jgraph_2.miz



    begin

    reserve T,T1,T2,S for non empty TopSpace;

    theorem :: JGRAPH_2:1

    

     Th1: for f be Function of T1, S, g be Function of T2, S, F1,F2 be Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = ( [#] T1) & F2 = ( [#] T2) & (( [#] T1) \/ ( [#] T2)) = ( [#] T) & F1 is closed & F2 is closed & f is continuous & g is continuous & (for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f . p) = (g . p)) holds ex h be Function of T, S st h = (f +* g) & h is continuous

    proof

      let f be Function of T1, S, g be Function of T2, S, F1,F2 be Subset of T;

      assume that

       A1: T1 is SubSpace of T and

       A2: T2 is SubSpace of T and

       A3: F1 = ( [#] T1) and

       A4: F2 = ( [#] T2) and

       A5: (( [#] T1) \/ ( [#] T2)) = ( [#] T) and

       A6: F1 is closed and

       A7: F2 is closed and

       A8: f is continuous and

       A9: g is continuous and

       A10: for p be object st p in (( [#] T1) /\ ( [#] T2)) holds (f . p) = (g . p);

      set h = (f +* g);

      

       A11: ( dom g) = the carrier of T2 by FUNCT_2:def 1

      .= ( [#] T2);

      

       A12: ( dom f) = the carrier of T1 by FUNCT_2:def 1

      .= ( [#] T1);

      

      then

       A13: ( dom h) = ( [#] T) by A5, A11, FUNCT_4:def 1

      .= the carrier of T;

      ( rng h) c= (( rng f) \/ ( rng g)) by FUNCT_4: 17;

      then

      reconsider h as Function of T, S by A13, FUNCT_2: 2, XBOOLE_1: 1;

      take h;

      thus h = (f +* g);

      for P be Subset of S st P is closed holds (h " P) is closed

      proof

        let P be Subset of S;

        set P3 = (f " P), P4 = (g " P);

        ( [#] T1) c= ( [#] T) by A5, XBOOLE_1: 7;

        then

        reconsider P1 = (f " P) as Subset of T by XBOOLE_1: 1;

        ( [#] T2) c= ( [#] T) by A5, XBOOLE_1: 7;

        then

        reconsider P2 = (g " P) as Subset of T by XBOOLE_1: 1;

        

         A14: ( dom h) = (( dom f) \/ ( dom g)) by FUNCT_4:def 1;

         A15:

        now

          let x be object;

          thus x in ((h " P) /\ ( [#] T2)) implies x in (g " P)

          proof

            assume

             A16: x in ((h " P) /\ ( [#] T2));

            then x in (h " P) by XBOOLE_0:def 4;

            then

             A17: (h . x) in P by FUNCT_1:def 7;

            (g . x) = (h . x) by A11, A16, FUNCT_4: 13;

            hence thesis by A11, A16, A17, FUNCT_1:def 7;

          end;

          assume

           A18: x in (g " P);

          then

           A19: x in ( dom g) by FUNCT_1:def 7;

          (g . x) in P by A18, FUNCT_1:def 7;

          then

           A20: (h . x) in P by A19, FUNCT_4: 13;

          x in ( dom h) by A14, A19, XBOOLE_0:def 3;

          then x in (h " P) by A20, FUNCT_1:def 7;

          hence x in ((h " P) /\ ( [#] T2)) by A18, XBOOLE_0:def 4;

        end;

        

         A21: for x be set st x in ( [#] T1) holds (h . x) = (f . x)

        proof

          let x be set such that

           A22: x in ( [#] T1);

          now

            per cases ;

              suppose

               A23: x in ( [#] T2);

              then x in (( [#] T1) /\ ( [#] T2)) by A22, XBOOLE_0:def 4;

              then (f . x) = (g . x) by A10;

              hence thesis by A11, A23, FUNCT_4: 13;

            end;

              suppose not x in ( [#] T2);

              hence thesis by A11, FUNCT_4: 11;

            end;

          end;

          hence thesis;

        end;

        now

          let x be object;

          thus x in ((h " P) /\ ( [#] T1)) implies x in (f " P)

          proof

            assume

             A24: x in ((h " P) /\ ( [#] T1));

            then x in (h " P) by XBOOLE_0:def 4;

            then

             A25: (h . x) in P by FUNCT_1:def 7;

            (f . x) = (h . x) by A21, A24;

            hence thesis by A12, A24, A25, FUNCT_1:def 7;

          end;

          assume

           A26: x in (f " P);

          then x in ( dom f) by FUNCT_1:def 7;

          then

           A27: x in ( dom h) by A14, XBOOLE_0:def 3;

          (f . x) in P by A26, FUNCT_1:def 7;

          then (h . x) in P by A21, A26;

          then x in (h " P) by A27, FUNCT_1:def 7;

          hence x in ((h " P) /\ ( [#] T1)) by A26, XBOOLE_0:def 4;

        end;

        then

         A28: ((h " P) /\ ( [#] T1)) = (f " P) by TARSKI: 2;

        assume

         A29: P is closed;

        then P3 is closed by A8, PRE_TOPC:def 6;

        then ex F01 be Subset of T st F01 is closed & P3 = (F01 /\ ( [#] T1)) by A1, PRE_TOPC: 13;

        then

         A30: P1 is closed by A3, A6;

        P4 is closed by A9, A29, PRE_TOPC:def 6;

        then ex F02 be Subset of T st F02 is closed & P4 = (F02 /\ ( [#] T2)) by A2, PRE_TOPC: 13;

        then

         A31: P2 is closed by A4, A7;

        (h " P) = ((h " P) /\ (( [#] T1) \/ ( [#] T2))) by A12, A11, A14, RELAT_1: 132, XBOOLE_1: 28

        .= (((h " P) /\ ( [#] T1)) \/ ((h " P) /\ ( [#] T2))) by XBOOLE_1: 23;

        then (h " P) = ((f " P) \/ (g " P)) by A28, A15, TARSKI: 2;

        hence thesis by A30, A31;

      end;

      hence thesis by PRE_TOPC:def 6;

    end;

    theorem :: JGRAPH_2:2

    

     Th2: for n be Element of NAT , q2 be Point of ( Euclid n), q be Point of ( TOP-REAL n), r be Real st q = q2 holds ( Ball (q2,r)) = { q3 where q3 be Point of ( TOP-REAL n) : |.(q - q3).| < r }

    proof

      let n be Element of NAT , q2 be Point of ( Euclid n), q be Point of ( TOP-REAL n), r be Real;

      assume

       A1: q = q2;

      

       A2: { q4 where q4 be Element of ( Euclid n) : ( dist (q2,q4)) < r } c= { q3 where q3 be Point of ( TOP-REAL n) : |.(q - q3).| < r }

      proof

        let x be object;

        assume x in { q4 where q4 be Element of ( Euclid n) : ( dist (q2,q4)) < r };

        then

        consider q4 be Element of ( Euclid n) such that

         A3: q4 = x & ( dist (q2,q4)) < r;

        reconsider q44 = q4 as Point of ( TOP-REAL n) by TOPREAL3: 8;

        ( dist (q2,q4)) = |.(q - q44).| by A1, JGRAPH_1: 28;

        hence thesis by A3;

      end;

      

       A4: { q3 where q3 be Point of ( TOP-REAL n) : |.(q - q3).| < r } c= { q4 where q4 be Element of ( Euclid n) : ( dist (q2,q4)) < r }

      proof

        let x be object;

        assume x in { q3 where q3 be Point of ( TOP-REAL n) : |.(q - q3).| < r };

        then

        consider q3 be Point of ( TOP-REAL n) such that

         A5: x = q3 & |.(q - q3).| < r;

        reconsider q34 = q3 as Point of ( Euclid n) by TOPREAL3: 8;

        ( dist (q2,q34)) = |.(q - q3).| by A1, JGRAPH_1: 28;

        hence thesis by A5;

      end;

      ( Ball (q2,r)) = { q4 where q4 be Element of ( Euclid n) : ( dist (q2,q4)) < r } by METRIC_1: 17;

      hence thesis by A2, A4;

    end;

    theorem :: JGRAPH_2:3

    

     Th3: (( 0. ( TOP-REAL 2)) `1 ) = 0 & (( 0. ( TOP-REAL 2)) `2 ) = 0 by EUCLID: 52, EUCLID: 54;

    theorem :: JGRAPH_2:4

    

     Th4: ( 1.REAL 2) = <*1, 1*> by FINSEQ_2: 61;

    theorem :: JGRAPH_2:5

    

     Th5: (( 1.REAL 2) `1 ) = 1 & (( 1.REAL 2) `2 ) = 1 by Th4, EUCLID: 52;

    theorem :: JGRAPH_2:6

    

     Th6: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ( dom proj1 ) = ( REAL 2)

    proof

      thus ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      hence thesis by EUCLID: 22;

    end;

    theorem :: JGRAPH_2:7

    

     Th7: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ( dom proj2 ) = ( REAL 2)

    proof

      thus ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      hence thesis by EUCLID: 22;

    end;

    theorem :: JGRAPH_2:8

    

     Th8: for p be Point of ( TOP-REAL 2) holds p = |[( proj1 . p), ( proj2 . p)]|

    proof

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

      p = |[(p `1 ), (p `2 )]| & (p `1 ) = ( proj1 . p) by EUCLID: 53, PSCOMP_1:def 5;

      hence thesis by PSCOMP_1:def 6;

    end;

    theorem :: JGRAPH_2:9

    

     Th9: for B be Subset of ( TOP-REAL 2) st B = {( 0. ( TOP-REAL 2))} holds (B ` ) <> {} & (the carrier of ( TOP-REAL 2) \ B) <> {}

    proof

      let B be Subset of ( TOP-REAL 2);

      assume

       A1: B = {( 0. ( TOP-REAL 2))};

      now

        assume |[ 0 , 1]| in B;

        then ( |[ 0 , 1]| `2 ) = 0 by A1, Th3, TARSKI:def 1;

        hence contradiction by EUCLID: 52;

      end;

      then |[ 0 , 1]| in (the carrier of ( TOP-REAL 2) \ B) by XBOOLE_0:def 5;

      hence thesis by SUBSET_1:def 4;

    end;

    theorem :: JGRAPH_2:10

    

     Th10: for X,Y be non empty TopSpace, f be Function of X, Y holds f is continuous iff for p be Point of X, V be Subset of Y st (f . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (f .: W) c= V

    proof

      let X,Y be non empty TopSpace, f be Function of X, Y;

      

       A1: ( [#] Y) <> {} ;

      

       A2: ( dom f) = the carrier of X by FUNCT_2:def 1;

      hereby

        assume

         A3: f is continuous;

        thus for p be Point of X, V be Subset of Y st (f . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (f .: W) c= V

        proof

          let p be Point of X, V be Subset of Y;

          assume (f . p) in V & V is open;

          then

           A4: (f " V) is open & p in (f " V) by A2, A1, A3, FUNCT_1:def 7, TOPS_2: 43;

          (f .: (f " V)) c= V by FUNCT_1: 75;

          hence thesis by A4;

        end;

      end;

      assume

       A5: for p be Point of X, V be Subset of Y st (f . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (f .: W) c= V;

      for G be Subset of Y st G is open holds (f " G) is open

      proof

        let G be Subset of Y;

        assume

         A6: G is open;

        for z be set holds z in (f " G) iff ex Q be Subset of X st Q is open & Q c= (f " G) & z in Q

        proof

          let z be set;

          now

            assume

             A7: z in (f " G);

            then

            reconsider p = z as Point of X;

            (f . z) in G by A7, FUNCT_1:def 7;

            then

            consider W be Subset of X such that

             A8: p in W & W is open and

             A9: (f .: W) c= G by A5, A6;

            

             A10: W c= (f " (f .: W)) by A2, FUNCT_1: 76;

            (f " (f .: W)) c= (f " G) by A9, RELAT_1: 143;

            hence ex Q be Subset of X st Q is open & Q c= (f " G) & z in Q by A8, A10, XBOOLE_1: 1;

          end;

          hence thesis;

        end;

        hence thesis by TOPS_1: 25;

      end;

      hence thesis by A1, TOPS_2: 43;

    end;

    theorem :: JGRAPH_2:11

    

     Th11: for p be Point of ( TOP-REAL 2), G be Subset of ( TOP-REAL 2) st G is open & p in G holds ex r be Real st r > 0 & { q where q be Point of ( TOP-REAL 2) : ((p `1 ) - r) < (q `1 ) & (q `1 ) < ((p `1 ) + r) & ((p `2 ) - r) < (q `2 ) & (q `2 ) < ((p `2 ) + r) } c= G

    proof

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

      assume that

       A1: G is open and

       A2: p in G;

      reconsider GG = G as Subset of the TopStruct of ( TOP-REAL 2);

      reconsider q2 = p as Point of ( Euclid 2) by TOPREAL3: 8;

      ( TopSpaceMetr ( Euclid 2)) = the TopStruct of ( TOP-REAL 2) & GG is open by A1, EUCLID:def 8, PRE_TOPC: 30;

      then

      consider r be Real such that

       A3: r > 0 and

       A4: ( Ball (q2,r)) c= GG by A2, TOPMETR: 15;

      set s = (r / ( sqrt 2));

      

       A5: ( Ball (q2,r)) = { q3 where q3 be Point of ( TOP-REAL 2) : |.(p - q3).| < r } by Th2;

      

       A6: { q where q be Point of ( TOP-REAL 2) : ((p `1 ) - s) < (q `1 ) & (q `1 ) < ((p `1 ) + s) & ((p `2 ) - s) < (q `2 ) & (q `2 ) < ((p `2 ) + s) } c= ( Ball (q2,r))

      proof

        let x be object;

        assume x in { q where q be Point of ( TOP-REAL 2) : ((p `1 ) - s) < (q `1 ) & (q `1 ) < ((p `1 ) + s) & ((p `2 ) - s) < (q `2 ) & (q `2 ) < ((p `2 ) + s) };

        then

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

         A7: q = x and

         A8: ((p `1 ) - s) < (q `1 ) and

         A9: (q `1 ) < ((p `1 ) + s) and

         A10: ((p `2 ) - s) < (q `2 ) and

         A11: (q `2 ) < ((p `2 ) + s);

        (((p `1 ) + s) - s) > ((q `1 ) - s) by A9, XREAL_1: 14;

        then

         A12: ((p `1 ) - (q `1 )) > (((q `1 ) + ( - s)) - (q `1 )) by XREAL_1: 14;

        (((p `2 ) + s) - s) > ((q `2 ) - s) by A11, XREAL_1: 14;

        then

         A13: ((p `2 ) - (q `2 )) > (((q `2 ) + ( - s)) - (q `2 )) by XREAL_1: 14;

        (((p `2 ) - s) + s) < ((q `2 ) + s) by A10, XREAL_1: 8;

        then ((p `2 ) - (q `2 )) < (((q `2 ) + s) - (q `2 )) by XREAL_1: 14;

        then

         A14: (((p `2 ) - (q `2 )) ^2 ) < (s ^2 ) by A13, SQUARE_1: 50;

        (s ^2 ) = ((r ^2 ) / (( sqrt 2) ^2 )) by XCMPLX_1: 76

        .= ((r ^2 ) / 2) by SQUARE_1:def 2;

        then

         A15: ((s ^2 ) + (s ^2 )) = (r ^2 );

        (((p `1 ) - s) + s) < ((q `1 ) + s) by A8, XREAL_1: 8;

        then ((p `1 ) - (q `1 )) < (((q `1 ) + s) - (q `1 )) by XREAL_1: 14;

        then

         A16: ((p - q) `2 ) = ((p `2 ) - (q `2 )) & (((p `1 ) - (q `1 )) ^2 ) < (s ^2 ) by A12, SQUARE_1: 50, TOPREAL3: 3;

        ( |.(p - q).| ^2 ) = ((((p - q) `1 ) ^2 ) + (((p - q) `2 ) ^2 )) & ((p - q) `1 ) = ((p `1 ) - (q `1 )) by JGRAPH_1: 29, TOPREAL3: 3;

        then ( |.(p - q).| ^2 ) < (r ^2 ) by A16, A14, A15, XREAL_1: 8;

        then |.(p - q).| < r by A3, SQUARE_1: 48;

        hence thesis by A5, A7;

      end;

      s > 0 by A3, XREAL_1: 139;

      hence thesis by A4, A6, XBOOLE_1: 1;

    end;

    theorem :: JGRAPH_2:12

    

     Th12: for X,Y,Z be non empty TopSpace, B be Subset of Y, C be Subset of Z, f be Function of X, Y, h be Function of (Y | B), (Z | C) st f is continuous & h is continuous & ( rng f) c= B & B <> {} & C <> {} holds ex g be Function of X, Z st g is continuous & g = (h * f)

    proof

      let X,Y,Z be non empty TopSpace, B be Subset of Y, C be Subset of Z, f be Function of X, Y, h be Function of (Y | B), (Z | C);

      assume that

       A1: f is continuous and

       A2: h is continuous and

       A3: ( rng f) c= B and

       A4: B <> {} and

       A5: C <> {} ;

      

       A6: the carrier of X = ( dom f) by FUNCT_2:def 1;

      the carrier of (Y | B) = ( [#] (Y | B))

      .= B by PRE_TOPC:def 5;

      then

      reconsider u = f as Function of X, (Y | B) by A3, A6, FUNCT_2: 2;

      reconsider V = B as non empty Subset of Y by A4;

      (Y | V) is non empty;

      then

      reconsider H = (Y | B) as non empty TopSpace;

      reconsider F = C as non empty Subset of Z by A5;

      reconsider k = u as Function of X, H;

      (Z | F) is non empty;

      then

      reconsider G = (Z | C) as non empty TopSpace;

      reconsider j = h as Function of H, G;

      

       A7: the carrier of (Z | C) = ( [#] (Z | C))

      .= C by PRE_TOPC:def 5;

      (j * k) is Function of X, G;

      then

      reconsider v = (h * u) as Function of X, Z by A7, FUNCT_2: 7;

      u is continuous by A1, TOPMETR: 6;

      then v is continuous by A2, A4, A5, PRE_TOPC: 26;

      hence thesis;

    end;

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

    definition

      :: JGRAPH_2:def1

      func Out_In_Sq -> Function of ( NonZero ( TOP-REAL 2)), ( NonZero ( TOP-REAL 2)) means

      : Def1: for p be Point of ( TOP-REAL 2) st p <> ( 0. ( TOP-REAL 2)) holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (it . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (it . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|);

      existence

      proof

        reconsider BP = ( NonZero ( TOP-REAL 2)) as non empty set by Th9;

        defpred P[ set, set] means (for p be Point of ( TOP-REAL 2) st p = $1 holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies $2 = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies $2 = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|));

        

         A1: for x be Element of BP holds ex y be Element of BP st P[x, y]

        proof

          let x be Element of BP;

          reconsider q = x as Point of ( TOP-REAL 2) by TARSKI:def 3;

          now

            per cases ;

              case

               A2: (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

              now

                assume |[(1 / (q `1 )), (((q `2 ) / (q `1 )) / (q `1 ))]| in {( 0. ( TOP-REAL 2))};

                then ( 0. ( TOP-REAL 2)) = |[(1 / (q `1 )), (((q `2 ) / (q `1 )) / (q `1 ))]| by TARSKI:def 1;

                then 0 = (1 / (q `1 )) by Th3, EUCLID: 52;

                then

                 A3: 0 = ((1 / (q `1 )) * (q `1 ));

                now

                  per cases ;

                    case

                     A4: (q `1 ) = 0 ;

                    then (q `2 ) = 0 by A2;

                    then q = ( 0. ( TOP-REAL 2)) by A4, EUCLID: 53, EUCLID: 54;

                    then q in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

                    hence contradiction by XBOOLE_0:def 5;

                  end;

                    case (q `1 ) <> 0 ;

                    hence contradiction by A3, XCMPLX_1: 87;

                  end;

                end;

                hence contradiction;

              end;

              then

              reconsider r = |[(1 / (q `1 )), (((q `2 ) / (q `1 )) / (q `1 ))]| as Element of BP by XBOOLE_0:def 5;

              for p be Point of ( TOP-REAL 2) st p = x holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies r = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies r = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) by A2;

              hence thesis;

            end;

              case

               A5: not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

              now

                assume |[(((q `1 ) / (q `2 )) / (q `2 )), (1 / (q `2 ))]| in {( 0. ( TOP-REAL 2))};

                then ( 0. ( TOP-REAL 2)) = |[(((q `1 ) / (q `2 )) / (q `2 )), (1 / (q `2 ))]| by TARSKI:def 1;

                then (( 0. ( TOP-REAL 2)) `2 ) = (1 / (q `2 )) by EUCLID: 52;

                then

                 A6: 0 = ((1 / (q `2 )) * (q `2 )) by Th3;

                (q `2 ) <> 0 by A5;

                hence contradiction by A6, XCMPLX_1: 87;

              end;

              then

              reconsider r = |[(((q `1 ) / (q `2 )) / (q `2 )), (1 / (q `2 ))]| as Element of BP by XBOOLE_0:def 5;

              for p be Point of ( TOP-REAL 2) st p = x holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies r = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies r = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) by A5;

              hence thesis;

            end;

          end;

          hence thesis;

        end;

        ex h be Function of BP, BP st for x be Element of BP holds P[x, (h . x)] from FUNCT_2:sch 3( A1);

        then

        consider h be Function of BP, BP such that

         A7: for x be Element of BP holds for p be Point of ( TOP-REAL 2) st p = x holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h . x) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h . x) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|);

        for p be Point of ( TOP-REAL 2) st p <> ( 0. ( TOP-REAL 2)) holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|)

        proof

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

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

          then not p in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

          then p in ( NonZero ( TOP-REAL 2)) by XBOOLE_0:def 5;

          hence thesis by A7;

        end;

        hence thesis;

      end;

      uniqueness

      proof

        let h1,h2 be Function of ( NonZero ( TOP-REAL 2)), ( NonZero ( TOP-REAL 2));

        assume that

         A8: for p be Point of ( TOP-REAL 2) st p <> ( 0. ( TOP-REAL 2)) holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h1 . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h1 . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) and

         A9: for p be Point of ( TOP-REAL 2) st p <> ( 0. ( TOP-REAL 2)) holds (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h2 . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (h2 . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|);

        for x be object st x in ( NonZero ( TOP-REAL 2)) holds (h1 . x) = (h2 . x)

        proof

          let x be object;

          assume

           A10: x in ( NonZero ( TOP-REAL 2));

          then

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

           not q in {( 0. ( TOP-REAL 2))} by A10, XBOOLE_0:def 5;

          then

           A11: q <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

          now

            per cases ;

              case

               A12: (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

              then (h1 . q) = |[(1 / (q `1 )), (((q `2 ) / (q `1 )) / (q `1 ))]| by A8, A11;

              hence thesis by A9, A11, A12;

            end;

              case

               A13: not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

              then (h1 . q) = |[(((q `1 ) / (q `2 )) / (q `2 )), (1 / (q `2 ))]| by A8, A11;

              hence thesis by A9, A11, A13;

            end;

          end;

          hence thesis;

        end;

        hence h1 = h2 by FUNCT_2: 12;

      end;

    end

    theorem :: JGRAPH_2:13

    

     Th13: for p be Point of ( TOP-REAL 2) st not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) holds (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))

    proof

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

      

