 = p4;

            p3 in ( rng f) & p4 in ( rng g) by A2, A4, A3, Lm16, BORSUK_1: 40, FUNCT_1:def 3;

            hence ( rng f) meets ( rng g) by A8, XBOOLE_0: 3;

          end;

        end;

        hence thesis;

      end;

        suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

        then

        consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

         A9: h is being_homeomorphism and

         A10: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

         A11: |[( - 1), 0 ]| = (h . p1) and

         A12: |[ 0 , 1]| = (h . p2) and

         A13: |[1, 0 ]| = (h . p3) and

         A14: |[ 0 , ( - 1)]| = (h . p4) by A1, Th67;

        reconsider f2 = (h * f), g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

        

         A15: ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm10;

        

         A16: ( rng g2) c= C0

        proof

          let y be object;

          assume y in ( rng g2);

          then

          consider x be object such that

           A17: x in ( dom g2) and

           A18: y = (g2 . x) by FUNCT_1:def 3;

          

           A19: (g . x) in ( rng g) by A3, A17, FUNCT_1:def 3;

          then

          reconsider qg = (g . x) as Point of ( TOP-REAL 2);

          (g . x) in C0 by A2, A19;

          then

           A20: ex q5 be Point of ( TOP-REAL 2) st q5 = (g . x) & |.q5.| >= 1 by A2;

          

           A21: |.(h . qg).| = |.qg.| by A10;

          (g2 . x) = (h . (g . x)) by A17, FUNCT_1: 12;

          hence thesis by A2, A18, A20, A21;

        end;

        

         A22: ( rng f2) c= C0

        proof

          let y be object;

          assume y in ( rng f2);

          then

          consider x be object such that

           A23: x in ( dom f2) and

           A24: y = (f2 . x) by FUNCT_1:def 3;

          

           A25: (f . x) in ( rng f) by A4, A23, FUNCT_1:def 3;

          then

          reconsider qf = (f . x) as Point of ( TOP-REAL 2);

          (f . x) in C0 by A2, A25;

          then

           A26: ex q5 be Point of ( TOP-REAL 2) st q5 = (f . x) & |.q5.| >= 1 by A2;

          

           A27: |.(h . qf).| = |.qf.| by A10;

          (f2 . x) = (h . (f . x)) by A23, FUNCT_1: 12;

          hence thesis by A2, A24, A26, A27;

        end;

        reconsider h1 = h as Function;

        reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

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

        { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

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

        

         A28: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

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

        { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

        then

        reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

        

         A29: ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm7;

        

         A30: ( - ( |[ 0 , 1]| `1 )) = 0 by Lm12;

        

         A31: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;

        then (g2 . 0 ) = |[ 0 , 1]| by A2, A12, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A32: (g2 . O) in KYP by A30, Lm13, Lm14;

        (g2 . 1) = |[ 0 , ( - 1)]| by A2, A14, A31, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A33: (g2 . I) in KYN by A15, Lm11, Lm14;

        

         A34: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;

        then (f2 . 1) = |[1, 0 ]| by A2, A13, Lm16, BORSUK_1: 40, FUNCT_1: 12;

        then

         A35: (f2 . I) in KXP by Lm9, Lm14;

        (f2 . 0 ) = |[( - 1), 0 ]| by A2, A11, A34, Lm15, BORSUK_1: 40, FUNCT_1: 12;

        then

         A36: (f2 . O) in KXN by A29, Lm8, Lm14;

        

         A37: h is one-to-one by A9, TOPS_2:def 5;

        f2 is continuous one-to-one & g2 is continuous one-to-one by A2, A9, Th5, Th6;

        then ( rng f2) meets ( rng g2) by A2, A22, A16, A36, A35, A33, A32, Th15;

        then

        consider x2 be object such that

         A38: x2 in ( rng f2) and

         A39: x2 in ( rng g2) by XBOOLE_0: 3;

        consider z3 be object such that

         A40: z3 in ( dom g2) and

         A41: x2 = (g2 . z3) by A39, FUNCT_1:def 3;

        

         A42: (g . z3) in ( rng g) by A3, A40, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A40, A41, FUNCT_1: 12

        .= (g . z3) by A37, A28, A42, FUNCT_1: 34;

        then

         A43: ((h1 " ) . x2) in ( rng g) by A3, A40, FUNCT_1:def 3;

        consider z2 be object such that

         A44: z2 in ( dom f2) and

         A45: x2 = (f2 . z2) by A38, FUNCT_1:def 3;

        

         A46: (f . z2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A44, A45, FUNCT_1: 12

        .= (f . z2) by A37, A28, A46, FUNCT_1: 34;

        then ((h1 " ) . x2) in ( rng f) by A4, A44, FUNCT_1:def 3;

        hence thesis by A43, XBOOLE_0: 3;

      end;

    end;