       A1: ( - (p `1 )) < (p `2 ) implies ( - ( - (p `1 ))) > ( - (p `2 )) by XREAL_1: 24;

      

       A2: ( - (p `1 )) > (p `2 ) implies ( - ( - (p `1 ))) < ( - (p `2 )) by XREAL_1: 24;

      assume not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

      hence thesis by A1, A2;

    end;

    theorem :: JGRAPH_2:14

    

     Th14: for p be Point of ( TOP-REAL 2) st p <> ( 0. ( TOP-REAL 2)) holds (((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) implies ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) & ( not ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) implies ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|)

    proof

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

      assume

       A1: p <> ( 0. ( TOP-REAL 2));

      hereby

        assume

         A2: (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ));

        now

          per cases by A2;

            case

             A3: (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 );

            now

              assume

               A4: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

               A5:

              now

                per cases by A4;

                  case (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 );

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A3, XXREAL_0: 1;

                end;

                  case (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

                  then ( - (p `2 )) >= ( - ( - (p `1 ))) by XREAL_1: 24;

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A3, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A5;

                  case

                   A6: (p `1 ) = (p `2 );

                  then (p `1 ) <> 0 by A1, EUCLID: 53, EUCLID: 54;

                  then (((p `1 ) / (p `2 )) / (p `2 )) = (1 / (p `1 )) by A6, XCMPLX_1: 60;

                  hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, A4, A6, Def1;

                end;

                  case

                   A7: (p `1 ) = ( - (p `2 ));

                  then

                   A8: (p `2 ) <> 0 by A1, EUCLID: 53, EUCLID: 54;

                  

                   A9: (((p `1 ) / (p `2 )) / (p `2 )) = (( - ((p `2 ) / (p `2 ))) / (p `2 )) by A7

                  .= (( - 1) / (p `2 )) by A8, XCMPLX_1: 60

                  .= (1 / (p `1 )) by A7, XCMPLX_1: 192;

                  ( - (p `1 )) = (p `2 ) by A7;

                  

                  then (1 / (p `2 )) = ( - (1 / (p `1 ))) by XCMPLX_1: 188

                  .= ( - (((p `2 ) / (p `1 )) / ( - (p `1 )))) by A7, A9, XCMPLX_1: 192

                  .= ( - ( - (((p `2 ) / (p `1 )) / (p `1 )))) by XCMPLX_1: 188

                  .= (((p `2 ) / (p `1 )) / (p `1 ));

                  hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, A4, A9, Def1;

                end;

              end;

              hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|;

            end;

            hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, Def1;

          end;

            case

             A10: (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ));

            now

              assume

               A11: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

               A12:

              now

                per cases by A11;

                  case (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 );

                  then ( - ( - (p `1 ))) >= ( - (p `2 )) by XREAL_1: 24;

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A10, XXREAL_0: 1;

                end;

                  case (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A10, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A12;

                  case

                   A13: (p `1 ) = (p `2 );

                  then (p `1 ) <> 0 by A1, EUCLID: 53, EUCLID: 54;

                  then (((p `1 ) / (p `2 )) / (p `2 )) = (1 / (p `1 )) by A13, XCMPLX_1: 60;

                  hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, A11, A13, Def1;

                end;

                  case

                   A14: (p `1 ) = ( - (p `2 ));

                  then

                   A15: (p `2 ) <> 0 by A1, EUCLID: 53, EUCLID: 54;

                  

                   A16: (((p `1 ) / (p `2 )) / (p `2 )) = (( - ((p `2 ) / (p `2 ))) / (p `2 )) by A14

                  .= (( - 1) / (p `2 )) by A15, XCMPLX_1: 60

                  .= (1 / (p `1 )) by A14, XCMPLX_1: 192;

                  ( - (p `1 )) = (p `2 ) by A14;

                  

                  then (1 / (p `2 )) = ( - (((p `1 ) / (p `2 )) / (p `2 ))) by A16, XCMPLX_1: 188

                  .= ( - (((p `2 ) / (p `1 )) / ( - (p `1 )))) by A14, XCMPLX_1: 191

                  .= ( - ( - (((p `2 ) / (p `1 )) / (p `1 )))) by XCMPLX_1: 188

                  .= (((p `2 ) / (p `1 )) / (p `1 ));

                  hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, A11, A16, Def1;

                end;

              end;

              hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|;

            end;

            hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by A1, Def1;

          end;

        end;

        hence ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|;

      end;

      hereby

        

         A17: ( - (p `2 )) > (p `1 ) implies ( - ( - (p `2 ))) < ( - (p `1 )) by XREAL_1: 24;

        

         A18: ( - (p `2 )) < (p `1 ) implies ( - ( - (p `2 ))) > ( - (p `1 )) by XREAL_1: 24;

        assume not ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )));

        hence ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| by A1, A18, A17, Def1;

      end;

    end;

    theorem :: JGRAPH_2:15

    

     Th15: for D be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | D) st K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } holds ( rng ( Out_In_Sq | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

    proof

      let D be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | D);

      

       A1: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= D by PRE_TOPC:def 5;

      then

      reconsider K00 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A2: K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) };

      

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K00) holds (q `1 ) <> 0

      proof

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

        

         A4: the carrier of (( TOP-REAL 2) | K00) = ( [#] (( TOP-REAL 2) | K00))

        .= K0 by PRE_TOPC:def 5;

        assume q in the carrier of (( TOP-REAL 2) | K00);

        then

         A5: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A4;

        now

          assume

           A6: (q `1 ) = 0 ;

          then (q `2 ) = 0 by A5;

          hence contradiction by A5, A6, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      let y be object;

      assume y in ( rng ( Out_In_Sq | K0));

      then

      consider x be object such that

       A7: x in ( dom ( Out_In_Sq | K0)) and

       A8: y = (( Out_In_Sq | K0) . x) by FUNCT_1:def 3;

      

       A9: x in (( dom Out_In_Sq ) /\ K0) by A7, RELAT_1: 61;

      then

       A10: x in K0 by XBOOLE_0:def 4;

      K0 c= the carrier of ( TOP-REAL 2) by A1, XBOOLE_1: 1;

      then

      reconsider p = x as Point of ( TOP-REAL 2) by A10;

      

       A11: ( Out_In_Sq . p) = y by A8, A10, FUNCT_1: 49;

      

       A12: ex px be Point of ( TOP-REAL 2) st x = px & ((px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 ))) & px <> ( 0. ( TOP-REAL 2)) by A2, A10;

      then

       A13: ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| by Def1;

      set p9 = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|;

      K00 = ( [#] (( TOP-REAL 2) | K00)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K00);

      then

       A14: p in the carrier of (( TOP-REAL 2) | K00) by A9, XBOOLE_0:def 4;

      

       A15: (p9 `1 ) = (1 / (p `1 )) by EUCLID: 52;

       A16:

      now

        assume p9 = ( 0. ( TOP-REAL 2));

        then ( 0 * (p `1 )) = ((1 / (p `1 )) * (p `1 )) by A15, EUCLID: 52, EUCLID: 54;

        hence contradiction by A14, A3, XCMPLX_1: 87;

      end;

      

       A17: (p `1 ) <> 0 by A14, A3;

      now

        per cases ;

          suppose

           A18: (p `1 ) >= 0 ;

          then ((p `2 ) / (p `1 )) <= ((p `1 ) / (p `1 )) & (( - (1 * (p `1 ))) / (p `1 )) <= ((p `2 ) / (p `1 )) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (1 * (p `1 ))) by A12, XREAL_1: 72;

          then

           A19: ((p `2 ) / (p `1 )) <= 1 & ((( - 1) * (p `1 )) / (p `1 )) <= ((p `2 ) / (p `1 )) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (1 * (p `1 ))) by A14, A3, XCMPLX_1: 60;

          then ((p `2 ) / (p `1 )) <= 1 & ( - 1) <= ((p `2 ) / (p `1 )) or ((p `2 ) / (p `1 )) >= 1 & ((p `2 ) / (p `1 )) <= ((( - 1) * (p `1 )) / (p `1 )) by A17, A18, XCMPLX_1: 89;

          then (( - 1) / (p `1 )) <= (((p `2 ) / (p `1 )) / (p `1 )) by A18, XREAL_1: 72;

          then

           A20: (((p `2 ) / (p `1 )) / (p `1 )) <= (1 / (p `1 )) & ( - (1 / (p `1 ))) <= (((p `2 ) / (p `1 )) / (p `1 )) or (((p `2 ) / (p `1 )) / (p `1 )) >= (1 / (p `1 )) & (((p `2 ) / (p `1 )) / (p `1 )) <= ( - (1 / (p `1 ))) by A17, A18, A19, XREAL_1: 72;

          (p9 `1 ) = (1 / (p `1 )) & (p9 `2 ) = (((p `2 ) / (p `1 )) / (p `1 )) by EUCLID: 52;

          hence y in K0 by A2, A11, A16, A13, A20;

        end;

          suppose

           A21: (p `1 ) < 0 ;

           A22:

          now

            per cases by A12;

              case that

               A23: (p `2 ) <= (p `1 ) and

               A24: ( - (1 * (p `1 ))) <= (p `2 );

              ((p `2 ) / (p `1 )) >= 1 by A21, A23, XREAL_1: 182;

              hence (((p `2 ) / (p `1 )) / (p `1 )) <= (1 / (p `1 )) by A21, XREAL_1: 73;

              (( - 1) * (p `1 )) <= (p `2 ) by A24;

              then ( - 1) >= ((p `2 ) / (p `1 )) by A21, XREAL_1: 78;

              then (( - 1) / (p `1 )) <= (((p `2 ) / (p `1 )) / (p `1 )) by A21, XREAL_1: 73;

              hence ( - (1 / (p `1 ))) <= (((p `2 ) / (p `1 )) / (p `1 ));

            end;

              case that

               A25: (p `2 ) >= (p `1 ) and

               A26: (p `2 ) <= ( - (1 * (p `1 )));

              ((p `2 ) / (p `1 )) <= 1 by A21, A25, XREAL_1: 186;

              hence (((p `2 ) / (p `1 )) / (p `1 )) >= (1 / (p `1 )) by A21, XREAL_1: 73;

              (( - 1) * (p `1 )) >= (p `2 ) by A26;

              then ( - 1) <= ((p `2 ) / (p `1 )) by A21, XREAL_1: 80;

              then (( - 1) / (p `1 )) >= (((p `2 ) / (p `1 )) / (p `1 )) by A21, XREAL_1: 73;

              hence ( - (1 / (p `1 ))) >= (((p `2 ) / (p `1 )) / (p `1 ));

            end;

          end;

          (p9 `1 ) = (1 / (p `1 )) & (p9 `2 ) = (((p `2 ) / (p `1 )) / (p `1 )) by EUCLID: 52;

          hence y in K0 by A2, A11, A16, A13, A22;

        end;

      end;

      then y in ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      hence thesis;

    end;

    theorem :: JGRAPH_2:16

    

     Th16: for D be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | D) st K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } holds ( rng ( Out_In_Sq | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

    proof

      let D be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | D);

      

       A1: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= D by PRE_TOPC:def 5;

      then

      reconsider K00 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A2: K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) };

      

       A3: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K00) holds (q `2 ) <> 0

      proof

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

        

         A4: the carrier of (( TOP-REAL 2) | K00) = ( [#] (( TOP-REAL 2) | K00))

        .= K0 by PRE_TOPC:def 5;

        assume q in the carrier of (( TOP-REAL 2) | K00);

        then

         A5: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A4;

        now

          assume

           A6: (q `2 ) = 0 ;

          then (q `1 ) = 0 by A5;

          hence contradiction by A5, A6, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      let y be object;

      assume y in ( rng ( Out_In_Sq | K0));

      then

      consider x be object such that

       A7: x in ( dom ( Out_In_Sq | K0)) and

       A8: y = (( Out_In_Sq | K0) . x) by FUNCT_1:def 3;

      x in (( dom Out_In_Sq ) /\ K0) by A7, RELAT_1: 61;

      then

       A9: x in K0 by XBOOLE_0:def 4;

      K0 c= the carrier of ( TOP-REAL 2) by A1, XBOOLE_1: 1;

      then

      reconsider p = x as Point of ( TOP-REAL 2) by A9;

      

       A10: ( Out_In_Sq . p) = y by A8, A9, FUNCT_1: 49;

      

       A11: ex px be Point of ( TOP-REAL 2) st x = px & ((px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 ))) & px <> ( 0. ( TOP-REAL 2)) by A2, A9;

      then

       A12: ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by Th14;

      

       A13: K00 = ( [#] (( TOP-REAL 2) | K00)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K00);

      set p9 = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|;

      

       A14: (p9 `2 ) = (1 / (p `2 )) by EUCLID: 52;

       A15:

      now

        assume p9 = ( 0. ( TOP-REAL 2));

        then ( 0 * (p `2 )) = ((1 / (p `2 )) * (p `2 )) by A14, EUCLID: 52, EUCLID: 54;

        hence contradiction by A9, A13, A3, XCMPLX_1: 87;

      end;

      

       A16: (p `2 ) <> 0 by A9, A13, A3;

      now

        per cases ;

          case

           A17: (p `2 ) >= 0 ;

          then ((p `1 ) / (p `2 )) <= ((p `2 ) / (p `2 )) & (( - (1 * (p `2 ))) / (p `2 )) <= ((p `1 ) / (p `2 )) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (1 * (p `2 ))) by A11, XREAL_1: 72;

          then

           A18: ((p `1 ) / (p `2 )) <= 1 & ((( - 1) * (p `2 )) / (p `2 )) <= ((p `1 ) / (p `2 )) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (1 * (p `2 ))) by A9, A13, A3, XCMPLX_1: 60;

          then ((p `1 ) / (p `2 )) <= 1 & ( - 1) <= ((p `1 ) / (p `2 )) or ((p `1 ) / (p `2 )) >= 1 & ((p `1 ) / (p `2 )) <= ((( - 1) * (p `2 )) / (p `2 )) by A16, A17, XCMPLX_1: 89;

          then (( - 1) / (p `2 )) <= (((p `1 ) / (p `2 )) / (p `2 )) by A17, XREAL_1: 72;

          then

           A19: (((p `1 ) / (p `2 )) / (p `2 )) <= (1 / (p `2 )) & ( - (1 / (p `2 ))) <= (((p `1 ) / (p `2 )) / (p `2 )) or (((p `1 ) / (p `2 )) / (p `2 )) >= (1 / (p `2 )) & (((p `1 ) / (p `2 )) / (p `2 )) <= ( - (1 / (p `2 ))) by A16, A17, A18, XREAL_1: 72;

          (p9 `2 ) = (1 / (p `2 )) & (p9 `1 ) = (((p `1 ) / (p `2 )) / (p `2 )) by EUCLID: 52;

          hence y in K0 by A2, A10, A15, A12, A19;

        end;

          case

           A20: (p `2 ) < 0 ;

          then (p `1 ) <= (p `2 ) & ( - (1 * (p `2 ))) <= (p `1 ) or ((p `1 ) / (p `2 )) <= ((p `2 ) / (p `2 )) & ((p `1 ) / (p `2 )) >= (( - (1 * (p `2 ))) / (p `2 )) by A11, XREAL_1: 73;

          then

           A21: (p `1 ) <= (p `2 ) & ( - (1 * (p `2 ))) <= (p `1 ) or ((p `1 ) / (p `2 )) <= 1 & ((p `1 ) / (p `2 )) >= ((( - 1) * (p `2 )) / (p `2 )) by A20, XCMPLX_1: 60;

          then ((p `1 ) / (p `2 )) >= 1 & ((( - 1) * (p `2 )) / (p `2 )) >= ((p `1 ) / (p `2 )) or ((p `1 ) / (p `2 )) <= 1 & ((p `1 ) / (p `2 )) >= ( - 1) by A20, XCMPLX_1: 89;

          then (( - 1) / (p `2 )) >= (((p `1 ) / (p `2 )) / (p `2 )) by A20, XREAL_1: 73;

          then

           A22: (((p `1 ) / (p `2 )) / (p `2 )) <= (1 / (p `2 )) & ( - (1 / (p `2 ))) <= (((p `1 ) / (p `2 )) / (p `2 )) or (((p `1 ) / (p `2 )) / (p `2 )) >= (1 / (p `2 )) & (((p `1 ) / (p `2 )) / (p `2 )) <= ( - (1 / (p `2 ))) by A20, A21, XREAL_1: 73;

          (p9 `2 ) = (1 / (p `2 )) & (p9 `1 ) = (((p `1 ) / (p `2 )) / (p `2 )) by EUCLID: 52;

          hence y in K0 by A2, A10, A15, A12, A22;

        end;

      end;

      then y in ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      hence thesis;

    end;

    

     Lm1: ( 0. ( TOP-REAL 2)) = ( 0.REAL 2) by EUCLID: 66;

    theorem :: JGRAPH_2:17

    

     Th17: for K0a be set, D be non empty Subset of ( TOP-REAL 2) st K0a = { p where p be Point of ( TOP-REAL 2) : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } & (D ` ) = {( 0. ( TOP-REAL 2))} holds K0a is non empty Subset of (( TOP-REAL 2) | D) & K0a is non empty Subset of ( TOP-REAL 2)

    proof

      

       A1: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

      let K0a be set, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A2: K0a = { p where p be Point of ( TOP-REAL 2) : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } and

       A3: (D ` ) = {( 0. ( TOP-REAL 2))};

      (( 1.REAL 2) `2 ) <= (( 1.REAL 2) `1 ) & ( - (( 1.REAL 2) `1 )) <= (( 1.REAL 2) `2 ) or (( 1.REAL 2) `2 ) >= (( 1.REAL 2) `1 ) & (( 1.REAL 2) `2 ) <= ( - (( 1.REAL 2) `1 )) by Th5;

      then

       A4: ( 1.REAL 2) in K0a by A2, A1;

      

       A5: K0a c= D

      proof

        let x be object;

        

         A6: D = ((D ` ) ` )

        .= ( NonZero ( TOP-REAL 2)) by A3, SUBSET_1:def 4;

        assume x in K0a;

        then

         A7: ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A7, A6, XBOOLE_0:def 5;

      end;

      the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= D by PRE_TOPC:def 5;

      hence K0a is non empty Subset of (( TOP-REAL 2) | D) by A4, A5;

      thus thesis by A4, A5, XBOOLE_1: 1;

    end;

    theorem :: JGRAPH_2:18

    

     Th18: for K0a be set, D be non empty Subset of ( TOP-REAL 2) st K0a = { p where p be Point of ( TOP-REAL 2) : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } & (D ` ) = {( 0. ( TOP-REAL 2))} holds K0a is non empty Subset of (( TOP-REAL 2) | D) & K0a is non empty Subset of ( TOP-REAL 2)

    proof

      

       A1: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

      let K0a be set, D be non empty Subset of ( TOP-REAL 2);

      assume that

       A2: K0a = { p where p be Point of ( TOP-REAL 2) : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } and

       A3: (D ` ) = {( 0. ( TOP-REAL 2))};

      (( 1.REAL 2) `1 ) <= (( 1.REAL 2) `2 ) & ( - (( 1.REAL 2) `2 )) <= (( 1.REAL 2) `1 ) or (( 1.REAL 2) `1 ) >= (( 1.REAL 2) `2 ) & (( 1.REAL 2) `1 ) <= ( - (( 1.REAL 2) `2 )) by Th5;

      then

       A4: ( 1.REAL 2) in K0a by A2, A1;

      

       A5: K0a c= D

      proof

        let x be object;

        

         A6: D = ((D ` ) ` )

        .= ( NonZero ( TOP-REAL 2)) by A3, SUBSET_1:def 4;

        assume x in K0a;

        then

         A7: ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A7, A6, XBOOLE_0:def 5;

      end;

      the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= D by PRE_TOPC:def 5;

      hence K0a is non empty Subset of (( TOP-REAL 2) | D) by A4, A5;

      thus thesis by A4, A5, XBOOLE_1: 1;

    end;

    theorem :: JGRAPH_2:19

    

     Th19: for X be non empty TopSpace, f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r1 + r2)) & g is continuous

    proof

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

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous;

      defpred P[ set, set] means (for r1,r2 be Real st (f1 . $1) = r1 & (f2 . $1) = r2 holds $2 = (r1 + r2));

      

       A3: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        let x be Element of X;

        reconsider r1 = (f1 . x) as Element of REAL by TOPMETR: 17;

        reconsider r2 = (f2 . x) as Element of REAL by TOPMETR: 17;

        set r3 = (r1 + r2);

        for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds r3 = (r1 + r2);

        hence ex y be Element of REAL st for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds y = (r1 + r2);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A3);

      then

      consider f be Function of the carrier of X, REAL such that

       A4: for x be Element of X holds for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds (f . x) = (r1 + r2);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        reconsider r2 = (f2 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A5: r0 > 0 and

         A6: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        reconsider G1 = ].(r1 - (r0 / 2)), (r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR: 17;

        

         A7: r1 < (r1 + (r0 / 2)) by A5, XREAL_1: 29, XREAL_1: 215;

        then (r1 - (r0 / 2)) < r1 by XREAL_1: 19;

        then

         A8: (f1 . p) in G1 by A7, XXREAL_1: 4;

        reconsider G2 = ].(r2 - (r0 / 2)), (r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR: 17;

        

         A9: r2 < (r2 + (r0 / 2)) by A5, XREAL_1: 29, XREAL_1: 215;

        then (r2 - (r0 / 2)) < r2 by XREAL_1: 19;

        then

         A10: (f2 . p) in G2 by A9, XXREAL_1: 4;

        G2 is open by JORDAN6: 35;

        then

        consider W2 be Subset of X such that

         A11: p in W2 & W2 is open and

         A12: (f2 .: W2) c= G2 by A2, A10, Th10;

        G1 is open by JORDAN6: 35;

        then

        consider W1 be Subset of X such that

         A13: p in W1 & W1 is open and

         A14: (f1 .: W1) c= G1 by A1, A8, Th10;

        set W = (W1 /\ W2);

        

         A15: (g0 .: W) c= ].(r - r0), (r + r0).[

        proof

          let x be object;

          assume x in (g0 .: W);

          then

          consider z be object such that

           A16: z in ( dom g0) and

           A17: z in W and

           A18: (g0 . z) = x by FUNCT_1:def 6;

          reconsider pz = z as Point of X by A16;

          reconsider aa2 = (f2 . pz) as Real;

          reconsider aa1 = (f1 . pz) as Real;

          

           A19: pz in the carrier of X;

          then

           A20: pz in ( dom f2) by FUNCT_2:def 1;

          z in W2 by A17, XBOOLE_0:def 4;

          then

           A21: (f2 . pz) in (f2 .: W2) by A20, FUNCT_1:def 6;

          then

           A22: (r2 - (r0 / 2)) < aa2 by A12, XXREAL_1: 4;

          

           A23: pz in ( dom f1) by A19, FUNCT_2:def 1;

          z in W1 by A17, XBOOLE_0:def 4;

          then

           A24: (f1 . pz) in (f1 .: W1) by A23, FUNCT_1:def 6;

          then (r1 - (r0 / 2)) < aa1 by A14, XXREAL_1: 4;

          then ((r1 - (r0 / 2)) + (r2 - (r0 / 2))) < (aa1 + aa2) by A22, XREAL_1: 8;

          then ((r1 + r2) - ((r0 / 2) + (r0 / 2))) < (aa1 + aa2);

          then

           A25: (r - r0) < (aa1 + aa2) by A4;

          

           A26: aa2 < (r2 + (r0 / 2)) by A12, A21, XXREAL_1: 4;

          

           A27: x = (aa1 + aa2) by A4, A18;

          then

          reconsider rx = x as Real;

          aa1 < (r1 + (r0 / 2)) by A14, A24, XXREAL_1: 4;

          then (aa1 + aa2) < ((r1 + (r0 / 2)) + (r2 + (r0 / 2))) by A26, XREAL_1: 8;

          then (aa1 + aa2) < ((r1 + r2) + ((r0 / 2) + (r0 / 2)));

          then rx < (r + r0) by A4, A27;

          hence thesis by A27, A25, XXREAL_1: 4;

        end;

        W is open & p in W by A13, A11, XBOOLE_0:def 4;

        hence thesis by A6, A15, XBOOLE_1: 1;

      end;

      then

       A28: g0 is continuous by Th10;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g0 . p) = (r1 + r2) by A4;

      hence thesis by A28;

    end;

    theorem :: JGRAPH_2:20

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

    proof

      let X be non empty TopSpace, a be Real;

      reconsider a1 = a as Element of R^1 by TOPMETR: 17, XREAL_0:def 1;

      set g1 = (the carrier of X --> a1);

      reconsider g0 = g1 as Function of X, R^1 ;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        set f1 = g0;

        let p be Point of X, V be Subset of R^1 ;

        assume that

         A1: (g0 . p) in V and V is open;

        set G1 = V;

        (f1 .: ( [#] X)) c= G1

        proof

          let y be object;

          assume y in (f1 .: ( [#] X));

          then ex x be object st x in ( dom f1) & x in ( [#] X) & y = (f1 . x) by FUNCT_1:def 6;

          then y = a by FUNCOP_1: 7;

          hence thesis by A1, FUNCOP_1: 7;

        end;

        hence thesis;

      end;

      then (for p be Point of X holds (g1 . p) = a) & g0 is continuous by Th10, FUNCOP_1: 7;

      hence thesis;

    end;

    theorem :: JGRAPH_2:21

    

     Th21: for X be non empty TopSpace, f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r1 - r2)) & g is continuous

    proof

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

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous;

      defpred P[ set, set] means (for r1,r2 be Real st (f1 . $1) = r1 & (f2 . $1) = r2 holds $2 = (r1 - r2));

      

       A3: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        let x be Element of X;

        reconsider r1 = (f1 . x) as Element of REAL by TOPMETR: 17;

        reconsider r2 = (f2 . x) as Element of REAL by TOPMETR: 17;

        set r3 = (r1 - r2);

        for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds r3 = (r1 - r2);

        hence ex y be Element of REAL st for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds y = (r1 - r2);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A3);

      then

      consider f be Function of the carrier of X, REAL such that

       A4: for x be Element of X holds for r1,r2 be Real st (f1 . x) = r1 & (f2 . x) = r2 holds (f . x) = (r1 - r2);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        reconsider r2 = (f2 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A5: r0 > 0 and

         A6: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        reconsider G1 = ].(r1 - (r0 / 2)), (r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR: 17;

        

         A7: r1 < (r1 + (r0 / 2)) by A5, XREAL_1: 29, XREAL_1: 215;

        then (r1 - (r0 / 2)) < r1 by XREAL_1: 19;

        then

         A8: (f1 . p) in G1 by A7, XXREAL_1: 4;

        reconsider G2 = ].(r2 - (r0 / 2)), (r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR: 17;

        

         A9: r2 < (r2 + (r0 / 2)) by A5, XREAL_1: 29, XREAL_1: 215;

        then (r2 - (r0 / 2)) < r2 by XREAL_1: 19;

        then

         A10: (f2 . p) in G2 by A9, XXREAL_1: 4;

        G2 is open by JORDAN6: 35;

        then

        consider W2 be Subset of X such that

         A11: p in W2 & W2 is open and

         A12: (f2 .: W2) c= G2 by A2, A10, Th10;

        G1 is open by JORDAN6: 35;

        then

        consider W1 be Subset of X such that

         A13: p in W1 & W1 is open and

         A14: (f1 .: W1) c= G1 by A1, A8, Th10;

        set W = (W1 /\ W2);

        

         A15: (g0 .: W) c= ].(r - r0), (r + r0).[

        proof

          let x be object;

          assume x in (g0 .: W);

          then

          consider z be object such that

           A16: z in ( dom g0) and

           A17: z in W and

           A18: (g0 . z) = x by FUNCT_1:def 6;

          reconsider pz = z as Point of X by A16;

          reconsider aa2 = (f2 . pz) as Real;

          reconsider aa1 = (f1 . pz) as Real;

          

           A19: pz in the carrier of X;

          then

           A20: pz in ( dom f1) by FUNCT_2:def 1;

          

           A21: pz in ( dom f2) by A19, FUNCT_2:def 1;

          z in W2 by A17, XBOOLE_0:def 4;

          then

           A22: (f2 . pz) in (f2 .: W2) by A21, FUNCT_1:def 6;

          then

           A23: (r2 - (r0 / 2)) < aa2 by A12, XXREAL_1: 4;

          

           A24: aa2 < (r2 + (r0 / 2)) by A12, A22, XXREAL_1: 4;

          z in W1 by A17, XBOOLE_0:def 4;

          then

           A25: (f1 . pz) in (f1 .: W1) by A20, FUNCT_1:def 6;

          then (r1 - (r0 / 2)) < aa1 by A14, XXREAL_1: 4;

          then ((r1 - (r0 / 2)) - (r2 + (r0 / 2))) < (aa1 - aa2) by A24, XREAL_1: 14;

          then ((r1 - r2) - ((r0 / 2) + (r0 / 2))) < (aa1 - aa2);

          then

           A26: (r - r0) < (aa1 - aa2) by A4;

          

           A27: x = (aa1 - aa2) by A4, A18;

          then

          reconsider rx = x as Real;

          aa1 < (r1 + (r0 / 2)) by A14, A25, XXREAL_1: 4;

          then (aa1 - aa2) < ((r1 + (r0 / 2)) - (r2 - (r0 / 2))) by A23, XREAL_1: 14;

          then (aa1 - aa2) < ((r1 - r2) + ((r0 / 2) + (r0 / 2)));

          then rx < (r + r0) by A4, A27;

          hence thesis by A27, A26, XXREAL_1: 4;

        end;

        W is open & p in W by A13, A11, XBOOLE_0:def 4;

        hence thesis by A6, A15, XBOOLE_1: 1;

      end;

      then

       A28: g0 is continuous by Th10;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g0 . p) = (r1 - r2) by A4;

      hence thesis by A28;

    end;

    theorem :: JGRAPH_2:22

    

     Th22: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (r1 * r1)) & g is continuous

    proof

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

      defpred P[ set, set] means (for r1 be Real st (f1 . $1) = r1 holds $2 = (r1 * r1));

      

       A1: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        let x be Element of X;

        reconsider r1 = (f1 . x) as Element of REAL by TOPMETR: 17;

        set r3 = (r1 * r1);

        for r1 be Real st (f1 . x) = r1 holds r3 = (r1 * r1);

        hence ex y be Element of REAL st for r1 be Real st (f1 . x) = r1 holds y = (r1 * r1);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A1);

      then

      consider f be Function of the carrier of X, REAL such that

       A2: for x be Element of X holds for r1 be Real st (f1 . x) = r1 holds (f . x) = (r1 * r1);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      assume

       A3: f1 is continuous;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A4: r0 > 0 and

         A5: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        

         A6: r = (r1 ^2 ) by A2;

        

         A7: r = (r1 * r1) by A2;

        then

         A8: 0 <= r;

        then

         A9: (( sqrt (r + r0)) ^2 ) = (r + r0) by A4, SQUARE_1:def 2;

        now

          per cases ;

            case

             A10: r1 >= 0 ;

            set r4 = (( sqrt (r + r0)) - ( sqrt r));

            reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

            

             A11: G1 is open by JORDAN6: 35;

            (r + r0) > r by A4, XREAL_1: 29;

            then ( sqrt (r + r0)) > ( sqrt r) by A7, SQUARE_1: 27;

            then

             A12: r4 > 0 by XREAL_1: 50;

            then

             A13: r1 < (r1 + r4) by XREAL_1: 29;

            then (r1 - r4) < r1 by XREAL_1: 19;

            then (f1 . p) in G1 by A13, XXREAL_1: 4;

            then

            consider W1 be Subset of X such that

             A14: p in W1 & W1 is open and

             A15: (f1 .: W1) c= G1 by A3, A11, Th10;

            

             A16: r1 = ( sqrt r) by A6, A10, SQUARE_1:def 2;

            set W = W1;

            

             A17: (r4 ^2 ) = (((( sqrt (r + r0)) ^2 ) - ((2 * ( sqrt (r + r0))) * ( sqrt r))) + (( sqrt r) ^2 ))

            .= (((r + r0) - ((2 * ( sqrt (r + r0))) * ( sqrt r))) + (( sqrt r) ^2 )) by A4, A8, SQUARE_1:def 2

            .= ((r + (r0 - ((2 * ( sqrt (r + r0))) * ( sqrt r)))) + r) by A8, SQUARE_1:def 2

            .= (((2 * r) + r0) - ((2 * ( sqrt (r + r0))) * ( sqrt r)));

            (g0 .: W) c= ].(r - r0), (r + r0).[

            proof

              let x be object;

              assume x in (g0 .: W);

              then

              consider z be object such that

               A18: z in ( dom g0) and

               A19: z in W and

               A20: (g0 . z) = x by FUNCT_1:def 6;

              reconsider pz = z as Point of X by A18;

              reconsider aa1 = (f1 . pz) as Real;

              pz in the carrier of X;

              then pz in ( dom f1) by FUNCT_2:def 1;

              then

               A21: (f1 . pz) in (f1 .: W1) by A19, FUNCT_1:def 6;

              then

               A22: (r1 - r4) < aa1 by A15, XXREAL_1: 4;

               A23:

              now

                per cases ;

                  case

                   A24: 0 <= (r1 - r4);

                  

                   A25: ((((r1 - r4) ^2 ) - (aa1 ^2 )) + ( - (2 * (r4 ^2 )))) <= ((((r1 - r4) ^2 ) - (aa1 ^2 )) + 0 ) by XREAL_1: 7;

                  ((r1 - r4) ^2 ) < (aa1 ^2 ) by A22, A24, SQUARE_1: 16;

                  then ((((r1 - r4) ^2 ) - (2 * (r4 ^2 ))) - (aa1 ^2 )) < 0 by A25, XREAL_1: 49;

                  hence (r - r0) < (aa1 * aa1) by A7, A16, A17, XREAL_1: 48;

                end;

                  case 0 > (r1 - r4);

                  then r1 < r4 by XREAL_1: 48;

                  then (r1 ^2 ) < (r4 ^2 ) by A10, SQUARE_1: 16;

                  then ((r1 ^2 ) - (r4 ^2 )) < 0 by XREAL_1: 49;

                  then (((r1 ^2 ) - (r4 ^2 )) - ((2 * r1) * r4)) < ( 0 - 0 ) by A10, A12;

                  hence (r - r0) < (aa1 * aa1) by A7, A16, A17;

                end;

              end;

              (( - r1) - r4) <= (r1 - r4) by A10, XREAL_1: 9;

              then ( - (r1 + r4)) < aa1 by A22, XXREAL_0: 2;

              then

               A26: (aa1 - ( - (r1 + r4))) > 0 by XREAL_1: 50;

              aa1 < (r1 + r4) by A15, A21, XXREAL_1: 4;

              then ((r1 + r4) - aa1) > 0 by XREAL_1: 50;

              then (((r1 + r4) - aa1) * ((r1 + r4) + aa1)) > 0 by A26, XREAL_1: 129;

              then (((r1 + r4) ^2 ) - (aa1 ^2 )) > 0 ;

              then

               A27: (aa1 ^2 ) < ((r1 + r4) ^2 ) by XREAL_1: 47;

              x = (aa1 * aa1) by A2, A20;

              hence thesis by A7, A16, A17, A27, A23, XXREAL_1: 4;

            end;

            hence thesis by A5, A14, XBOOLE_1: 1;

          end;

            case

             A28: r1 < 0 ;

            set r4 = (( sqrt (r + r0)) - ( sqrt r));

            reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

            

             A29: G1 is open by JORDAN6: 35;

            (r + r0) > r by A4, XREAL_1: 29;

            then ( sqrt (r + r0)) > ( sqrt r) by A7, SQUARE_1: 27;

            then

             A30: r4 > 0 by XREAL_1: 50;

            then

             A31: r1 < (r1 + r4) by XREAL_1: 29;

            then (r1 - r4) < r1 by XREAL_1: 19;

            then (f1 . p) in G1 by A31, XXREAL_1: 4;

            then

            consider W1 be Subset of X such that

             A32: p in W1 & W1 is open and

             A33: (f1 .: W1) c= G1 by A3, A29, Th10;

            

             A34: (( - r1) ^2 ) = (r1 ^2 );

            then

             A35: ( - r1) = ( sqrt r) by A7, A28, SQUARE_1: 22;

            set W = W1;

            

             A36: (r4 ^2 ) = (((r + r0) - ((2 * ( sqrt (r + r0))) * ( sqrt r))) + (( sqrt r) ^2 )) by A9

            .= ((r + (r0 - ((2 * ( sqrt (r + r0))) * ( sqrt r)))) + r) by A7, A28, A34, SQUARE_1: 22

            .= (((2 * r) + r0) - ((2 * ( sqrt (r + r0))) * ( sqrt r)));

            then

             A37: (( - ((2 * r1) * r4)) + (r4 ^2 )) = r0 by A7, A35;

            (g0 .: W) c= ].(r - r0), (r + r0).[

            proof

              let x be object;

              assume x in (g0 .: W);

              then

              consider z be object such that

               A38: z in ( dom g0) and

               A39: z in W and

               A40: (g0 . z) = x by FUNCT_1:def 6;

              reconsider pz = z as Point of X by A38;

              reconsider aa1 = (f1 . pz) as Real;

              pz in the carrier of X;

              then pz in ( dom f1) by FUNCT_2:def 1;

              then

               A41: (f1 . pz) in (f1 .: W1) by A39, FUNCT_1:def 6;

              then

               A42: aa1 < (r1 + r4) by A33, XXREAL_1: 4;

               A43:

              now

                per cases ;

                  case

                   A44: 0 >= (r1 + r4);

                  

                   A45: ((((r1 + r4) ^2 ) - (aa1 ^2 )) + ( - (2 * (r4 ^2 )))) <= ((((r1 + r4) ^2 ) - (aa1 ^2 )) + 0 ) by XREAL_1: 7;

                  ( - aa1) > ( - (r1 + r4)) by A42, XREAL_1: 24;

                  then (( - (r1 + r4)) ^2 ) < (( - aa1) ^2 ) by A44, SQUARE_1: 16;

                  then ((((r1 + r4) ^2 ) - (2 * (r4 ^2 ))) - (aa1 ^2 )) < 0 by A45, XREAL_1: 49;

                  hence (r - r0) < (aa1 * aa1) by A7, A37, XREAL_1: 48;

                end;

                  case 0 < (r1 + r4);

                  then ( 0 + ( - r1)) < ((r1 + r4) + ( - r1)) by XREAL_1: 8;

                  then (( - r1) ^2 ) < (r4 ^2 ) by A28, SQUARE_1: 16;

                  then ((r1 ^2 ) - (r1 ^2 )) > ((r1 ^2 ) - (r4 ^2 )) by XREAL_1: 15;

                  then (((r1 ^2 ) - (r4 ^2 )) + ((2 * r1) * r4)) < ( 0 + 0 ) by A28, A30;

                  hence (r - r0) < (aa1 * aa1) by A7, A35, A36;

                end;

              end;

              (r1 - r4) < aa1 by A33, A41, XXREAL_1: 4;

              then (aa1 - (r1 - r4)) > 0 by XREAL_1: 50;

              then ( - (( - aa1) + (r1 - r4))) > 0 ;

              then

               A46: ((r1 - r4) + ( - aa1)) < 0 ;

              (( - r1) - r4) >= (r1 - r4) by A28, XREAL_1: 9;

              then ( - (( - r1) - r4)) <= ( - (r1 - r4)) by XREAL_1: 24;

              then ( - (r1 - r4)) > aa1 by A42, XXREAL_0: 2;

              then (( - (r1 - r4)) + (r1 - r4)) > (aa1 + (r1 - r4)) by XREAL_1: 8;

              then (((r1 - r4) - aa1) * ((r1 - r4) + aa1)) > 0 by A46, XREAL_1: 130;

              then (((r1 - r4) ^2 ) - (aa1 ^2 )) > 0 ;

              then

               A47: (aa1 ^2 ) < ((r1 - r4) ^2 ) by XREAL_1: 47;

              x = (aa1 * aa1) by A2, A40;

              hence thesis by A7, A37, A47, A43, XXREAL_1: 4;

            end;

            hence thesis by A5, A32, XBOOLE_1: 1;

          end;

        end;

        hence thesis;

      end;

      then

       A48: g0 is continuous by Th10;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g0 . p) = (r1 * r1) by A2;

      hence thesis by A48;

    end;

    theorem :: JGRAPH_2:23

    

     Th23: for X be non empty TopSpace, f1 be Function of X, R^1 , a be Real st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (a * r1)) & g is continuous

    proof

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

      defpred P[ set, set] means (for r1 be Real st (f1 . $1) = r1 holds $2 = (a * r1));

      

       A1: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        let x be Element of X;

        reconsider r1 = (f1 . x) as Real;

        reconsider r3 = (a * r1) as Element of REAL by XREAL_0:def 1;

        for r1 be Real st (f1 . x) = r1 holds r3 = (a * r1);

        hence ex y be Element of REAL st for r1 be Real st (f1 . x) = r1 holds y = (a * r1);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A1);

      then

      consider f be Function of the carrier of X, REAL such that

       A2: for x be Element of X holds for r1 be Real st (f1 . x) = r1 holds (f . x) = (a * r1);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      

       A3: for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g0 . p) = (a * r1) by A2;

      assume

       A4: f1 is continuous;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A5: r0 > 0 and

         A6: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        

         A7: r = (a * r1) by A2;

        

         A8: r = (a * r1) by A2;

        now

          per cases ;

            case

             A9: a >= 0 ;

            now

              per cases by A9;

                case

                 A10: a > 0 ;

                set r4 = (r0 / a);

                reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

                

                 A11: r1 < (r1 + r4) by A5, A10, XREAL_1: 29, XREAL_1: 139;

                then (r1 - r4) < r1 by XREAL_1: 19;

                then

                 A12: (f1 . p) in G1 by A11, XXREAL_1: 4;

                G1 is open by JORDAN6: 35;

                then

                consider W1 be Subset of X such that

                 A13: p in W1 & W1 is open and

                 A14: (f1 .: W1) c= G1 by A4, A12, Th10;

                set W = W1;

                (g0 .: W) c= ].(r - r0), (r + r0).[

                proof

                  let x be object;

                  assume x in (g0 .: W);

                  then

                  consider z be object such that

                   A15: z in ( dom g0) and

                   A16: z in W and

                   A17: (g0 . z) = x by FUNCT_1:def 6;

                  reconsider pz = z as Point of X by A15;

                  reconsider aa1 = (f1 . pz) as Real;

                  

                   A18: x = (a * aa1) by A2, A17;

                  pz in the carrier of X;

                  then pz in ( dom f1) by FUNCT_2:def 1;

                  then

                   A19: (f1 . pz) in (f1 .: W1) by A16, FUNCT_1:def 6;

                  then (r1 - r4) < aa1 by A14, XXREAL_1: 4;

                  then

                   A20: (a * (r1 - r4)) < (a * aa1) by A10, XREAL_1: 68;

                  reconsider rx = x as Real by A17;

                  

                   A21: (a * (r1 + r4)) = ((a * r1) + (a * r4))

                  .= (r + r0) by A7, A10, XCMPLX_1: 87;

                  

                   A22: (a * (r1 - r4)) = ((a * r1) - (a * r4))

                  .= (r - r0) by A7, A10, XCMPLX_1: 87;

                  aa1 < (r1 + r4) by A14, A19, XXREAL_1: 4;

                  then rx < (r + r0) by A10, A18, A21, XREAL_1: 68;

                  hence thesis by A18, A20, A22, XXREAL_1: 4;

                end;

                hence thesis by A6, A13, XBOOLE_1: 1;

              end;

                case

                 A23: a = 0 ;

                set r4 = r0;

                reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

                

                 A24: r1 < (r1 + r4) by A5, XREAL_1: 29;

                then (r1 - r4) < r1 by XREAL_1: 19;

                then

                 A25: (f1 . p) in G1 by A24, XXREAL_1: 4;

                G1 is open by JORDAN6: 35;

                then

                consider W1 be Subset of X such that

                 A26: p in W1 & W1 is open and (f1 .: W1) c= G1 by A4, A25, Th10;

                set W = W1;

                (g0 .: W) c= ].(r - r0), (r + r0).[

                proof

                  let x be object;

                  assume x in (g0 .: W);

                  then

                  consider z be object such that

                   A27: z in ( dom g0) and z in W and

                   A28: (g0 . z) = x by FUNCT_1:def 6;

                  reconsider pz = z as Point of X by A27;

                  reconsider aa1 = (f1 . pz) as Real;

                  x = (a * aa1) by A2, A28

                  .= 0 by A23;

                  hence thesis by A5, A8, A23, XXREAL_1: 4;

                end;

                hence thesis by A6, A26, XBOOLE_1: 1;

              end;

            end;

            hence thesis;

          end;

            case

             A29: a < 0 ;

            set r4 = (r0 / ( - a));

            reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

            ( - a) > 0 by A29, XREAL_1: 58;

            then

             A30: r1 < (r1 + r4) by A5, XREAL_1: 29, XREAL_1: 139;

            then (r1 - r4) < r1 by XREAL_1: 19;

            then

             A31: (f1 . p) in G1 by A30, XXREAL_1: 4;

            G1 is open by JORDAN6: 35;

            then

            consider W1 be Subset of X such that

             A32: p in W1 & W1 is open and

             A33: (f1 .: W1) c= G1 by A4, A31, Th10;

            set W = W1;

            ( - a) <> 0 by A29;

            then

             A34: (( - a) * r4) = r0 by XCMPLX_1: 87;

            (g0 .: W) c= ].(r - r0), (r + r0).[

            proof

              let x be object;

              assume x in (g0 .: W);

              then

              consider z be object such that

               A35: z in ( dom g0) and

               A36: z in W and

               A37: (g0 . z) = x by FUNCT_1:def 6;

              reconsider pz = z as Point of X by A35;

              reconsider aa1 = (f1 . pz) as Real;

              pz in the carrier of X;

              then pz in ( dom f1) by FUNCT_2:def 1;

              then

               A38: (f1 . pz) in (f1 .: W1) by A36, FUNCT_1:def 6;

              then (r1 - r4) < aa1 by A33, XXREAL_1: 4;

              then

               A39: (a * aa1) < (a * (r1 - r4)) by A29, XREAL_1: 69;

              

               A40: (a * (r1 + r4)) = ((a * r1) - ( - (a * r4)))

              .= (r - r0) by A3, A34;

              

               A41: (a * (r1 - r4)) = ((a * r1) + ( - (a * r4)))

              .= (r + r0) by A3, A34;

              aa1 < (r1 + r4) by A33, A38, XXREAL_1: 4;

              then

               A42: (r - r0) < (a * aa1) by A29, A40, XREAL_1: 69;

              x = (a * aa1) by A2, A37;

              hence thesis by A39, A41, A42, XXREAL_1: 4;

            end;

            hence thesis by A6, A32, XBOOLE_1: 1;

          end;

        end;

        hence thesis;

      end;

      then g0 is continuous by Th10;

      hence thesis by A3;

    end;

    theorem :: JGRAPH_2:24

    

     Th24: for X be non empty TopSpace, f1 be Function of X, R^1 , a be Real st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (r1 + a)) & g is continuous

    proof

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

      defpred P[ set, set] means (for r1 be Real st (f1 . $1) = r1 holds $2 = (r1 + a));

      

       A1: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        reconsider r2 = a as Element of REAL by XREAL_0:def 1;

        let x be Element of X;

        reconsider r1 = (f1 . x) as Element of REAL by TOPMETR: 17;

        set r3 = (r1 + r2);

        for r1 be Real st (f1 . x) = r1 holds r3 = (r1 + r2);

        hence ex y be Element of REAL st for r1 be Real st (f1 . x) = r1 holds y = (r1 + a);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A1);

      then

      consider f be Function of the carrier of X, REAL such that

       A2: for x be Element of X holds for r1 be Real st (f1 . x) = r1 holds (f . x) = (r1 + a);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      assume

       A3: f1 is continuous;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A4: r0 > 0 and

         A5: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        set r4 = r0;

        reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

        

         A6: r1 < (r1 + r4) by A4, XREAL_1: 29;

        then (r1 - r4) < r1 by XREAL_1: 19;

        then

         A7: (f1 . p) in G1 by A6, XXREAL_1: 4;

        G1 is open by JORDAN6: 35;

        then

        consider W1 be Subset of X such that

         A8: p in W1 & W1 is open and

         A9: (f1 .: W1) c= G1 by A3, A7, Th10;

        set W = W1;

        (g0 .: W) c= ].(r - r0), (r + r0).[

        proof

          let x be object;

          assume x in (g0 .: W);

          then

          consider z be object such that

           A10: z in ( dom g0) and

           A11: z in W and

           A12: (g0 . z) = x by FUNCT_1:def 6;

          reconsider pz = z as Point of X by A10;

          reconsider aa1 = (f1 . pz) as Real;

          pz in the carrier of X;

          then pz in ( dom f1) by FUNCT_2:def 1;

          then

           A13: (f1 . pz) in (f1 .: W1) by A11, FUNCT_1:def 6;

          then (r1 - r4) < aa1 by A9, XXREAL_1: 4;

          then

           A14: ((r1 - r4) + a) < (aa1 + a) by XREAL_1: 8;

          

           A15: ((r1 - r4) + a) = ((r1 + a) - r4)

          .= (r - r0) by A2;

          aa1 < (r1 + r4) by A9, A13, XXREAL_1: 4;

          then

           A16: ((r1 + r4) + a) > (aa1 + a) by XREAL_1: 8;

          x = (aa1 + a) by A2, A12;

          hence thesis by A16, A14, A15, XXREAL_1: 4;

        end;

        hence thesis by A5, A8, XBOOLE_1: 1;

      end;

      then

       A17: g0 is continuous by Th10;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g0 . p) = (r1 + a) by A2;

      hence thesis by A17;

    end;

    theorem :: JGRAPH_2:25

    

     Th25: for X be non empty TopSpace, f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r1 * r2)) & g is continuous

    proof

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

      assume

       A1: f1 is continuous & f2 is continuous;

      then

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

       A2: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g1 . p) = (r1 + r2) and

       A3: g1 is continuous by Th19;

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

       A4: for p be Point of X, r1 be Real st (g1 . p) = r1 holds (g3 . p) = (r1 * r1) and

       A5: g3 is continuous by A3, Th22;

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

       A6: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = (r1 - r2) and

       A7: g2 is continuous by A1, Th21;

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

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

       A9: g4 is continuous by A7, Th22;

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

       A10: for p be Point of X, r1,r2 be Real st (g3 . p) = r1 & (g4 . p) = r2 holds (g5 . p) = (r1 - r2) and

       A11: g5 is continuous by A5, A9, Th21;

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

       A12: for p be Point of X, r1 be Real st (g5 . p) = r1 holds (g6 . p) = ((1 / 4) * r1) and

       A13: g6 is continuous by A11, Th23;

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

      proof

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

        assume

         A14: (f1 . p) = r1 & (f2 . p) = r2;

        then (g2 . p) = (r1 - r2) by A6;

        then

         A15: (g4 . p) = ((r1 - r2) ^2 ) by A8;

        (g1 . p) = (r1 + r2) by A2, A14;

        then (g3 . p) = ((r1 + r2) ^2 ) by A4;

        then (g5 . p) = (((r1 + r2) ^2 ) - ((r1 - r2) ^2 )) by A10, A15;

        

        then (g6 . p) = ((1 / 4) * ((((r1 ^2 ) + ((2 * r1) * r2)) + (r2 ^2 )) - ((r1 - r2) ^2 ))) by A12

        .= (r1 * r2);

        hence thesis;

      end;

      hence thesis by A13;

    end;

    theorem :: JGRAPH_2:26

    

     Th26: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous & (for q be Point of X holds (f1 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (1 / r1)) & g is continuous

    proof

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

      assume that

       A1: f1 is continuous and

       A2: for q be Point of X holds (f1 . q) <> 0 ;

      defpred P[ set, set] means (for r1 be Real st (f1 . $1) = r1 holds $2 = (1 / r1));

      

       A3: for x be Element of X holds ex y be Element of REAL st P[x, y]

      proof

        let x be Element of X;

        reconsider r1 = (f1 . x) as Element of REAL by TOPMETR: 17;

        reconsider r3 = (1 / r1) as Element of REAL by XREAL_0:def 1;

        take r3;

        thus for r1 be Real st (f1 . x) = r1 holds r3 = (1 / r1);

      end;

      ex f be Function of the carrier of X, REAL st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A3);

      then

      consider f be Function of the carrier of X, REAL such that

       A4: for x be Element of X holds for r1 be Real st (f1 . x) = r1 holds (f . x) = (1 / r1);

      reconsider g0 = f as Function of X, R^1 by TOPMETR: 17;

      for p be Point of X, V be Subset of R^1 st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

      proof

        let p be Point of X, V be Subset of R^1 ;

        reconsider r = (g0 . p) as Real;

        reconsider r1 = (f1 . p) as Real;

        assume (g0 . p) in V & V is open;

        then

        consider r0 be Real such that

         A5: r0 > 0 and

         A6: ].(r - r0), (r + r0).[ c= V by FRECHET: 8;

        

         A7: r = (1 / r1) by A4;

        

         A8: r1 <> 0 by A2;

        now

          per cases ;

            case

             A9: r1 >= 0 ;

            set r4 = ((r0 / r) / (r + r0));

            reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

            (r0 / r) > 0 by A5, A8, A7, A9, XREAL_1: 139;

            then

             A10: r1 < (r1 + r4) by A5, A7, A9, XREAL_1: 29, XREAL_1: 139;

            then (r1 - r4) < r1 by XREAL_1: 19;

            then

             A11: (f1 . p) in G1 by A10, XXREAL_1: 4;

            

             A12: (r / (r + r0)) > 0 by A5, A8, A7, A9, XREAL_1: 139;

            G1 is open by JORDAN6: 35;

            then

            consider W1 be Subset of X such that

             A13: p in W1 & W1 is open and

             A14: (f1 .: W1) c= G1 by A1, A11, Th10;

            set W = W1;

            (r1 - r4) = ((1 / r) - ((r0 / (r + r0)) / r)) by A7

            .= ((1 - (r0 / (r + r0))) / r)

            .= ((((r + r0) / (r + r0)) - (r0 / (r + r0))) / r) by A5, A7, A9, XCMPLX_1: 60

            .= ((((r + r0) - r0) / (r + r0)) / r)

            .= ((r / (r + r0)) / r);

            then

             A15: (r1 - r4) > 0 by A8, A7, A9, A12, XREAL_1: 139;

            (g0 .: W) c= ].(r - r0), (r + r0).[

            proof

               0 < (r0 ^2 ) by A5, SQUARE_1: 12;

              then (r0 * r) < ((r0 * r) + ((r0 * r0) + (r0 * r0))) by XREAL_1: 29;

              then ((r0 * r) - ((r0 * r0) + (r0 * r0))) < (r0 * r) by XREAL_1: 19;

              then (((r0 * r) - ((r0 * r0) + (r0 * r0))) + (r * r)) < ((r * r) + (r0 * r)) by XREAL_1: 8;

              then (((r - r0) * ((r + r0) + r0)) / ((r + r0) + r0)) < ((r * (r + r0)) / ((r + r0) + r0)) by A5, A7, A9, XREAL_1: 74;

              then (r - r0) < ((r * (r + r0)) / ((r + r0) + r0)) by A5, A7, A9, XCMPLX_1: 89;

              then (r - r0) < (r / (((r + r0) + r0) / (r + r0))) by XCMPLX_1: 77;

              then (r - r0) < (r / (((r + r0) / (r + r0)) + (r0 / (r + r0))));

              then (r - r0) < ((r * 1) / (1 + (r0 / (r + r0)))) by A5, A7, A9, XCMPLX_1: 60;

              then

               A16: (r - r0) < (1 / ((1 + (r0 / (r + r0))) / r)) by XCMPLX_1: 77;

              let x be object;

              assume x in (g0 .: W);

              then

              consider z be object such that

               A17: z in ( dom g0) and

               A18: z in W and

               A19: (g0 . z) = x by FUNCT_1:def 6;

              reconsider pz = z as Point of X by A17;

              reconsider aa1 = (f1 . pz) as Real;

              

               A20: x = (1 / aa1) by A4, A19;

              pz in the carrier of X;

              then pz in ( dom f1) by FUNCT_2:def 1;

              then

               A21: (f1 . pz) in (f1 .: W1) by A18, FUNCT_1:def 6;

              then

               A22: (r1 - r4) < aa1 by A14, XXREAL_1: 4;

              then

               A23: (1 / aa1) < (1 / (r1 - r4)) by A15, XREAL_1: 88;

              aa1 < (r1 + r4) by A14, A21, XXREAL_1: 4;

              then (1 / ((1 / r) + r4)) < (1 / aa1) by A7, A15, A22, XREAL_1: 76;

              then

               A24: (r - r0) < (1 / aa1) by A16, XXREAL_0: 2;

              (1 / (r1 - r4)) = (1 / (r1 - ((r0 * (r " )) / (r + r0))))

              .= (1 / (r1 - ((r0 * (1 / r)) / (r + r0))))

              .= (1 / (r1 - (r0 / ((r + r0) / r1)))) by A7, XCMPLX_1: 77

              .= (1 / ((r1 * 1) - (r1 * (r0 / (r + r0))))) by XCMPLX_1: 81

              .= (1 / ((1 - (r0 / (r + r0))) * r1))

              .= (1 / ((((r + r0) / (r + r0)) - (r0 / (r + r0))) * r1)) by A5, A7, A9, XCMPLX_1: 60

              .= (1 / ((((r + r0) - r0) / (r + r0)) * r1))

              .= (1 / (r / ((r + r0) / r1))) by XCMPLX_1: 81

              .= (1 / ((r * r1) / (r + r0))) by XCMPLX_1: 77

              .= (((r + r0) / (r * r1)) * 1) by XCMPLX_1: 80

              .= ((r + r0) / 1) by A8, A7, XCMPLX_0:def 7

              .= (r + r0);

              hence thesis by A20, A24, A23, XXREAL_1: 4;

            end;

            hence thesis by A6, A13, XBOOLE_1: 1;

          end;

            case

             A25: r1 < 0 ;

            set r4 = ((r0 / ( - r)) / (( - r) + r0));

            reconsider G1 = ].(r1 - r4), (r1 + r4).[ as Subset of R^1 by TOPMETR: 17;

            

             A26: G1 is open by JORDAN6: 35;

            

             A27: 0 < ( - r) by A7, A25, XREAL_1: 58;

            then (( - r) / (( - r) + r0)) > 0 by A5, XREAL_1: 139;

            then ( - (r / (( - r) + r0))) > 0 ;

            then

             A28: (r / (( - r) + r0)) < 0 ;

            (r0 / ( - r)) > 0 by A5, A27, XREAL_1: 139;

            then

             A29: r1 < (r1 + r4) by A5, A7, A25, XREAL_1: 29, XREAL_1: 139;

            then (r1 - r4) < r1 by XREAL_1: 19;

            then (f1 . p) in G1 by A29, XXREAL_1: 4;

            then

            consider W1 be Subset of X such that

             A30: p in W1 & W1 is open and

             A31: (f1 .: W1) c= G1 by A1, A26, Th10;

            set W = W1;

            (r1 * (( - r) * (1 / ( - r)))) = (r1 * 1) by A27, XCMPLX_1: 87;

            then (( - (r * r1)) * (1 / ( - r))) = r1;

            then

             A32: (( - 1) * (1 / ( - r))) = r1 by A2, A7, XCMPLX_1: 87;

            

            then (r1 + r4) = (( - (1 / ( - r))) + ((r0 / (( - r) + r0)) / ( - r)))

            .= ((( - 1) / ( - r)) + ((r0 / (( - r) + r0)) / ( - r)))

            .= ((( - 1) + (r0 / (( - r) + r0))) / ( - r))

            .= ((( - ((( - r) + r0) / (( - r) + r0))) + (r0 / (( - r) + r0))) / ( - r)) by A5, A7, A25, XCMPLX_1: 60

            .= (((( - (( - r) + r0)) / (( - r) + r0)) + (r0 / (( - r) + r0))) / ( - r))

            .= ((((r - r0) + r0) / (( - r) + r0)) / ( - r))

            .= ((r / (( - r) + r0)) / ( - r));

            then

             A33: (r1 + r4) < 0 by A27, A28, XREAL_1: 141;

            (g0 .: W) c= ].(r - r0), (r + r0).[

            proof

               0 < (r0 ^2 ) by A5, SQUARE_1: 12;

              then (r0 * ( - r)) < ((r0 * ( - r)) + ((r0 * r0) + (r0 * r0))) by XREAL_1: 29;

              then ((r0 * ( - r)) - ((r0 * r0) + (r0 * r0))) < (r0 * ( - r)) by XREAL_1: 19;

              then (((r0 * ( - r)) - ((r0 * r0) + (r0 * r0))) + (( - r) * ( - r))) < ((r0 * ( - r)) + (( - r) * ( - r))) by XREAL_1: 8;

              then (((( - r) - r0) * ((( - r) + r0) + r0)) / ((( - r) + r0) + r0)) < ((( - r) * (( - r) + r0)) / ((( - r) + r0) + r0)) by A5, A7, A25, XREAL_1: 74;

              then (( - r) - r0) < ((( - r) * (( - r) + r0)) / ((( - r) + r0) + r0)) by A5, A7, A25, XCMPLX_1: 89;

              then (( - r) - r0) < (( - r) / (((( - r) + r0) + r0) / (( - r) + r0))) by XCMPLX_1: 77;

              then (( - r) - r0) < (( - r) / (((( - r) + r0) / (( - r) + r0)) + (r0 / (( - r) + r0))));

              then (( - r) - r0) < ((( - r) * 1) / (1 + (r0 / (( - r) + r0)))) by A5, A7, A25, XCMPLX_1: 60;

              then (( - r) - r0) < (1 / ((1 + (r0 / (( - r) + r0))) / ( - r))) by XCMPLX_1: 77;

              then ( - (r + r0)) < (1 / ((1 / ( - r)) + r4));

              then (r + r0) > ( - (1 / ((1 / ( - r)) + r4))) by XREAL_1: 25;

              then

               A34: (r + r0) > (1 / ( - ((1 / ( - r)) + r4))) by XCMPLX_1: 188;

              let x be object;

              assume x in (g0 .: W);

              then

              consider z be object such that

               A35: z in ( dom g0) and

               A36: z in W and

               A37: (g0 . z) = x by FUNCT_1:def 6;

              reconsider pz = z as Point of X by A35;

              reconsider aa1 = (f1 . pz) as Real;

              

               A38: x = (1 / aa1) by A4, A37;

              pz in the carrier of X;

              then pz in ( dom f1) by FUNCT_2:def 1;

              then

               A39: (f1 . pz) in (f1 .: W1) by A36, FUNCT_1:def 6;

              then

               A40: aa1 < (r1 + r4) by A31, XXREAL_1: 4;

              then

               A41: (1 / aa1) > (1 / (r1 + r4)) by A33, XREAL_1: 87;

              (r1 - r4) < aa1 by A31, A39, XXREAL_1: 4;

              then (1 / (( - (1 / ( - r))) - r4)) > (1 / aa1) by A32, A33, A40, XREAL_1: 99;

              then

               A42: (r + r0) > (1 / aa1) by A34, XXREAL_0: 2;

              (1 / (r1 + r4)) = (1 / (r1 + ((r0 * (( - r) " )) / (( - r) + r0))))

              .= (1 / (r1 + ((r0 * (1 / ( - r))) / (( - r) + r0))))

              .= (1 / (r1 + (( - (r1 * r0)) / (( - r) + r0)))) by A32

              .= (1 / (r1 + ( - ((r1 * r0) / (( - r) + r0)))))

              .= (1 / (r1 - ((r1 * r0) / (( - r) + r0))))

              .= (1 / (r1 - (r0 / ((( - r) + r0) / r1)))) by XCMPLX_1: 77

              .= (1 / ((r1 * 1) - (r1 * (r0 / (( - r) + r0))))) by XCMPLX_1: 81

              .= (1 / (r1 * (1 - (r0 / (( - r) + r0)))))

              .= (1 / ((((( - r) + r0) / (( - r) + r0)) - (r0 / (( - r) + r0))) * r1)) by A5, A7, A25, XCMPLX_1: 60

              .= (1 / ((((( - r) + r0) - r0) / ( - (r - r0))) * r1))

              .= (1 / (( - (((( - r) + r0) - r0) / (r - r0))) * r1)) by XCMPLX_1: 188

              .= (1 / ((((( - r) + r0) - r0) / (r - r0)) * ( - r1)))

              .= (1 / (( - r) / ((r - r0) / ( - r1)))) by XCMPLX_1: 81

              .= (1 / ((( - r) * ( - r1)) / (r - r0))) by XCMPLX_1: 77

              .= (((r - r0) / (( - r) * ( - r1))) * 1) by XCMPLX_1: 80

              .= ((r - r0) / (( - r) * (( - r) " ))) by A32

              .= ((r - r0) / 1) by A27, XCMPLX_0:def 7

              .= (r - r0);

              hence thesis by A38, A42, A41, XXREAL_1: 4;

            end;

            hence thesis by A6, A30, XBOOLE_1: 1;

          end;

        end;

        hence thesis;

      end;

      then

       A43: g0 is continuous by Th10;

      for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g0 . p) = (1 / r1) by A4;

      hence thesis by A43;

    end;

    theorem :: JGRAPH_2:27

    

     Th27: for X be non empty TopSpace, f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = (r1 / r2)) & g is continuous

    proof

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

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous & for q be Point of X holds (f2 . q) <> 0 ;

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

       A3: for p be Point of X, r2 be Real st (f2 . p) = r2 holds (g1 . p) = (1 / r2) and

       A4: g1 is continuous by A2, Th26;

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

       A5: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (g1 . p) = r2 holds (g2 . p) = (r1 * r2) and

       A6: g2 is continuous by A1, A4, Th25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = (r1 / r2)

      proof

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

        assume that

         A7: (f1 . p) = r1 and

         A8: (f2 . p) = r2;

        (g1 . p) = (1 / r2) by A3, A8;

        

        then (g2 . p) = (r1 * (1 / r2)) by A5, A7

        .= (r1 / r2);

        hence thesis;

      end;

      hence thesis by A6;

    end;

    theorem :: JGRAPH_2:28

    

     Th28: for X be non empty TopSpace, f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous & (for q be Point of X holds (f2 . q) <> 0 ) holds ex g be Function of X, R^1 st (for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g . p) = ((r1 / r2) / r2)) & g is continuous

    proof

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

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous & for q be Point of X holds (f2 . q) <> 0 ;

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

       A3: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = (r1 / r2) and

       A4: g2 is continuous by A1, A2, Th27;

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

       A5: for p be Point of X, r1,r2 be Real st (g2 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (r1 / r2) and

       A6: g3 is continuous by A2, A4, Th27;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = ((r1 / r2) / r2)

      proof

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

        assume that

         A7: (f1 . p) = r1 and

         A8: (f2 . p) = r2;

        (g2 . p) = (r1 / r2) by A3, A7, A8;

        hence thesis by A5, A8;

      end;

      hence thesis by A6;

    end;

    theorem :: JGRAPH_2:29

    

     Th29: for K0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), R^1 st (for p be Point of (( TOP-REAL 2) | K0) holds (f . p) = ( proj1 . p)) holds f is continuous

    proof

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

      let K0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), R^1 ;

      

       A1: ( dom f) = the carrier of (( TOP-REAL 2) | K0) & (the carrier of ( TOP-REAL 2) /\ K0) = K0 by FUNCT_2:def 1, XBOOLE_1: 28;

      

       A2: g is continuous by JORDAN5A: 27;

      assume for p be Point of (( TOP-REAL 2) | K0) holds (f . p) = ( proj1 . p);

      then

       A3: for x be object st x in ( dom f) holds (f . x) = ( proj1 . x);

      the carrier of (( TOP-REAL 2) | K0) = ( [#] (( TOP-REAL 2) | K0))

      .= K0 by PRE_TOPC:def 5;

      then f = (g | K0) by A1, A3, Th6, FUNCT_1: 46;

      hence thesis by A2, TOPMETR: 7;

    end;

    theorem :: JGRAPH_2:30

    

     Th30: for K0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), R^1 st (for p be Point of (( TOP-REAL 2) | K0) holds (f . p) = ( proj2 . p)) holds f is continuous

    proof

      let K0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), R^1 ;

      

       A1: ( dom f) = the carrier of (( TOP-REAL 2) | K0) & (the carrier of ( TOP-REAL 2) /\ K0) = K0 by FUNCT_2:def 1, XBOOLE_1: 28;

      assume for p be Point of (( TOP-REAL 2) | K0) holds (f . p) = ( proj2 . p);

      then

       A2: for x be object st x in ( dom f) holds (f . x) = ( proj2 . x);

      the carrier of (( TOP-REAL 2) | K0) = ( [#] (( TOP-REAL 2) | K0))

      .= K0 by PRE_TOPC:def 5;

      then f = ( proj2 | K0) by A1, A2, Th7, FUNCT_1: 46;

      hence thesis by JORDAN5A: 27;

    end;

    theorem :: JGRAPH_2:31

    

     Th31: for K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (1 / (p `1 ))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ) holds f is continuous

    proof

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

      assume that

       A1: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (1 / (p `1 )) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      reconsider g1 = ( proj1 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

      .= K1 by PRE_TOPC:def 5;

      

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) = ( proj1 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj1 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      

       A5: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0

      proof

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

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

        then

        reconsider q2 = q as Point of ( TOP-REAL 2) by A3;

        (g1 . q) = ( proj1 . q) by A4

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence thesis by A2;

      end;

      g1 is continuous by A4, Th29;

      then

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

       A6: for q be Point of (( TOP-REAL 2) | K1), r2 be Real st (g1 . q) = r2 holds (g3 . q) = (1 / r2) and

       A7: g3 is continuous by A5, Th26;

      

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

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

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

        x in ( [#] (( TOP-REAL 2) | K1)) by A9;

        then x in K1 by PRE_TOPC:def 5;

        then

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

        

         A10: (g1 . s) = ( proj1 . s) & ( proj1 . r) = (r `1 ) by A4, PSCOMP_1:def 5;

        (f . r) = (1 / (r `1 )) by A1, A9;

        hence thesis by A6, A10;

      end;

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

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

      hence thesis by A7, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_2:32

    

     Th32: for K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (1 / (p `2 ))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <> 0 ) holds f is continuous

    proof

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

      assume that

       A1: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (1 / (p `2 )) and

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

      reconsider g1 = ( proj2 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

      .= K1 by PRE_TOPC:def 5;

      

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) = ( proj2 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj2 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      

       A5: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0

      proof

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

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

        then

        reconsider q2 = q as Point of ( TOP-REAL 2) by A3;

        (g1 . q) = ( proj2 . q) by A4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence thesis by A2;

      end;

      g1 is continuous by A4, Th30;

      then

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

       A6: for q be Point of (( TOP-REAL 2) | K1), r2 be Real st (g1 . q) = r2 holds (g3 . q) = (1 / r2) and

       A7: g3 is continuous by A5, Th26;

      

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

      proof

        let x be object;

        assume

         A9: x in ( dom f);

        then

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

        x in ( [#] (( TOP-REAL 2) | K1)) by A9;

        then x in K1 by PRE_TOPC:def 5;

        then

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

        

         A10: (g1 . s) = ( proj2 . s) & ( proj2 . r) = (r `2 ) by A4, PSCOMP_1:def 6;

        (f . r) = (1 / (r `2 )) by A1, A9;

        hence thesis by A6, A10;

      end;

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

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

      hence thesis by A7, A8, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_2:33

    

     Th33: for K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (((p `2 ) / (p `1 )) / (p `1 ))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ) holds f is continuous

    proof

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

      assume that

       A1: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (((p `2 ) / (p `1 )) / (p `1 )) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      reconsider g2 = ( proj2 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      reconsider g1 = ( proj1 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

      .= K1 by PRE_TOPC:def 5;

      

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) = ( proj1 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj1 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      then

       A5: g1 is continuous by Th29;

      

       A6: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0

      proof

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

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

        then

        reconsider q2 = q as Point of ( TOP-REAL 2) by A3;

        (g1 . q) = ( proj1 . q) by A4

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence thesis by A2;

      end;

      

       A7: for q be Point of (( TOP-REAL 2) | K1) holds (g2 . q) = ( proj2 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj2 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      then g2 is continuous by Th30;

      then

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

       A8: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = ((r1 / r2) / r2) and

       A9: g3 is continuous by A5, A6, Th28;

      

       A10: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A11: x in ( dom f);

        then

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

        x in ( [#] (( TOP-REAL 2) | K1)) by A11;

        then x in K1 by PRE_TOPC:def 5;

        then

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

        

         A12: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A13: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by A7, A4;

        (f . r) = (((r `2 ) / (r `1 )) / (r `1 )) by A1, A11;

        hence thesis by A8, A13, A12;

      end;

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

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

      hence thesis by A9, A10, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_2:34

    

     Th34: for K1 be non empty Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K1), R^1 st (for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (((p `1 ) / (p `2 )) / (p `2 ))) & (for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <> 0 ) holds f is continuous

    proof

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

      assume that

       A1: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f . p) = (((p `1 ) / (p `2 )) / (p `2 )) and

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

      reconsider g2 = ( proj1 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      reconsider g1 = ( proj2 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

      .= K1 by PRE_TOPC:def 5;

      

       A4: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) = ( proj2 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj2 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      then

       A5: g1 is continuous by Th30;

      

       A6: for q be Point of (( TOP-REAL 2) | K1) holds (g1 . q) <> 0

      proof

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

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

        then

        reconsider q2 = q as Point of ( TOP-REAL 2) by A3;

        (g1 . q) = ( proj2 . q) by A4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence thesis by A2;

      end;

      

       A7: for q be Point of (( TOP-REAL 2) | K1) holds (g2 . q) = ( proj1 . q)

      proof

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

        q in the carrier of (( TOP-REAL 2) | K1) & ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        then q in (( dom proj1 ) /\ K1) by A3, XBOOLE_0:def 4;

        hence thesis by FUNCT_1: 48;

      end;

      then g2 is continuous by Th29;

      then

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

       A8: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = ((r1 / r2) / r2) and

       A9: g3 is continuous by A5, A6, Th28;

      

       A10: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A11: x in ( dom f);

        then

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

        x in ( [#] (( TOP-REAL 2) | K1)) by A11;

        then x in K1 by PRE_TOPC:def 5;

        then

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

        

         A12: ( proj1 . r) = (r `1 ) & ( proj2 . r) = (r `2 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A13: (g2 . s) = ( proj1 . s) & (g1 . s) = ( proj2 . s) by A7, A4;

        (f . r) = (((r `1 ) / (r `2 )) / (r `2 )) by A1, A11;

        hence thesis by A8, A13, A12;

      end;

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

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

      hence thesis by A9, A10, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_2:35

    

     Th35: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0), f1,f2 be Function of (( TOP-REAL 2) | K0), R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & (for x,y,r,s be Real st |[x, y]| in K0 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|) holds f is continuous

    proof

      let K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0), f1,f2 be Function of (( TOP-REAL 2) | K0), R^1 ;

      assume that

       A1: f1 is continuous and

       A2: f2 is continuous and

       A3: K0 <> {} and

       A4: B0 <> {} and

       A5: for x,y,r,s be Real st |[x, y]| in K0 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|;

      reconsider B1 = B0 as non empty Subset of ( TOP-REAL 2) by A4;

      reconsider K1 = K0 as non empty Subset of ( TOP-REAL 2) by A3;

      reconsider X = (( TOP-REAL 2) | K1), Y = (( TOP-REAL 2) | B1) as non empty TopSpace;

      reconsider f0 = f as Function of X, Y;

      for r be Point of X, V be Subset of Y st (f0 . r) in V & V is open holds ex W be Subset of X st r in W & W is open & (f0 .: W) c= V

      proof

        let r be Point of X, V be Subset of Y;

        assume that

         A6: (f0 . r) in V and

         A7: V is open;

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

         A8: V2 is open and

         A9: V = (V2 /\ ( [#] Y)) by A7, TOPS_2: 24;

        

         A10: (V2 /\ ( [#] Y)) c= V2 by XBOOLE_1: 17;

        then (f0 . r) in V2 by A6, A9;

        then

        reconsider p = (f0 . r) as Point of ( TOP-REAL 2);

        consider r2 be Real such that

         A11: r2 > 0 and

         A12: { q where q be Point of ( TOP-REAL 2) : ((p `1 ) - r2) < (q `1 ) & (q `1 ) < ((p `1 ) + r2) & ((p `2 ) - r2) < (q `2 ) & (q `2 ) < ((p `2 ) + r2) } c= V2 by A6, A8, A9, A10, Th11;

        reconsider G1 = ].((p `1 ) - r2), ((p `1 ) + r2).[, G2 = ].((p `2 ) - r2), ((p `2 ) + r2).[ as Subset of R^1 by TOPMETR: 17;

        

         A13: G1 is open by JORDAN6: 35;

        reconsider r3 = (f1 . r), r4 = (f2 . r) as Real;

        

         A14: the carrier of X = ( [#] X)

        .= K0 by PRE_TOPC:def 5;

        then r in K0;

        then

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

        

         A15: r = |[(pr `1 ), (pr `2 )]| by EUCLID: 53;

        then

         A16: (f0 . |[(pr `1 ), (pr `2 )]|) = |[r3, r4]| by A5, A14;

        

         A17: (p `2 ) < ((p `2 ) + r2) by A11, XREAL_1: 29;

        then ((p `2 ) - r2) < (p `2 ) by XREAL_1: 19;

        then (p `2 ) in ].((p `2 ) - r2), ((p `2 ) + r2).[ by A17, XXREAL_1: 4;

        then G2 is open & (f2 . r) in G2 by A15, A16, EUCLID: 52, JORDAN6: 35;

        then

        consider W2 be Subset of X such that

         A18: r in W2 and

         A19: W2 is open and

         A20: (f2 .: W2) c= G2 by A2, Th10;

        

         A21: (p `1 ) < ((p `1 ) + r2) by A11, XREAL_1: 29;

        then ((p `1 ) - r2) < (p `1 ) by XREAL_1: 19;

        then (p `1 ) in ].((p `1 ) - r2), ((p `1 ) + r2).[ by A21, XXREAL_1: 4;

        then (f1 . r) in ].((p `1 ) - r2), ((p `1 ) + r2).[ by A15, A16, EUCLID: 52;

        then

        consider W1 be Subset of X such that

         A22: r in W1 and

         A23: W1 is open and

         A24: (f1 .: W1) c= G1 by A1, A13, Th10;

        reconsider W5 = (W1 /\ W2) as Subset of X;

        (f2 .: W5) c= (f2 .: W2) by RELAT_1: 123, XBOOLE_1: 17;

        then

         A25: (f2 .: W5) c= G2 by A20;

        (f1 .: W5) c= (f1 .: W1) by RELAT_1: 123, XBOOLE_1: 17;

        then

         A26: (f1 .: W5) c= G1 by A24;

        

         A27: (f0 .: W5) c= V

        proof

          let v be object;

          assume

           A28: v in (f0 .: W5);

          then

          reconsider q2 = v as Point of Y;

          consider k be object such that

           A29: k in ( dom f0) and

           A30: k in W5 and

           A31: q2 = (f0 . k) by A28, FUNCT_1:def 6;

          the carrier of X = ( [#] X)

          .= K0 by PRE_TOPC:def 5;

          then k in K0 by A29;

          then

          reconsider r8 = k as Point of ( TOP-REAL 2);

          

           A32: ( dom f0) = the carrier of (( TOP-REAL 2) | K1) by FUNCT_2:def 1

          .= ( [#] (( TOP-REAL 2) | K1))

          .= K0 by PRE_TOPC:def 5;

          then

           A33: |[(r8 `1 ), (r8 `2 )]| in K0 by A29, EUCLID: 53;

          

           A34: ( dom f2) = the carrier of (( TOP-REAL 2) | K0) by FUNCT_2:def 1

          .= ( [#] (( TOP-REAL 2) | K0))

          .= K0 by PRE_TOPC:def 5;

          

           A35: ( dom f1) = the carrier of (( TOP-REAL 2) | K0) by FUNCT_2:def 1

          .= ( [#] (( TOP-REAL 2) | K0))

          .= K0 by PRE_TOPC:def 5;

          reconsider r7 = (f1 . |[(r8 `1 ), (r8 `2 )]|), s7 = (f2 . |[(r8 `1 ), (r8 `2 )]|) as Real;

          

           A36: |[(r8 `1 ), (r8 `2 )]| in W5 by A30, EUCLID: 53;

          then (f1 . |[(r8 `1 ), (r8 `2 )]|) in (f1 .: W5) by A33, A35, FUNCT_1:def 6;

          then

           A37: ((p `1 ) - r2) < r7 & r7 < ((p `1 ) + r2) by A26, XXREAL_1: 4;

          (f2 . |[(r8 `1 ), (r8 `2 )]|) in (f2 .: W5) by A33, A34, A36, FUNCT_1:def 6;

          then

           A38: ((p `2 ) - r2) < s7 & s7 < ((p `2 ) + r2) by A25, XXREAL_1: 4;

          k = |[(r8 `1 ), (r8 `2 )]| by EUCLID: 53;

          then

           A39: v = |[r7, s7]| by A5, A29, A31, A32;

          ( |[r7, s7]| `1 ) = r7 & ( |[r7, s7]| `2 ) = s7 by EUCLID: 52;

          then q2 in ( [#] Y) & v in { q3 where q3 be Point of ( TOP-REAL 2) : ((p `1 ) - r2) < (q3 `1 ) & (q3 `1 ) < ((p `1 ) + r2) & ((p `2 ) - r2) < (q3 `2 ) & (q3 `2 ) < ((p `2 ) + r2) } by A39, A37, A38;

          hence thesis by A9, A12, XBOOLE_0:def 4;

        end;

        r in W5 by A22, A18, XBOOLE_0:def 4;

        hence thesis by A23, A19, A27;

      end;

      hence thesis by Th10;

    end;

    theorem :: JGRAPH_2:36

    

     Th36: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

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

      

       A1: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

      assume

       A2: f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) };

      

       A3: K0 c= B0

      proof

        let x be object;

        assume

         A4: x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A2, A4, XBOOLE_0:def 5;

      end;

      (( 1.REAL 2) `2 ) <= (( 1.REAL 2) `1 ) & ( - (( 1.REAL 2) `1 )) <= (( 1.REAL 2) `2 ) or (( 1.REAL 2) `2 ) >= (( 1.REAL 2) `1 ) & (( 1.REAL 2) `2 ) <= ( - (( 1.REAL 2) `1 )) by Th5;

      then

       A5: ( 1.REAL 2) in K0 by A2, A1;

      then

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

      

       A6: K1 c= ( NonZero ( TOP-REAL 2))

      proof

        let z be object;

        assume

         A7: z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A7, XBOOLE_0:def 5;

      end;

      

       A8: ( dom ( Out_In_Sq | K1)) c= ( dom ( proj2 * ( Out_In_Sq | K1)))

      proof

        let x be object;

        assume

         A9: x in ( dom ( Out_In_Sq | K1));

        then x in (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61;

        then x in ( dom Out_In_Sq ) by XBOOLE_0:def 4;

        then ( Out_In_Sq . x) in ( rng Out_In_Sq ) by FUNCT_1: 3;

        then

         A10: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ( Out_In_Sq . x) in the carrier of ( TOP-REAL 2) by FUNCT_2:def 1, XBOOLE_0:def 5;

        (( Out_In_Sq | K1) . x) = ( Out_In_Sq . x) by A9, FUNCT_1: 47;

        hence thesis by A9, A10, FUNCT_1: 11;

      end;

      

       A11: ( rng ( proj2 * ( Out_In_Sq | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A12: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

      

       A13: ( dom ( Out_In_Sq | K1)) c= ( dom ( proj1 * ( Out_In_Sq | K1)))

      proof

        let x be object;

        assume

         A14: x in ( dom ( Out_In_Sq | K1));

        then x in (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61;

        then x in ( dom Out_In_Sq ) by XBOOLE_0:def 4;

        then ( Out_In_Sq . x) in ( rng Out_In_Sq ) by FUNCT_1: 3;

        then

         A15: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ( Out_In_Sq . x) in the carrier of ( TOP-REAL 2) by FUNCT_2:def 1, XBOOLE_0:def 5;

        (( Out_In_Sq | K1) . x) = ( Out_In_Sq . x) by A14, FUNCT_1: 47;

        hence thesis by A14, A15, FUNCT_1: 11;

      end;

      

       A16: ( rng ( proj1 * ( Out_In_Sq | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj1 * ( Out_In_Sq | K1))) c= ( dom ( Out_In_Sq | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ( Out_In_Sq | K1))) = ( dom ( Out_In_Sq | K1)) by A13

      .= (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K1) by A12, FUNCT_2:def 1

      .= K1 by A6, XBOOLE_1: 28

      .= ( [#] (( TOP-REAL 2) | K1)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K1);

      then

      reconsider g1 = ( proj1 * ( Out_In_Sq | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A16, FUNCT_2: 2;

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

      proof

        

         A17: K1 c= ( NonZero ( TOP-REAL 2))

        proof

          let z be object;

          assume

           A18: z in K1;

          then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

          then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

          hence thesis by A18, XBOOLE_0:def 5;

        end;

        

         A19: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

        

         A20: ( dom ( Out_In_Sq | K1)) = (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

        .= (( NonZero ( TOP-REAL 2)) /\ K1) by A19, FUNCT_2:def 1

        .= K1 by A17, XBOOLE_1: 28;

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

        

         A21: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        assume

         A22: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A21;

        then

         A23: ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| by Def1;

        (( Out_In_Sq | K1) . p) = ( Out_In_Sq . p) by A22, A21, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) by A22, A20, A21, A23, FUNCT_1: 13

        .= ( |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| `1 ) by PSCOMP_1:def 5

        .= (1 / (p `1 )) by EUCLID: 52;

        hence thesis;

      end;

      then

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

       A24: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = (1 / (p `1 ));

      ( dom ( proj2 * ( Out_In_Sq | K1))) c= ( dom ( Out_In_Sq | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ( Out_In_Sq | K1))) = ( dom ( Out_In_Sq | K1)) by A8

      .= (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K1) by A12, FUNCT_2:def 1

      .= K1 by A6, XBOOLE_1: 28

      .= ( [#] (( TOP-REAL 2) | K1)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K1);

      then

      reconsider g2 = ( proj2 * ( Out_In_Sq | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A11, FUNCT_2: 2;

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

      proof

        

         A25: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

        

         A26: ( dom ( Out_In_Sq | K1)) = (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

        .= (( NonZero ( TOP-REAL 2)) /\ K1) by A25, FUNCT_2:def 1

        .= K1 by A6, XBOOLE_1: 28;

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

        

         A27: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        assume

         A28: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A27;

        then

         A29: ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| by Def1;

        (( Out_In_Sq | K1) . p) = ( Out_In_Sq . p) by A28, A27, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|) by A28, A26, A27, A29, FUNCT_1: 13

        .= ( |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| `2 ) by PSCOMP_1:def 6

        .= (((p `2 ) / (p `1 )) / (p `1 )) by EUCLID: 52;

        hence thesis;

      end;

      then

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

       A30: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = (((p `2 ) / (p `1 )) / (p `1 ));

      

       A31: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0

      proof

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

        

         A32: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

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

        then

         A33: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A32;

        now

          assume

           A34: (q `1 ) = 0 ;

          then (q `2 ) = 0 by A33;

          hence contradiction by A33, A34, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      then

       A35: f1 is continuous by A24, Th31;

      

       A36: for x,y,r,s be Real st |[x, y]| in K1 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that

         A37: |[x, y]| in K1 and

         A38: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A39: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        then

         A40: (f1 . p99) = (1 / (p99 `1 )) by A24, A37;

        

         A41: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A37;

        then ((p99 `2 ) <= (p99 `1 ) & ( - (p99 `1 )) <= (p99 `2 ) or (p99 `2 ) >= (p99 `1 ) & (p99 `2 ) <= ( - (p99 `1 ))) implies ( Out_In_Sq . p99) = |[(1 / (p99 `1 )), (((p99 `2 ) / (p99 `1 )) / (p99 `1 ))]| by Def1;

        

        then (( Out_In_Sq | K0) . |[x, y]|) = |[(1 / (p99 `1 )), (((p99 `2 ) / (p99 `1 )) / (p99 `1 ))]| by A37, A41, FUNCT_1: 49

        .= |[r, s]| by A30, A37, A38, A39, A40;

        hence thesis by A2;

      end;

      f2 is continuous by A31, A30, Th33;

      hence thesis by A5, A3, A35, A36, Th35;

    end;

    theorem :: JGRAPH_2:37

    

     Th37: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous

    proof

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

      

       A1: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

      assume

       A2: f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) };

      

       A3: K0 c= B0

      proof

        let x be object;

        assume

         A4: x in K0;

        then ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A2, A4, XBOOLE_0:def 5;

      end;

      (( 1.REAL 2) `1 ) <= (( 1.REAL 2) `2 ) & ( - (( 1.REAL 2) `2 )) <= (( 1.REAL 2) `1 ) or (( 1.REAL 2) `1 ) >= (( 1.REAL 2) `2 ) & (( 1.REAL 2) `1 ) <= ( - (( 1.REAL 2) `2 )) by Th5;

      then

       A5: ( 1.REAL 2) in K0 by A2, A1;

      then

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

      

       A6: K1 c= ( NonZero ( TOP-REAL 2))

      proof

        let z be object;

        assume

         A7: z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

        then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A7, XBOOLE_0:def 5;

      end;

      

       A8: ( dom ( Out_In_Sq | K1)) c= ( dom ( proj1 * ( Out_In_Sq | K1)))

      proof

        let x be object;

        assume

         A9: x in ( dom ( Out_In_Sq | K1));

        then x in (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61;

        then x in ( dom Out_In_Sq ) by XBOOLE_0:def 4;

        then ( Out_In_Sq . x) in ( rng Out_In_Sq ) by FUNCT_1: 3;

        then

         A10: ( dom proj1 ) = the carrier of ( TOP-REAL 2) & ( Out_In_Sq . x) in the carrier of ( TOP-REAL 2) by FUNCT_2:def 1, XBOOLE_0:def 5;

        (( Out_In_Sq | K1) . x) = ( Out_In_Sq . x) by A9, FUNCT_1: 47;

        hence thesis by A9, A10, FUNCT_1: 11;

      end;

      

       A11: ( rng ( proj1 * ( Out_In_Sq | K1))) c= the carrier of R^1 by TOPMETR: 17;

      

       A12: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

      

       A13: ( dom ( Out_In_Sq | K1)) c= ( dom ( proj2 * ( Out_In_Sq | K1)))

      proof

        let x be object;

        assume

         A14: x in ( dom ( Out_In_Sq | K1));

        then x in (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61;

        then x in ( dom Out_In_Sq ) by XBOOLE_0:def 4;

        then ( Out_In_Sq . x) in ( rng Out_In_Sq ) by FUNCT_1: 3;

        then

         A15: ( dom proj2 ) = the carrier of ( TOP-REAL 2) & ( Out_In_Sq . x) in the carrier of ( TOP-REAL 2) by FUNCT_2:def 1, XBOOLE_0:def 5;

        (( Out_In_Sq | K1) . x) = ( Out_In_Sq . x) by A14, FUNCT_1: 47;

        hence thesis by A14, A15, FUNCT_1: 11;

      end;

      

       A16: ( rng ( proj2 * ( Out_In_Sq | K1))) c= the carrier of R^1 by TOPMETR: 17;

      ( dom ( proj2 * ( Out_In_Sq | K1))) c= ( dom ( Out_In_Sq | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * ( Out_In_Sq | K1))) = ( dom ( Out_In_Sq | K1)) by A13

      .= (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K1) by A12, FUNCT_2:def 1

      .= K1 by A6, XBOOLE_1: 28

      .= ( [#] (( TOP-REAL 2) | K1)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K1);

      then

      reconsider g1 = ( proj2 * ( Out_In_Sq | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A16, FUNCT_2: 2;

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

      proof

        

         A17: K1 c= ( NonZero ( TOP-REAL 2))

        proof

          let z be object;

          assume

           A18: z in K1;

          then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A2;

          then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

          hence thesis by A18, XBOOLE_0:def 5;

        end;

        

         A19: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

        

         A20: ( dom ( Out_In_Sq | K1)) = (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

        .= (( NonZero ( TOP-REAL 2)) /\ K1) by A19, FUNCT_2:def 1

        .= K1 by A17, XBOOLE_1: 28;

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

        

         A21: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        assume

         A22: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A21;

        then

         A23: ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by Th14;

        (( Out_In_Sq | K1) . p) = ( Out_In_Sq . p) by A22, A21, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) by A22, A20, A21, A23, FUNCT_1: 13

        .= ( |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| `2 ) by PSCOMP_1:def 6

        .= (1 / (p `2 )) by EUCLID: 52;

        hence thesis;

      end;

      then

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

       A24: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = (1 / (p `2 ));

      ( dom ( proj1 * ( Out_In_Sq | K1))) c= ( dom ( Out_In_Sq | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * ( Out_In_Sq | K1))) = ( dom ( Out_In_Sq | K1)) by A8

      .= (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K1) by A12, FUNCT_2:def 1

      .= K1 by A6, XBOOLE_1: 28

      .= ( [#] (( TOP-REAL 2) | K1)) by PRE_TOPC:def 5

      .= the carrier of (( TOP-REAL 2) | K1);

      then

      reconsider g2 = ( proj1 * ( Out_In_Sq | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by A11, FUNCT_2: 2;

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

      proof

        

         A25: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

        

         A26: ( dom ( Out_In_Sq | K1)) = (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

        .= (( NonZero ( TOP-REAL 2)) /\ K1) by A25, FUNCT_2:def 1

        .= K1 by A6, XBOOLE_1: 28;

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

        

         A27: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        assume

         A28: p in the carrier of (( TOP-REAL 2) | K1);

        then ex p3 be Point of ( TOP-REAL 2) st p = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A27;

        then

         A29: ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by Th14;

        (( Out_In_Sq | K1) . p) = ( Out_In_Sq . p) by A28, A27, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|) by A28, A26, A27, A29, FUNCT_1: 13

        .= ( |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| `1 ) by PSCOMP_1:def 5

        .= (((p `1 ) / (p `2 )) / (p `2 )) by EUCLID: 52;

        hence thesis;

      end;

      then

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

       A30: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = (((p `1 ) / (p `2 )) / (p `2 ));

      

       A31: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `2 ) <> 0

      proof

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

        

         A32: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

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

        then

         A33: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A32;

        now

          assume

           A34: (q `2 ) = 0 ;

          then (q `1 ) = 0 by A33;

          hence contradiction by A33, A34, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      then

       A35: f1 is continuous by A24, Th32;

      

       A36: for x,y,s,r be Real st |[x, y]| in K1 & s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|) holds (f . |[x, y]|) = |[s, r]|

      proof

        let x,y,s,r be Real;

        assume that

         A37: |[x, y]| in K1 and

         A38: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A39: ex p3 be Point of ( TOP-REAL 2) st p99 = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A2, A37;

        

         A40: the carrier of (( TOP-REAL 2) | K1) = ( [#] (( TOP-REAL 2) | K1))

        .= K1 by PRE_TOPC:def 5;

        then

         A41: (f1 . p99) = (1 / (p99 `2 )) by A24, A37;

        (( Out_In_Sq | K0) . |[x, y]|) = ( Out_In_Sq . |[x, y]|) by A37, FUNCT_1: 49

        .= |[(((p99 `1 ) / (p99 `2 )) / (p99 `2 )), (1 / (p99 `2 ))]| by A39, Th14

        .= |[s, r]| by A30, A37, A38, A40, A41;

        hence thesis by A2;

      end;

      f2 is continuous by A31, A30, Th34;

      hence thesis by A5, A3, A35, A36, Th35;

    end;

    scheme :: JGRAPH_2:sch1

    TopSubset { P[ set] } :

{ p where p be Point of ( TOP-REAL 2) : P[p] } is Subset of ( TOP-REAL 2);

      { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2)

      proof

        let x be object;

        assume x in { p where p be Point of ( TOP-REAL 2) : P[p] };

        then ex p be Point of ( TOP-REAL 2) st p = x & P[p];

        hence thesis;

      end;

      hence thesis;

    end;

    scheme :: JGRAPH_2:sch2

    TopCompl { P[ set], K() -> Subset of ( TOP-REAL 2) } :

(K() ` ) = { p where p be Point of ( TOP-REAL 2) : not P[p] }

      provided

       A1: K() = { p where p be Point of ( TOP-REAL 2) : P[p] };

      thus (K() ` ) c= { p where p be Point of ( TOP-REAL 2) : not P[p] }

      proof

        let x be object;

        assume

         A2: x in (K() ` );

        then

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

        x in (the carrier of ( TOP-REAL 2) \ K()) by A2, SUBSET_1:def 4;

        then not x in K() by XBOOLE_0:def 5;

        then not P[qx] by A1;

        hence thesis;

      end;

      let x be object;

      assume x in { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] };

      then

       A3: ex p7 be Point of ( TOP-REAL 2) st p7 = x & not P[p7];

      then not ex q7 be Point of ( TOP-REAL 2) st x = q7 & P[q7];

      then not x in K() by A1;

      then x in (the carrier of ( TOP-REAL 2) \ K()) by A3, XBOOLE_0:def 5;

      hence thesis by SUBSET_1:def 4;

    end;

     Lm2:

    now

      let p01,p02,px1,px2 be Real;

      set r0 = ((p01 - p02) / 4);

      assume ((p01 - px1) - (p02 - px2)) <= (r0 - ( - r0));

      then ((p01 - p02) - (px1 - px2)) <= (r0 + r0);

      then (p01 - p02) <= ((px1 - px2) + (r0 + r0)) by XREAL_1: 20;

      then ((p01 - p02) - ((p01 - p02) / 2)) <= (px1 - px2) by XREAL_1: 20;

      hence ((p01 - p02) / 2) <= (px1 - px2);

    end;

    scheme :: JGRAPH_2:sch3

    ClosedSubset { F,G( Point of ( TOP-REAL 2)) -> Real } :

{ p where p be Point of ( TOP-REAL 2) : F(p) <= G(p) } is closed Subset of ( TOP-REAL 2)

      provided

       A1: for p,q be Point of ( TOP-REAL 2) holds F(-) = (F(p) - F(q)) & G(-) = (G(p) - G(q))

       and

       A2: for p,q be Point of ( TOP-REAL 2) holds ( |.(p - q).| ^2 ) = ((F(-) ^2 ) + (G(-) ^2 ));

      defpred P[ Point of ( TOP-REAL 2)] means F($1) <= G($1);

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } as Subset of ( TOP-REAL 2) from TopSubset;

      

       A3: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

      then

      reconsider K21 = (K2 ` ) as Subset of ( TopSpaceMetr ( Euclid 2));

      

       A4: K2 = { p7 where p7 be Point of ( TOP-REAL 2) : P[p7] };

      

       A5: (K2 ` ) = { p7 where p7 be Point of ( TOP-REAL 2) : not P[p7] } from TopCompl( A4);

      for p be Point of ( Euclid 2) st p in K21 holds ex r be Real st r > 0 & ( Ball (p,r)) c= K21

      proof

        let p be Point of ( Euclid 2);

        assume

         A6: p in K21;

        then

        reconsider p0 = p as Point of ( TOP-REAL 2);

        set r0 = ((F(p0) - G(p0)) / 4);

        ex p7 be Point of ( TOP-REAL 2) st p0 = p7 & F(p7) > G(p7) by A5, A6;

        then

         A7: (F(p0) - G(p0)) > 0 by XREAL_1: 50;

        then

         A8: ((F(p0) - G(p0)) / 2) > 0 by XREAL_1: 139;

        ( Ball (p,r0)) c= (K2 ` )

        proof

          let x be object;

          

           A9: ( Ball (p,r0)) = { q where q be Element of ( Euclid 2) : ( dist (p,q)) < r0 } by METRIC_1: 17;

          assume

           A10: x in ( Ball (p,r0));

          then

          reconsider px = x as Point of ( TOP-REAL 2) by TOPREAL3: 8;

          consider q be Element of ( Euclid 2) such that

           A11: q = x and

           A12: ( dist (p,q)) < r0 by A10, A9;

          ( dist (p,q)) = |.(p0 - px).| by A11, JGRAPH_1: 28;

          then

           A13: ( |.(p0 - px).| ^2 ) <= (r0 ^2 ) by A12, SQUARE_1: 15;

          

           A14: F(-) = (F(p0) - F(px)) by A1;

          

           A15: ( |.(p0 - px).| ^2 ) = ((F(-) ^2 ) + (G(-) ^2 )) by A2;

          ( 0 + (F(-) ^2 )) <= ((G(-) ^2 ) + (F(-) ^2 )) by XREAL_1: 7;

          then (F(-) ^2 ) <= (r0 ^2 ) by A15, A13, XXREAL_0: 2;

          then

           A16: (F(p0) - F(px)) <= r0 by A7, A14, SQUARE_1: 47;

          

           A17: G(-) = (G(p0) - G(px)) by A1;

          ((G(-) ^2 ) + 0 ) <= ((G(-) ^2 ) + (F(-) ^2 )) by XREAL_1: 7;

          then (G(-) ^2 ) <= (r0 ^2 ) by A15, A13, XXREAL_0: 2;

          then ( - r0) <= (G(p0) - G(px)) by A7, A17, SQUARE_1: 47;

          then ((F(p0) - F(px)) - (G(p0) - G(px))) <= (r0 - ( - r0)) by A16, XREAL_1: 13;

          then (F(px) - G(px)) > 0 by A8, Lm2;

          then F(px) > G(px) by XREAL_1: 47;

          hence thesis by A5;

        end;

        hence thesis by A7, XREAL_1: 139;

      end;

      then K21 is open by TOPMETR: 15;

      then (K2 ` ) is open by A3, PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    deffunc F( Point of ( TOP-REAL 2)) = ($1 `1 );

    deffunc G( Point of ( TOP-REAL 2)) = ($1 `2 );

    

     Lm3: for p,q be Point of ( TOP-REAL 2) holds F(-) = ( F(p) - F(q)) & G(-) = ( G(p) - G(q)) by TOPREAL3: 3;

    

     Lm4: for p,q be Point of ( TOP-REAL 2) holds ( |.(p - q).| ^2 ) = (( F(-) ^2 ) + ( G(-) ^2 )) by JGRAPH_1: 29;

    

     Lm5: { p7 where p7 be Point of ( TOP-REAL 2) : F(p7) <= G(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm3, Lm4);

    

     Lm6: for p,q be Point of ( TOP-REAL 2) holds G(-) = ( G(p) - G(q)) & F(-) = ( F(p) - F(q)) by TOPREAL3: 3;

    

     Lm7: for p,q be Point of ( TOP-REAL 2) holds ( |.(p - q).| ^2 ) = (( G(-) ^2 ) + ( F(-) ^2 )) by JGRAPH_1: 29;

    

     Lm8: { p7 where p7 be Point of ( TOP-REAL 2) : G(p7) <= F(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm6, Lm7);

    deffunc H( Point of ( TOP-REAL 2)) = ( - ($1 `1 ));

    deffunc I( Point of ( TOP-REAL 2)) = ( - ($1 `2 ));

     Lm9:

    now

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

      

      thus H(-) = ( - ((p `1 ) - (q `1 ))) by TOPREAL3: 3

      .= ( H(p) - H(q));

      thus G(-) = ( G(p) - G(q)) by TOPREAL3: 3;

    end;

     Lm10:

    now

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

      ( H(-) ^2 ) = ( F(-) ^2 );

      hence ( |.(p - q).| ^2 ) = (( H(-) ^2 ) + ( G(-) ^2 )) by JGRAPH_1: 29;

    end;

    

     Lm11: { p7 where p7 be Point of ( TOP-REAL 2) : H(p7) <= G(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm9, Lm10);

     Lm12:

    now

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

      thus G(-) = ( G(p) - G(q)) by TOPREAL3: 3;

      

      thus H(-) = ( - ((p `1 ) - (q `1 ))) by TOPREAL3: 3

      .= ( H(p) - H(q));

    end;

     Lm13:

    now

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

      (( - ((p - q) `1 )) ^2 ) = (((p - q) `1 ) ^2 );

      hence ( |.(p - q).| ^2 ) = (( G(-) ^2 ) + ( H(-) ^2 )) by JGRAPH_1: 29;

    end;

    

     Lm14: { p7 where p7 be Point of ( TOP-REAL 2) : G(p7) <= H(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm12, Lm13);

     Lm15:

    now

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

      

      thus I(-) = ( - ((p `2 ) - (q `2 ))) by TOPREAL3: 3

      .= ( I(p) - I(q));

      thus F(-) = ( F(p) - F(q)) by TOPREAL3: 3;

    end;

     Lm16:

    now

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

      (( - ((p - q) `2 )) ^2 ) = (((p - q) `2 ) ^2 );

      hence ( |.(p - q).| ^2 ) = (( I(-) ^2 ) + ( F(-) ^2 )) by JGRAPH_1: 29;

    end;

    

     Lm17: { p7 where p7 be Point of ( TOP-REAL 2) : I(p7) <= F(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm15, Lm16);

     Lm18:

    now

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

      thus F(-) = ( F(p) - F(q)) by TOPREAL3: 3;

      

      thus I(-) = ( - ((p `2 ) - (q `2 ))) by TOPREAL3: 3

      .= ( I(p) - I(q));

    end;

     Lm19:

    now

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

      ( I(-) ^2 ) = ( G(-) ^2 );

      hence ( |.(p - q).| ^2 ) = (( F(-) ^2 ) + ( I(-) ^2 )) by JGRAPH_1: 29;

    end;

    

     Lm20: { p7 where p7 be Point of ( TOP-REAL 2) : F(p7) <= I(p7) } is closed Subset of ( TOP-REAL 2) from ClosedSubset( Lm18, Lm19);

    theorem :: JGRAPH_2:38

    

     Th38: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous & K0 is closed

    proof

      reconsider K5 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= ( - (p7 `1 )) } as closed Subset of ( TOP-REAL 2) by Lm14;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by Lm5;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `1 )) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by Lm11;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by Lm8;

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

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

      the carrier of (( TOP-REAL 2) | B0) = ( [#] (( TOP-REAL 2) | B0))

      .= B0 by PRE_TOPC:def 5;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then

       A3: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

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

      

       A4: (K1 /\ B0) c= K0

      proof

        let x be object;

        assume

         A5: x in (K1 /\ B0);

        then x in B0 by XBOOLE_0:def 4;

        then not x in {( 0. ( TOP-REAL 2))} by A1, XBOOLE_0:def 5;

        then

         A6: not x = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        x in K1 by A5, XBOOLE_0:def 4;

        then ex p7 be Point of ( TOP-REAL 2) st p7 = x & ((p7 `2 ) <= (p7 `1 ) & ( - (p7 `1 )) <= (p7 `2 ) or (p7 `2 ) >= (p7 `1 ) & (p7 `2 ) <= ( - (p7 `1 )));

        hence thesis by A1, A6;

      end;

      

       A7: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A8: x in ((K2 /\ K3) \/ (K4 /\ K5));

        now

          per cases by A8, XBOOLE_0:def 3;

            case

             A9: x in (K2 /\ K3);

            then x in K3 by XBOOLE_0:def 4;

            then

             A10: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `1 )) <= (p8 `2 );

            x in K2 by A9, XBOOLE_0:def 4;

            then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) <= (p7 `1 );

            hence thesis by A10;

          end;

            case

             A11: x in (K4 /\ K5);

            then x in K5 by XBOOLE_0:def 4;

            then

             A12: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `2 ) <= ( - (p8 `1 ));

            x in K4 by A11, XBOOLE_0:def 4;

            then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) >= (p7 `1 );

            hence thesis by A12;

          end;

        end;

        hence thesis;

      end;

      K1 c= ((K2 /\ K3) \/ (K4 /\ K5))

      proof

        let x be object;

        assume x in K1;

        then ex p be Point of ( TOP-REAL 2) st p = x & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

        then x in K2 & x in K3 or x in K4 & x in K5;

        then x in (K2 /\ K3) or x in (K4 /\ K5) by XBOOLE_0:def 4;

        hence thesis by XBOOLE_0:def 3;

      end;

      then K1 = ((K2 /\ K3) \/ (K4 /\ K5)) by A7;

      then

       A13: K1 is closed;

      K0 c= (K1 /\ B0)

      proof

        let x be object;

        assume x in K0;

        then

         A14: ex p be Point of ( TOP-REAL 2) st x = p & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        then

         A15: x in B0 by A1, A14, XBOOLE_0:def 5;

        x in K1 by A14;

        hence thesis by A15, XBOOLE_0:def 4;

      end;

      

      then K0 = (K1 /\ B0) by A4

      .= (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by PRE_TOPC:def 5;

      hence thesis by A1, A3, A13, Th36, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_2:39

    

     Th39: for B0 be Subset of ( TOP-REAL 2), K0 be Subset of (( TOP-REAL 2) | B0), f be Function of ((( TOP-REAL 2) | B0) | K0), (( TOP-REAL 2) | B0) st f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } holds f is continuous & K0 is closed

    proof

      reconsider K5 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= ( - (p7 `2 )) } as closed Subset of ( TOP-REAL 2) by Lm20;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by Lm8;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `2 )) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by Lm17;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by Lm5;

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

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

      the carrier of (( TOP-REAL 2) | B0) = ( [#] (( TOP-REAL 2) | B0))

      .= B0 by PRE_TOPC:def 5;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      assume

       A1: f = ( Out_In_Sq | K0) & B0 = ( NonZero ( TOP-REAL 2)) & K0 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) };

      K0 c= B0

      proof

        let x be object;

        assume x in K0;

        then

         A2: ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A1, A2, XBOOLE_0:def 5;

      end;

      then

       A3: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by PRE_TOPC: 7;

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

      

       A4: (K1 /\ B0) c= K0

      proof

        let x be object;

        assume

         A5: x in (K1 /\ B0);

        then x in B0 by XBOOLE_0:def 4;

        then not x in {( 0. ( TOP-REAL 2))} by A1, XBOOLE_0:def 5;

        then

         A6: not x = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        x in K1 by A5, XBOOLE_0:def 4;

        then ex p7 be Point of ( TOP-REAL 2) st p7 = x & ((p7 `1 ) <= (p7 `2 ) & ( - (p7 `2 )) <= (p7 `1 ) or (p7 `1 ) >= (p7 `2 ) & (p7 `1 ) <= ( - (p7 `2 )));

        hence thesis by A1, A6;

      end;

      

       A7: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A8: x in ((K2 /\ K3) \/ (K4 /\ K5));

        now

          per cases by A8, XBOOLE_0:def 3;

            case

             A9: x in (K2 /\ K3);

            then x in K3 by XBOOLE_0:def 4;

            then

             A10: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `2 )) <= (p8 `1 );

            x in K2 by A9, XBOOLE_0:def 4;

            then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `1 ) <= (p7 `2 );

            hence thesis by A10;

          end;

            case

             A11: x in (K4 /\ K5);

            then x in K5 by XBOOLE_0:def 4;

            then

             A12: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `1 ) <= ( - (p8 `2 ));

            x in K4 by A11, XBOOLE_0:def 4;

            then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `1 ) >= (p7 `2 );

            hence thesis by A12;

          end;

        end;

        hence thesis;

      end;

      K1 c= ((K2 /\ K3) \/ (K4 /\ K5))

      proof

        let x be object;

        assume x in K1;

        then ex p be Point of ( TOP-REAL 2) st p = x & ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )));

        then x in K2 & x in K3 or x in K4 & x in K5;

        then x in (K2 /\ K3) or x in (K4 /\ K5) by XBOOLE_0:def 4;

        hence thesis by XBOOLE_0:def 3;

      end;

      then K1 = ((K2 /\ K3) \/ (K4 /\ K5)) by A7;

      then

       A13: K1 is closed;

      K0 c= (K1 /\ B0)

      proof

        let x be object;

        assume x in K0;

        then

         A14: ex p be Point of ( TOP-REAL 2) st x = p & ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) by A1;

        then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        then

         A15: x in B0 by A1, A14, XBOOLE_0:def 5;

        x in K1 by A14;

        hence thesis by A15, XBOOLE_0:def 4;

      end;

      

      then K0 = (K1 /\ B0) by A4

      .= (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by PRE_TOPC:def 5;

      hence thesis by A1, A3, A13, Th37, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_2:40

    

     Th40: for D be non empty Subset of ( TOP-REAL 2) st (D ` ) = {( 0. ( TOP-REAL 2))} holds ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = Out_In_Sq & h is continuous

    proof

      set Y1 = |[( - 1), 1]|;

      reconsider B0 = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

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

      assume

       A1: (D ` ) = {( 0. ( TOP-REAL 2))};

      

      then

       A2: D = (B0 ` )

      .= ( NonZero ( TOP-REAL 2)) by SUBSET_1:def 4;

      

       A3: { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D)

      proof

        let x be object;

        assume x in { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) };

        then

         A4: ex p st x = p & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2));

        now

          assume not x in D;

          then x in (the carrier of ( TOP-REAL 2) \ D) by A4, XBOOLE_0:def 5;

          then x in (D ` ) by SUBSET_1:def 4;

          hence contradiction by A1, A4, TARSKI:def 1;

        end;

        then x in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

        hence thesis;

      end;

      

       A5: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

      

       A6: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

      (( 1.REAL 2) `2 ) <= (( 1.REAL 2) `1 ) & ( - (( 1.REAL 2) `1 )) <= (( 1.REAL 2) `2 ) or (( 1.REAL 2) `2 ) >= (( 1.REAL 2) `1 ) & (( 1.REAL 2) `2 ) <= ( - (( 1.REAL 2) `1 )) by Th5;

      then ( 1.REAL 2) in { p where p be Point of ( TOP-REAL 2) : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } by A6;

      then

      reconsider K0 = { p : ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A3;

      

       A7: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5

      .= the carrier of ((( TOP-REAL 2) | D) | K0);

      

       A8: { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } c= the carrier of (( TOP-REAL 2) | D)

      proof

        let x be object;

        assume x in { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) };

        then

         A9: ex p st x = p & ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2));

        now

          assume not x in D;

          then x in (the carrier of ( TOP-REAL 2) \ D) by A9, XBOOLE_0:def 5;

          then x in (D ` ) by SUBSET_1:def 4;

          hence contradiction by A1, A9, TARSKI:def 1;

        end;

        then x in ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

        hence thesis;

      end;

      (Y1 `1 ) = ( - 1) & (Y1 `2 ) = 1 by EUCLID: 52;

      then Y1 in { p where p be Point of ( TOP-REAL 2) : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } by Th3;

      then

      reconsider K1 = { p : ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) & p <> ( 0. ( TOP-REAL 2)) } as non empty Subset of (( TOP-REAL 2) | D) by A8;

      

       A10: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5

      .= the carrier of ((( TOP-REAL 2) | D) | K1);

      

       A11: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= D by PRE_TOPC:def 5;

      

       A12: ( rng ( Out_In_Sq | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        reconsider K10 = K1 as Subset of ( TOP-REAL 2) by A11, XBOOLE_1: 1;

        let y be object;

        

         A13: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K10) holds (q `2 ) <> 0

        proof

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

          

           A14: the carrier of (( TOP-REAL 2) | K10) = ( [#] (( TOP-REAL 2) | K10))

          .= K1 by PRE_TOPC:def 5;

          assume q in the carrier of (( TOP-REAL 2) | K10);

          then

           A15: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & (p3 `1 ) <= ( - (p3 `2 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A14;

          now

            assume

             A16: (q `2 ) = 0 ;

            then (q `1 ) = 0 by A15;

            hence contradiction by A15, A16, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

        assume y in ( rng ( Out_In_Sq | K1));

        then

        consider x be object such that

         A17: x in ( dom ( Out_In_Sq | K1)) and

         A18: y = (( Out_In_Sq | K1) . x) by FUNCT_1:def 3;

        

         A19: x in (( dom Out_In_Sq ) /\ K1) by A17, RELAT_1: 61;

        then

         A20: x in K1 by XBOOLE_0:def 4;

        K1 c= the carrier of ( TOP-REAL 2) by A11, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A20;

        

         A21: ( Out_In_Sq . p) = y by A18, A20, FUNCT_1: 49;

        set p9 = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]|;

        K10 = ( [#] (( TOP-REAL 2) | K10)) by PRE_TOPC:def 5

        .= the carrier of (( TOP-REAL 2) | K10);

        then

         A22: p in the carrier of (( TOP-REAL 2) | K10) by A19, XBOOLE_0:def 4;

         A23:

        now

          assume p9 = ( 0. ( TOP-REAL 2));

          then (p9 `2 ) = 0 by EUCLID: 52, EUCLID: 54;

          then ( 0 * (p `2 )) = ((1 / (p `2 )) * (p `2 )) by EUCLID: 52;

          hence contradiction by A22, A13, XCMPLX_1: 87;

        end;

        

         A24: ex px be Point of ( TOP-REAL 2) st x = px & ((px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 ))) & px <> ( 0. ( TOP-REAL 2)) by A20;

        then

         A25: ( Out_In_Sq . p) = |[(((p `1 ) / (p `2 )) / (p `2 )), (1 / (p `2 ))]| by Th14;

        now

          per cases ;

            case

             A26: (p `2 ) >= 0 ;

            then ((p `1 ) / (p `2 )) <= ((p `2 ) / (p `2 )) & (( - (1 * (p `2 ))) / (p `2 )) <= ((p `1 ) / (p `2 )) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (1 * (p `2 ))) by A24, XREAL_1: 72;

            then

             A27: ((p `1 ) / (p `2 )) <= 1 & ((( - 1) * (p `2 )) / (p `2 )) <= ((p `1 ) / (p `2 )) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (1 * (p `2 ))) by A22, A13, XCMPLX_1: 60;

            then

             A28: ((p `1 ) / (p `2 )) <= 1 & ( - 1) <= ((p `1 ) / (p `2 )) or ((p `1 ) / (p `2 )) >= 1 & ((p `1 ) / (p `2 )) <= ((( - 1) * (p `2 )) / (p `2 )) by A22, A13, A26, XCMPLX_1: 89, XREAL_1: 72;

            ((p `1 ) / (p `2 )) <= 1 & ( - 1) <= ((p `1 ) / (p `2 )) or ((p `1 ) / (p `2 )) >= ((p `2 ) / (p `2 )) & (p `1 ) <= ( - (1 * (p `2 ))) by A22, A13, A26, A27, XCMPLX_1: 89;

            then (( - 1) / (p `2 )) <= (((p `1 ) / (p `2 )) / (p `2 )) by A26, XREAL_1: 72;

            then

             A29: (((p `1 ) / (p `2 )) / (p `2 )) <= (1 / (p `2 )) & ( - (1 / (p `2 ))) <= (((p `1 ) / (p `2 )) / (p `2 )) or (((p `1 ) / (p `2 )) / (p `2 )) >= (1 / (p `2 )) & (((p `1 ) / (p `2 )) / (p `2 )) <= ( - (1 / (p `2 ))) by A26, A28, XREAL_1: 72;

            (p9 `2 ) = (1 / (p `2 )) & (p9 `1 ) = (((p `1 ) / (p `2 )) / (p `2 )) by EUCLID: 52;

            hence y in K1 by A21, A23, A25, A29;

          end;

            case

             A30: (p `2 ) < 0 ;

            then (p `1 ) <= (p `2 ) & ( - (1 * (p `2 ))) <= (p `1 ) or ((p `1 ) / (p `2 )) <= ((p `2 ) / (p `2 )) & ((p `1 ) / (p `2 )) >= (( - (1 * (p `2 ))) / (p `2 )) by A24, XREAL_1: 73;

            then

             A31: (p `1 ) <= (p `2 ) & ( - (1 * (p `2 ))) <= (p `1 ) or ((p `1 ) / (p `2 )) <= 1 & ((p `1 ) / (p `2 )) >= ((( - 1) * (p `2 )) / (p `2 )) by A30, XCMPLX_1: 60;

            then ((p `1 ) / (p `2 )) >= 1 & ((( - 1) * (p `2 )) / (p `2 )) >= ((p `1 ) / (p `2 )) or ((p `1 ) / (p `2 )) <= 1 & ((p `1 ) / (p `2 )) >= ( - 1) by A30, XCMPLX_1: 89;

            then (( - 1) / (p `2 )) >= (((p `1 ) / (p `2 )) / (p `2 )) by A30, XREAL_1: 73;

            then

             A32: (((p `1 ) / (p `2 )) / (p `2 )) <= (1 / (p `2 )) & ( - (1 / (p `2 ))) <= (((p `1 ) / (p `2 )) / (p `2 )) or (((p `1 ) / (p `2 )) / (p `2 )) >= (1 / (p `2 )) & (((p `1 ) / (p `2 )) / (p `2 )) <= ( - (1 / (p `2 ))) by A30, A31, XREAL_1: 73;

            (p9 `2 ) = (1 / (p `2 )) & (p9 `1 ) = (((p `1 ) / (p `2 )) / (p `2 )) by EUCLID: 52;

            hence y in K1 by A21, A23, A25, A32;

          end;

        end;

        then y in ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

        hence thesis;

      end;

      

       A33: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A34: x in D;

        then

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

         not x in {( 0. ( TOP-REAL 2))} by A2, A34, XBOOLE_0:def 5;

        then ((px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 ))) & px <> ( 0. ( TOP-REAL 2)) or ((px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 ))) & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1, XREAL_1: 26;

        then x in K0 or x in K1;

        hence thesis by XBOOLE_0:def 3;

      end;

      

       A35: ( NonZero ( TOP-REAL 2)) <> {} by Th9;

      

       A36: K1 c= ( NonZero ( TOP-REAL 2))

      proof

        let z be object;

        assume z in K1;

        then

         A37: ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2));

        then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A37, XBOOLE_0:def 5;

      end;

      

       A38: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= ( NonZero ( TOP-REAL 2)) by A2, PRE_TOPC:def 5;

      

       A39: ( rng ( Out_In_Sq | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        reconsider K00 = K0 as Subset of ( TOP-REAL 2) by A11, XBOOLE_1: 1;

        let y be object;

        

         A40: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K00) holds (q `1 ) <> 0

        proof

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

          

           A41: the carrier of (( TOP-REAL 2) | K00) = ( [#] (( TOP-REAL 2) | K00))

          .= K0 by PRE_TOPC:def 5;

          assume q in the carrier of (( TOP-REAL 2) | K00);

          then

           A42: ex p3 be Point of ( TOP-REAL 2) st q = p3 & ((p3 `2 ) <= (p3 `1 ) & ( - (p3 `1 )) <= (p3 `2 ) or (p3 `2 ) >= (p3 `1 ) & (p3 `2 ) <= ( - (p3 `1 ))) & p3 <> ( 0. ( TOP-REAL 2)) by A41;

          now

            assume

             A43: (q `1 ) = 0 ;

            then (q `2 ) = 0 by A42;

            hence contradiction by A42, A43, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

        assume y in ( rng ( Out_In_Sq | K0));

        then

        consider x be object such that

         A44: x in ( dom ( Out_In_Sq | K0)) and

         A45: y = (( Out_In_Sq | K0) . x) by FUNCT_1:def 3;

        

         A46: x in (( dom Out_In_Sq ) /\ K0) by A44, RELAT_1: 61;

        then

         A47: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A11, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A47;

        

         A48: ( Out_In_Sq . p) = y by A45, A47, FUNCT_1: 49;

        set p9 = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]|;

        K00 = ( [#] (( TOP-REAL 2) | K00)) by PRE_TOPC:def 5

        .= the carrier of (( TOP-REAL 2) | K00);

        then

         A49: p in the carrier of (( TOP-REAL 2) | K00) by A46, XBOOLE_0:def 4;

        

         A50: (p9 `1 ) = (1 / (p `1 )) by EUCLID: 52;

         A51:

        now

          assume p9 = ( 0. ( TOP-REAL 2));

          then ( 0 * (p `1 )) = ((1 / (p `1 )) * (p `1 )) by A50, EUCLID: 52, EUCLID: 54;

          hence contradiction by A49, A40, XCMPLX_1: 87;

        end;

        

         A52: ex px be Point of ( TOP-REAL 2) st x = px & ((px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 ))) & px <> ( 0. ( TOP-REAL 2)) by A47;

        then

         A53: ( Out_In_Sq . p) = |[(1 / (p `1 )), (((p `2 ) / (p `1 )) / (p `1 ))]| by Def1;

        

         A54: (p `1 ) <> 0 by A49, A40;

        now

          per cases ;

            case

             A55: (p `1 ) >= 0 ;

            ((p `2 ) / (p `1 )) <= ((p `1 ) / (p `1 )) & (( - (1 * (p `1 ))) / (p `1 )) <= ((p `2 ) / (p `1 )) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (1 * (p `1 ))) by A52, A55, XREAL_1: 72;

            then

             A56: ((p `2 ) / (p `1 )) <= 1 & ((( - 1) * (p `1 )) / (p `1 )) <= ((p `2 ) / (p `1 )) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (1 * (p `1 ))) by A49, A40, XCMPLX_1: 60;

            then ((p `2 ) / (p `1 )) <= 1 & ( - 1) <= ((p `2 ) / (p `1 )) or ((p `2 ) / (p `1 )) >= ((p `1 ) / (p `1 )) & (p `2 ) <= ( - (1 * (p `1 ))) by A49, A40, A55, XCMPLX_1: 89;

            then (( - 1) / (p `1 )) <= (((p `2 ) / (p `1 )) / (p `1 )) by A55, XREAL_1: 72;

            then

             A57: (((p `2 ) / (p `1 )) / (p `1 )) <= (1 / (p `1 )) & ( - (1 / (p `1 ))) <= (((p `2 ) / (p `1 )) / (p `1 )) or (((p `2 ) / (p `1 )) / (p `1 )) >= (1 / (p `1 )) & (((p `2 ) / (p `1 )) / (p `1 )) <= ( - (1 / (p `1 ))) by A54, A55, A56, XREAL_1: 72;

            (p9 `1 ) = (1 / (p `1 )) & (p9 `2 ) = (((p `2 ) / (p `1 )) / (p `1 )) by EUCLID: 52;

            hence y in K0 by A48, A51, A53, A57;

          end;

            case

             A58: (p `1 ) < 0 ;

            

             A59: ( - (1 / (p `1 ))) = (( - 1) / (p `1 ));

            (p `2 ) <= (p `1 ) & ( - (1 * (p `1 ))) <= (p `2 ) or ((p `2 ) / (p `1 )) <= ((p `1 ) / (p `1 )) & ((p `2 ) / (p `1 )) >= (( - (1 * (p `1 ))) / (p `1 )) by A52, A58, XREAL_1: 73;

            then

             A60: (p `2 ) <= (p `1 ) & ( - (1 * (p `1 ))) <= (p `2 ) or ((p `2 ) / (p `1 )) <= 1 & ((p `2 ) / (p `1 )) >= ((( - 1) * (p `1 )) / (p `1 )) by A58, XCMPLX_1: 60;

            then ((p `2 ) / (p `1 )) >= ((p `1 ) / (p `1 )) & ( - (1 * (p `1 ))) <= (p `2 ) or ((p `2 ) / (p `1 )) <= 1 & ((p `2 ) / (p `1 )) >= ( - 1) by A58, XCMPLX_1: 89;

            then (( - 1) / (p `1 )) >= (((p `2 ) / (p `1 )) / (p `1 )) by A58, XREAL_1: 73;

            then

             A61: (((p `2 ) / (p `1 )) / (p `1 )) <= (1 / (p `1 )) & (( - 1) / (p `1 )) <= (((p `2 ) / (p `1 )) / (p `1 )) or (((p `2 ) / (p `1 )) / (p `1 )) >= (1 / (p `1 )) & (((p `2 ) / (p `1 )) / (p `1 )) <= ( - (1 / (p `1 ))) by A58, A60, XREAL_1: 73;

            (p9 `1 ) = (1 / (p `1 )) & (p9 `2 ) = (((p `2 ) / (p `1 )) / (p `1 )) by EUCLID: 52;

            hence y in K0 by A48, A51, A53, A61, A59;

          end;

        end;

        then y in ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

        hence thesis;

      end;

      

       A62: K0 c= ( NonZero ( TOP-REAL 2))

      proof

        let z be object;

        assume z in K0;

        then

         A63: ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2));

        then not z in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        hence thesis by A63, XBOOLE_0:def 5;

      end;

      ( dom ( Out_In_Sq | K0)) = (( dom Out_In_Sq ) /\ K0) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K0) by A5, FUNCT_2:def 1

      .= K0 by A62, XBOOLE_1: 28;

      then

      reconsider f = ( Out_In_Sq | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A7, A39, FUNCT_2: 2, XBOOLE_1: 1;

      

       A64: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      ( dom ( Out_In_Sq | K1)) = (( dom Out_In_Sq ) /\ K1) by RELAT_1: 61

      .= (( NonZero ( TOP-REAL 2)) /\ K1) by A35, FUNCT_2:def 1

      .= K1 by A36, XBOOLE_1: 28;

      then

      reconsider g = ( Out_In_Sq | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A10, A12, FUNCT_2: 2, XBOOLE_1: 1;

      

       A65: ( dom g) = K1 by A10, FUNCT_2:def 1;

      g = ( Out_In_Sq | K1);

      then

       A66: K1 is closed by A2, Th39;

      

       A67: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A68: for x be object st x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1))) holds (f . x) = (g . x)

      proof

        let x be object;

        assume

         A69: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A67, XBOOLE_0:def 4;

        then (f . x) = ( Out_In_Sq . x) by FUNCT_1: 49;

        hence thesis by A64, A69, FUNCT_1: 49;

      end;

      f = ( Out_In_Sq | K0);

      then

       A70: K0 is closed by A2, Th38;

      

       A71: ( dom f) = K0 by A7, FUNCT_2:def 1;

      D = ( [#] (( TOP-REAL 2) | D)) by PRE_TOPC:def 5;

      then

       A72: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A67, A64, A33;

      

       A73: f is continuous & g is continuous by A2, Th38, Th39;

      then

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

       A74: h = (f +* g) and h is continuous by A67, A64, A72, A70, A66, A68, Th1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A75: f tolerates g by A68, A71, A65, PARTFUN1:def 4;

      

       A76: for x be object st x in ( dom h) holds (h . x) = ( Out_In_Sq . x)

      proof

        let x be object;

        assume

         A77: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A38, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A38, A77, XBOOLE_0:def 5;

        then

         A78: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        now

          per cases ;

            case

             A79: x in K0;

            (h . p) = ((g +* f) . p) by A74, A75, FUNCT_4: 34

            .= (f . p) by A71, A79, FUNCT_4: 13;

            hence thesis by A79, FUNCT_1: 49;

          end;

            case not x in K0;

            then not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) by A78;

            then (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )) by XREAL_1: 26;

            then

             A80: x in K1 by A78;

            then ( Out_In_Sq . p) = (g . p) by FUNCT_1: 49;

            hence thesis by A74, A65, A80, FUNCT_4: 13;

          end;

        end;

        hence thesis;

      end;

      ( dom h) = the carrier of (( TOP-REAL 2) | D) & ( dom Out_In_Sq ) = the carrier of (( TOP-REAL 2) | D) by A38, FUNCT_2:def 1;

      then (f +* g) = Out_In_Sq by A74, A76, FUNCT_1: 2;

      hence thesis by A67, A64, A72, A70, A73, A66, A68, Th1;

    end;

    theorem :: JGRAPH_2:41

    

     Th41: for B,K0,Kb be Subset of ( TOP-REAL 2) st B = {( 0. ( TOP-REAL 2))} & K0 = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & Kb = { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 } holds ex f be Function of (( TOP-REAL 2) | (B ` )), (( TOP-REAL 2) | (B ` )) st f is continuous & f is one-to-one & (for t be Point of ( TOP-REAL 2) st t in K0 & t <> ( 0. ( TOP-REAL 2)) holds not (f . t) in (K0 \/ Kb)) & (for r be Point of ( TOP-REAL 2) st not r in (K0 \/ Kb) holds (f . r) in K0) & for s be Point of ( TOP-REAL 2) st s in Kb holds (f . s) = s

    proof

      set K1a = { p8 where p8 be Point of ( TOP-REAL 2) : ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) };

      set K0a = { p8 where p8 be Point of ( TOP-REAL 2) : ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) };

      let B,K0,Kb be Subset of ( TOP-REAL 2);

      assume

       A1: B = {( 0. ( TOP-REAL 2))} & K0 = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & Kb = { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 };

      then

      reconsider D = (B ` ) as non empty Subset of ( TOP-REAL 2) by Th9;

      

       A2: (D ` ) = {( 0. ( TOP-REAL 2))} by A1;

      

       A3: (B ` ) = ( NonZero ( TOP-REAL 2)) by A1, SUBSET_1:def 4;

      

       A4: for t be Point of ( TOP-REAL 2) st t in K0 & t <> ( 0. ( TOP-REAL 2)) holds not ( Out_In_Sq . t) in (K0 \/ Kb)

      proof

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

        assume that

         A5: t in K0 and

         A6: t <> ( 0. ( TOP-REAL 2));

        

         A7: ex p3 be Point of ( TOP-REAL 2) st p3 = t & ( - 1) < (p3 `1 ) & (p3 `1 ) < 1 & ( - 1) < (p3 `2 ) & (p3 `2 ) < 1 by A1, A5;

        now

          assume

           A8: ( Out_In_Sq . t) in (K0 \/ Kb);

          now

            per cases by A8, XBOOLE_0:def 3;

              case ( Out_In_Sq . t) in K0;

              then

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

               A9: p4 = ( Out_In_Sq . t) and

               A10: ( - 1) < (p4 `1 ) and

               A11: (p4 `1 ) < 1 and

               A12: ( - 1) < (p4 `2 ) and

               A13: (p4 `2 ) < 1 by A1;

              now

                per cases ;

                  case

                   A14: (t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 ));

                  then ( Out_In_Sq . t) = |[(1 / (t `1 )), (((t `2 ) / (t `1 )) / (t `1 ))]| by A6, Def1;

                  then

                   A15: (p4 `1 ) = (1 / (t `1 )) by A9, EUCLID: 52;

                  now

                    per cases ;

                      case

                       A16: (t `1 ) >= 0 ;

                      now

                        per cases by A16;

                          case

                           A17: (t `1 ) > 0 ;

                          then ((1 / (t `1 )) * (t `1 )) < (1 * (t `1 )) by A11, A15, XREAL_1: 68;

                          hence contradiction by A7, A17, XCMPLX_1: 87;

                        end;

                          case

                           A18: (t `1 ) = 0 ;

                          then (t `2 ) = 0 by A14;

                          hence contradiction by A6, A18, EUCLID: 53, EUCLID: 54;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A19: (t `1 ) < 0 ;

                      then (( - 1) * (t `1 )) > ((1 / (t `1 )) * (t `1 )) by A10, A15, XREAL_1: 69;

                      then (( - 1) * (t `1 )) > 1 by A19, XCMPLX_1: 87;

                      then ( - ( - (t `1 ))) <= ( - 1) by XREAL_1: 24;

                      hence contradiction by A7;

                    end;

                  end;

                  hence contradiction;

                end;

                  case

                   A20: not ((t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 )));

                  then ( Out_In_Sq . t) = |[(((t `1 ) / (t `2 )) / (t `2 )), (1 / (t `2 ))]| by A6, Def1;

                  then

                   A21: (p4 `2 ) = (1 / (t `2 )) by A9, EUCLID: 52;

                  now

                    per cases ;

                      case

                       A22: (t `2 ) >= 0 ;

                      now

                        per cases by A22;

                          case

                           A23: (t `2 ) > 0 ;

                          then ((1 / (t `2 )) * (t `2 )) < (1 * (t `2 )) by A13, A21, XREAL_1: 68;

                          hence contradiction by A7, A23, XCMPLX_1: 87;

                        end;

                          case (t `2 ) = 0 ;

                          hence contradiction by A20;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A24: (t `2 ) < 0 ;

                      then (( - 1) * (t `2 )) > ((1 / (t `2 )) * (t `2 )) by A12, A21, XREAL_1: 69;

                      then (( - 1) * (t `2 )) > 1 by A24, XCMPLX_1: 87;

                      then ( - ( - (t `2 ))) <= ( - 1) by XREAL_1: 24;

                      hence contradiction by A7;

                    end;

                  end;

                  hence contradiction;

                end;

              end;

              hence contradiction;

            end;

              case ( Out_In_Sq . t) in Kb;

              then

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

               A25: p4 = ( Out_In_Sq . t) and

               A26: ( - 1) = (p4 `1 ) & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1 or (p4 `1 ) = 1 & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1 or ( - 1) = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1 or 1 = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1 by A1;

              now

                per cases ;

                  case

                   A27: (t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 ));

                  then

                   A28: ( Out_In_Sq . t) = |[(1 / (t `1 )), (((t `2 ) / (t `1 )) / (t `1 ))]| by A6, Def1;

                  then

                   A29: (p4 `1 ) = (1 / (t `1 )) by A25, EUCLID: 52;

                  now

                    per cases by A26;

                      case ( - 1) = (p4 `1 ) & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1;

                      then

                       A30: ((t `1 ) * ((t `1 ) " )) = ( - (t `1 )) by A29;

                      now

                        per cases ;

                          case (t `1 ) <> 0 ;

                          then ( - (t `1 )) = 1 by A30, XCMPLX_0:def 7;

                          hence contradiction by A7;

                        end;

                          case

                           A31: (t `1 ) = 0 ;

                          then (t `2 ) = 0 by A27;

                          hence contradiction by A6, A31, EUCLID: 53, EUCLID: 54;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case (p4 `1 ) = 1 & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1;

                      then

                       A32: ((t `1 ) * ((t `1 ) " )) = (t `1 ) by A29;

                      now

                        per cases ;

                          case (t `1 ) <> 0 ;

                          hence contradiction by A7, A32, XCMPLX_0:def 7;

                        end;

                          case

                           A33: (t `1 ) = 0 ;

                          then (t `2 ) = 0 by A27;

                          hence contradiction by A6, A33, EUCLID: 53, EUCLID: 54;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A34: ( - 1) = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1;

                      reconsider K01 = K0a as non empty Subset of (( TOP-REAL 2) | D) by A2, Th17;

                      

                       A35: the carrier of ((( TOP-REAL 2) | D) | K01) = ( [#] ((( TOP-REAL 2) | D) | K01))

                      .= K01 by PRE_TOPC:def 5;

                      

                       A36: ( dom ( Out_In_Sq | K01)) = (( dom Out_In_Sq ) /\ K01) by RELAT_1: 61

                      .= (D /\ K01) by A3, FUNCT_2:def 1

                      .= (( [#] (( TOP-REAL 2) | D)) /\ K01) by PRE_TOPC:def 5

                      .= (the carrier of (( TOP-REAL 2) | D) /\ K01)

                      .= K01 by XBOOLE_1: 28;

                      t in K01 by A6, A27;

                      then

                       A37: (( Out_In_Sq | K01) . t) in ( rng ( Out_In_Sq | K01)) by A36, FUNCT_1: 3;

                      ( rng ( Out_In_Sq | K01)) c= the carrier of ((( TOP-REAL 2) | D) | K01) by Th15;

                      then

                       A38: (( Out_In_Sq | K01) . t) in the carrier of ((( TOP-REAL 2) | D) | K01) by A37;

                      t in K01 by A6, A27;

                      then ( Out_In_Sq . t) in K0a by A38, A35, FUNCT_1: 49;

                      then

                       A39: ex p5 be Point of ( TOP-REAL 2) st p5 = p4 & ((p5 `2 ) <= (p5 `1 ) & ( - (p5 `1 )) <= (p5 `2 ) or (p5 `2 ) >= (p5 `1 ) & (p5 `2 ) <= ( - (p5 `1 ))) & p5 <> ( 0. ( TOP-REAL 2)) by A25;

                      now

                        per cases by A34, A39, XREAL_1: 24;

                          case

                           A40: (p4 `1 ) >= 1;

                          

                          then (((t `2 ) / (t `1 )) / (t `1 )) = (((t `2 ) / (t `1 )) * 1) by A29, A34, XXREAL_0: 1

                          .= ((t `2 ) * 1) by A29, A34, A40, XXREAL_0: 1

                          .= (t `2 );

                          hence contradiction by A7, A25, A28, A34, EUCLID: 52;

                        end;

                          case

                           A41: ( - 1) >= (p4 `1 );

                          

                          then (((t `2 ) / (t `1 )) / (t `1 )) = (((t `2 ) / (t `1 )) * ( - 1)) by A29, A34, XXREAL_0: 1

                          .= ( - ((t `2 ) / (t `1 )))

                          .= ( - ((t `2 ) * ( - 1))) by A29, A34, A41, XXREAL_0: 1

                          .= (t `2 );

                          hence contradiction by A7, A25, A28, A34, EUCLID: 52;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A42: 1 = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1;

                      reconsider K01 = K0a as non empty Subset of (( TOP-REAL 2) | D) by A2, Th17;

                      t in K01 by A6, A27;

                      then

                       A43: ( Out_In_Sq . t) = (( Out_In_Sq | K01) . t) by FUNCT_1: 49;

                      ( dom ( Out_In_Sq | K01)) = (( dom Out_In_Sq ) /\ K01) by RELAT_1: 61

                      .= (D /\ K01) by A3, FUNCT_2:def 1

                      .= (( [#] (( TOP-REAL 2) | D)) /\ K01) by PRE_TOPC:def 5

                      .= (the carrier of (( TOP-REAL 2) | D) /\ K01)

                      .= K01 by XBOOLE_1: 28;

                      then t in ( dom ( Out_In_Sq | K01)) by A6, A27;

                      then

                       A44: (( Out_In_Sq | K01) . t) in ( rng ( Out_In_Sq | K01)) by FUNCT_1: 3;

                      ( rng ( Out_In_Sq | K01)) c= the carrier of ((( TOP-REAL 2) | D) | K01) by Th15;

                      then

                       A45: (( Out_In_Sq | K01) . t) in the carrier of ((( TOP-REAL 2) | D) | K01) by A44;

                      the carrier of ((( TOP-REAL 2) | D) | K01) = ( [#] ((( TOP-REAL 2) | D) | K01))

                      .= K01 by PRE_TOPC:def 5;

                      then

                       A46: ex p5 be Point of ( TOP-REAL 2) st p5 = p4 & ((p5 `2 ) <= (p5 `1 ) & ( - (p5 `1 )) <= (p5 `2 ) or (p5 `2 ) >= (p5 `1 ) & (p5 `2 ) <= ( - (p5 `1 ))) & p5 <> ( 0. ( TOP-REAL 2)) by A25, A45, A43;

                      now

                        per cases by A42, A46, XREAL_1: 25;

                          case

                           A47: (p4 `1 ) >= 1;

                          

                          then (((t `2 ) / (t `1 )) / (t `1 )) = (((t `2 ) / (t `1 )) * 1) by A29, A42, XXREAL_0: 1

                          .= ((t `2 ) * 1) by A29, A42, A47, XXREAL_0: 1

                          .= (t `2 );

                          hence contradiction by A7, A25, A28, A42, EUCLID: 52;

                        end;

                          case

                           A48: ( - 1) >= (p4 `1 );

                          

                          then (((t `2 ) / (t `1 )) / (t `1 )) = (((t `2 ) / (t `1 )) * ( - 1)) by A29, A42, XXREAL_0: 1

                          .= ( - ((t `2 ) / (t `1 )))

                          .= ( - ((t `2 ) * ( - 1))) by A29, A42, A48, XXREAL_0: 1

                          .= (t `2 );

                          hence contradiction by A7, A25, A28, A42, EUCLID: 52;

                        end;

                      end;

                      hence contradiction;

                    end;

                  end;

                  hence contradiction;

                end;

                  case

                   A49: not ((t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 )));

                  then

                   A50: ( Out_In_Sq . t) = |[(((t `1 ) / (t `2 )) / (t `2 )), (1 / (t `2 ))]| by A6, Def1;

                  then

                   A51: (p4 `2 ) = (1 / (t `2 )) by A25, EUCLID: 52;

                  now

                    per cases by A26;

                      case ( - 1) = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1;

                      then

                       A52: ((t `2 ) * ((t `2 ) " )) = ( - (t `2 )) by A51;

                      now

                        per cases ;

                          case (t `2 ) <> 0 ;

                          then ( - (t `2 )) = 1 by A52, XCMPLX_0:def 7;

                          hence contradiction by A7;

                        end;

                          case (t `2 ) = 0 ;

                          hence contradiction by A49;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case (p4 `2 ) = 1 & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1;

                      then

                       A53: ((t `2 ) * ((t `2 ) " )) = (t `2 ) by A51;

                      now

                        per cases ;

                          case (t `2 ) <> 0 ;

                          hence contradiction by A7, A53, XCMPLX_0:def 7;

                        end;

                          case (t `2 ) = 0 ;

                          hence contradiction by A49;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A54: ( - 1) = (p4 `1 ) & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1;

                      reconsider K11 = K1a as non empty Subset of (( TOP-REAL 2) | D) by A2, Th18;

                      

                       A55: ( dom ( Out_In_Sq | K11)) = (( dom Out_In_Sq ) /\ K11) by RELAT_1: 61

                      .= (D /\ K11) by A3, FUNCT_2:def 1

                      .= (( [#] (( TOP-REAL 2) | D)) /\ K11) by PRE_TOPC:def 5

                      .= (the carrier of (( TOP-REAL 2) | D) /\ K11)

                      .= K11 by XBOOLE_1: 28;

                      

                       A56: (t `1 ) <= (t `2 ) & ( - (t `2 )) <= (t `1 ) or (t `1 ) >= (t `2 ) & (t `1 ) <= ( - (t `2 )) by A49, Th13;

                      then t in K11 by A6;

                      then

                       A57: ( Out_In_Sq . t) = (( Out_In_Sq | K11) . t) by FUNCT_1: 49;

                      t in K11 by A6, A56;

                      then

                       A58: (( Out_In_Sq | K11) . t) in ( rng ( Out_In_Sq | K11)) by A55, FUNCT_1: 3;

                      ( rng ( Out_In_Sq | K11)) c= the carrier of ((( TOP-REAL 2) | D) | K11) by Th16;

                      then

                       A59: (( Out_In_Sq | K11) . t) in the carrier of ((( TOP-REAL 2) | D) | K11) by A58;

                      the carrier of ((( TOP-REAL 2) | D) | K11) = ( [#] ((( TOP-REAL 2) | D) | K11))

                      .= K11 by PRE_TOPC:def 5;

                      then

                       A60: ex p5 be Point of ( TOP-REAL 2) st p5 = p4 & ((p5 `1 ) <= (p5 `2 ) & ( - (p5 `2 )) <= (p5 `1 ) or (p5 `1 ) >= (p5 `2 ) & (p5 `1 ) <= ( - (p5 `2 ))) & p5 <> ( 0. ( TOP-REAL 2)) by A25, A59, A57;

                      now

                        per cases by A54, A60, XREAL_1: 24;

                          case

                           A61: (p4 `2 ) >= 1;

                          

                          then (((t `1 ) / (t `2 )) / (t `2 )) = (((t `1 ) / (t `2 )) * 1) by A51, A54, XXREAL_0: 1

                          .= ((t `1 ) * 1) by A51, A54, A61, XXREAL_0: 1

                          .= (t `1 );

                          hence contradiction by A7, A25, A50, A54, EUCLID: 52;

                        end;

                          case

                           A62: ( - 1) >= (p4 `2 );

                          

                          then (((t `1 ) / (t `2 )) / (t `2 )) = (((t `1 ) / (t `2 )) * ( - 1)) by A51, A54, XXREAL_0: 1

                          .= ( - ((t `1 ) / (t `2 )))

                          .= ( - ((t `1 ) * ( - 1))) by A51, A54, A62, XXREAL_0: 1

                          .= (t `1 );

                          hence contradiction by A7, A25, A50, A54, EUCLID: 52;

                        end;

                      end;

                      hence contradiction;

                    end;

                      case

                       A63: 1 = (p4 `1 ) & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1;

                      reconsider K11 = K1a as non empty Subset of (( TOP-REAL 2) | D) by A2, Th18;

                      

                       A64: the carrier of ((( TOP-REAL 2) | D) | K11) = ( [#] ((( TOP-REAL 2) | D) | K11))

                      .= K11 by PRE_TOPC:def 5;

                      

                       A65: ( dom ( Out_In_Sq | K11)) = (( dom Out_In_Sq ) /\ K11) by RELAT_1: 61

                      .= (D /\ K11) by A3, FUNCT_2:def 1

                      .= (( [#] (( TOP-REAL 2) | D)) /\ K11) by PRE_TOPC:def 5

                      .= (the carrier of (( TOP-REAL 2) | D) /\ K11)

                      .= K11 by XBOOLE_1: 28;

                      

                       A66: (t `1 ) <= (t `2 ) & ( - (t `2 )) <= (t `1 ) or (t `1 ) >= (t `2 ) & (t `1 ) <= ( - (t `2 )) by A49, Th13;

                      then t in K11 by A6;

                      then

                       A67: (( Out_In_Sq | K11) . t) in ( rng ( Out_In_Sq | K11)) by A65, FUNCT_1: 3;

                      ( rng ( Out_In_Sq | K11)) c= the carrier of ((( TOP-REAL 2) | D) | K11) by Th16;

                      then

                       A68: (( Out_In_Sq | K11) . t) in the carrier of ((( TOP-REAL 2) | D) | K11) by A67;

                      t in K11 by A6, A66;

                      then ( Out_In_Sq . t) in K1a by A68, A64, FUNCT_1: 49;

                      then

                       A69: ex p5 be Point of ( TOP-REAL 2) st p5 = p4 & ((p5 `1 ) <= (p5 `2 ) & ( - (p5 `2 )) <= (p5 `1 ) or (p5 `1 ) >= (p5 `2 ) & (p5 `1 ) <= ( - (p5 `2 ))) & p5 <> ( 0. ( TOP-REAL 2)) by A25;

                      now

                        per cases by A63, A69, XREAL_1: 25;

                          case

                           A70: (p4 `2 ) >= 1;

                          

                          then (((t `1 ) / (t `2 )) / (t `2 )) = (((t `1 ) / (t `2 )) * 1) by A51, A63, XXREAL_0: 1

                          .= ((t `1 ) * 1) by A51, A63, A70, XXREAL_0: 1

                          .= (t `1 );

                          hence contradiction by A7, A25, A50, A63, EUCLID: 52;

                        end;

                          case

                           A71: ( - 1) >= (p4 `2 );

                          

                          then (((t `1 ) / (t `2 )) / (t `2 )) = (((t `1 ) / (t `2 )) * ( - 1)) by A51, A63, XXREAL_0: 1

                          .= ( - ((t `1 ) / (t `2 )))

                          .= ( - ((t `1 ) * ( - 1))) by A51, A63, A71, XXREAL_0: 1

                          .= (t `1 );

                          hence contradiction by A7, A25, A50, A63, EUCLID: 52;

                        end;

                      end;

                      hence contradiction;

                    end;

                  end;

                  hence contradiction;

                end;

              end;

              hence contradiction;

            end;

          end;

          hence contradiction;

        end;

        hence thesis;

      end;

      

       A72: for t be Point of ( TOP-REAL 2) st not t in (K0 \/ Kb) holds ( Out_In_Sq . t) in K0

      proof

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

        assume

         A73: not t in (K0 \/ Kb);

        then

         A74: not t in K0 by XBOOLE_0:def 3;

        then

         A75: not t = ( 0. ( TOP-REAL 2)) by A1, Th3;

        then not t in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

        then t in ( NonZero ( TOP-REAL 2)) by XBOOLE_0:def 5;

        then ( Out_In_Sq . t) in ( NonZero ( TOP-REAL 2)) by FUNCT_2: 5;

        then

        reconsider p4 = ( Out_In_Sq . t) as Point of ( TOP-REAL 2);

        

         A76: not t in Kb by A73, XBOOLE_0:def 3;

        now

          per cases ;

            case

             A77: (t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 ));

             A78:

            now

              per cases ;

                case

                 A79: (t `1 ) > 0 ;

                now

                  per cases ;

                    case

                     A80: (t `2 ) > 0 ;

                    ( - 1) >= (t `1 ) or (t `1 ) >= 1 or ( - 1) >= (t `2 ) or (t `2 ) >= 1 by A1, A74;

                    then

                     A81: (t `1 ) >= 1 by A77, A79, A80, XXREAL_0: 2;

                     not (t `1 ) = 1 by A1, A76, A77;

                    then

                     A82: (t `1 ) > 1 by A81, XXREAL_0: 1;

                    then (t `1 ) < ((t `1 ) ^2 ) by SQUARE_1: 14;

                    then (t `2 ) < ((t `1 ) ^2 ) by A77, A79, XXREAL_0: 2;

                    then ((t `2 ) / (t `1 )) < (((t `1 ) ^2 ) / (t `1 )) by A79, XREAL_1: 74;

                    then ((t `2 ) / (t `1 )) < (t `1 ) by A79, XCMPLX_1: 89;

                    then

                     A83: (((t `2 ) / (t `1 )) / (t `1 )) < ((t `1 ) / (t `1 )) by A79, XREAL_1: 74;

                     0 < ((t `2 ) / (t `1 )) by A79, A80, XREAL_1: 139;

                    then

                     A84: ((( - 1) * (t `1 )) / (t `1 )) < (((t `2 ) / (t `1 )) / (t `1 )) by A79, XREAL_1: 74;

                    ((t `1 ) / (t `1 )) > (1 / (t `1 )) by A82, XREAL_1: 74;

                    hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1 by A79, A84, A83, XCMPLX_1: 60, XCMPLX_1: 89;

                  end;

                    case

                     A85: (t `2 ) <= 0 ;

                     A86:

                    now

                      assume (t `1 ) < 1;

                      then ( - 1) >= (t `2 ) by A1, A74, A79, A85;

                      then ( - (t `1 )) <= ( - 1) by A77, A79, XXREAL_0: 2;

                      hence (t `1 ) >= 1 by XREAL_1: 24;

                    end;

                     not (t `1 ) = 1 by A1, A76, A77;

                    then

                     A87: (t `1 ) > 1 by A86, XXREAL_0: 1;

                    then

                     A88: (t `1 ) < ((t `1 ) ^2 ) by SQUARE_1: 14;

                    ( - ( - (t `1 ))) >= ( - (t `2 )) by A77, A79, XREAL_1: 24;

                    then ((t `1 ) ^2 ) > ( - (t `2 )) by A88, XXREAL_0: 2;

                    then (((t `1 ) ^2 ) / (t `1 )) > (( - (t `2 )) / (t `1 )) by A79, XREAL_1: 74;

                    then (t `1 ) > ( - ((t `2 ) / (t `1 ))) by A79, XCMPLX_1: 89;

                    then ( - (t `1 )) < ( - ( - ((t `2 ) / (t `1 )))) by XREAL_1: 24;

                    then

                     A89: ((( - 1) * (t `1 )) / (t `1 )) < (((t `2 ) / (t `1 )) / (t `1 )) by A79, XREAL_1: 74;

                    ((t `1 ) / (t `1 )) > (1 / (t `1 )) by A87, XREAL_1: 74;

                    hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1 by A79, A85, A89, XCMPLX_1: 60, XCMPLX_1: 89;

                  end;

                end;

                hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1;

              end;

                case

                 A90: (t `1 ) <= 0 ;

                now

                  per cases by A90;

                    case

                     A91: (t `1 ) = 0 ;

                    then (t `2 ) = 0 by A77;

                    hence contradiction by A1, A74, A91;

                  end;

                    case

                     A92: (t `1 ) < 0 ;

                    now

                      per cases ;

                        case

                         A93: (t `2 ) > 0 ;

                        ( - 1) >= (t `1 ) or (t `1 ) >= 1 or ( - 1) >= (t `2 ) or (t `2 ) >= 1 by A1, A74;

                        then (t `1 ) <= ( - 1) or 1 <= ( - (t `1 )) by A77, A92, XXREAL_0: 2;

                        then

                         A94: (t `1 ) <= ( - 1) or ( - 1) >= ( - ( - (t `1 ))) by XREAL_1: 24;

                         not (t `1 ) = ( - 1) by A1, A76, A77;

                        then

                         A95: (t `1 ) < ( - 1) by A94, XXREAL_0: 1;

                        then ((t `1 ) / (t `1 )) > (( - 1) / (t `1 )) by XREAL_1: 75;

                        then

                         A96: ( - ((t `1 ) / (t `1 ))) < ( - (( - 1) / (t `1 ))) by XREAL_1: 24;

                        ( - (t `1 )) < ((t `1 ) ^2 ) by A95, SQUARE_1: 46;

                        then (t `2 ) < ((t `1 ) ^2 ) by A77, A92, XXREAL_0: 2;

                        then ((t `2 ) / (t `1 )) > (((t `1 ) ^2 ) / (t `1 )) by A92, XREAL_1: 75;

                        then ((t `2 ) / (t `1 )) > (t `1 ) by A92, XCMPLX_1: 89;

                        then

                         A97: (((t `2 ) / (t `1 )) / (t `1 )) < ((t `1 ) / (t `1 )) by A92, XREAL_1: 75;

                         0 > ((t `2 ) / (t `1 )) by A92, A93, XREAL_1: 142;

                        then ((( - 1) * (t `1 )) / (t `1 )) < (((t `2 ) / (t `1 )) / (t `1 )) by A92, XREAL_1: 75;

                        hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1 by A92, A96, A97, XCMPLX_1: 60;

                      end;

                        case

                         A98: (t `2 ) <= 0 ;

                        then ( - 1) >= (t `1 ) or ( - 1) >= (t `2 ) by A1, A74, A92;

                        then

                         A99: (t `1 ) <= ( - 1) by A77, A92, XXREAL_0: 2;

                         not (t `1 ) = ( - 1) by A1, A76, A77;

                        then

                         A100: (t `1 ) < ( - 1) by A99, XXREAL_0: 1;

                        then

                         A101: ( - (t `1 )) < ((t `1 ) ^2 ) by SQUARE_1: 46;

                        (t `1 ) <= (t `2 ) by A77, A92;

                        then ( - (t `1 )) >= ( - (t `2 )) by XREAL_1: 24;

                        then ((t `1 ) ^2 ) > ( - (t `2 )) by A101, XXREAL_0: 2;

                        then (((t `1 ) ^2 ) / (t `1 )) < (( - (t `2 )) / (t `1 )) by A92, XREAL_1: 75;

                        then (t `1 ) < ( - ((t `2 ) / (t `1 ))) by A92, XCMPLX_1: 89;

                        then ( - (t `1 )) > ( - ( - ((t `2 ) / (t `1 )))) by XREAL_1: 24;

                        then

                         A102: ((( - 1) * (t `1 )) / (t `1 )) < (((t `2 ) / (t `1 )) / (t `1 )) by A92, XREAL_1: 75;

                        ((t `1 ) / (t `1 )) > (( - 1) / (t `1 )) by A100, XREAL_1: 75;

                        then 1 > (( - 1) / (t `1 )) by A92, XCMPLX_1: 60;

                        then ( - 1) < ( - (( - 1) / (t `1 ))) by XREAL_1: 24;

                        hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1 by A92, A98, A102, XCMPLX_1: 89;

                      end;

                    end;

                    hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1;

                  end;

                end;

                hence ( - 1) < (1 / (t `1 )) & (1 / (t `1 )) < 1 & ( - 1) < (((t `2 ) / (t `1 )) / (t `1 )) & (((t `2 ) / (t `1 )) / (t `1 )) < 1;

              end;

            end;

            ( Out_In_Sq . t) = |[(1 / (t `1 )), (((t `2 ) / (t `1 )) / (t `1 ))]| by A75, A77, Def1;

            then (p4 `1 ) = (1 / (t `1 )) & (p4 `2 ) = (((t `2 ) / (t `1 )) / (t `1 )) by EUCLID: 52;

            hence thesis by A1, A78;

          end;

            case

             A103: not ((t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 )));

            then

             A104: (t `1 ) <= (t `2 ) & ( - (t `2 )) <= (t `1 ) or (t `1 ) >= (t `2 ) & (t `1 ) <= ( - (t `2 )) by Th13;

             A105:

            now

              per cases ;

                case

                 A106: (t `2 ) > 0 ;

                now

                  per cases ;

                    case

                     A107: (t `1 ) > 0 ;

                    

                     A108: ( - 1) >= (t `2 ) or (t `2 ) >= 1 or ( - 1) >= (t `1 ) or (t `1 ) >= 1 by A1, A74;

                     not (t `2 ) = 1 by A1, A76, A103, A107;

                    then

                     A109: (t `2 ) > 1 by A103, A106, A107, A108, XXREAL_0: 1, XXREAL_0: 2;

                    then (t `2 ) < ((t `2 ) ^2 ) by SQUARE_1: 14;

                    then (t `1 ) < ((t `2 ) ^2 ) by A103, A106, XXREAL_0: 2;

                    then ((t `1 ) / (t `2 )) < (((t `2 ) ^2 ) / (t `2 )) by A106, XREAL_1: 74;

                    then ((t `1 ) / (t `2 )) < (t `2 ) by A106, XCMPLX_1: 89;

                    then

                     A110: (((t `1 ) / (t `2 )) / (t `2 )) < ((t `2 ) / (t `2 )) by A106, XREAL_1: 74;

                     0 < ((t `1 ) / (t `2 )) by A106, A107, XREAL_1: 139;

                    then

                     A111: ((( - 1) * (t `2 )) / (t `2 )) < (((t `1 ) / (t `2 )) / (t `2 )) by A106, XREAL_1: 74;

                    ((t `2 ) / (t `2 )) > (1 / (t `2 )) by A109, XREAL_1: 74;

                    hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1 by A106, A111, A110, XCMPLX_1: 60, XCMPLX_1: 89;

                  end;

                    case

                     A112: (t `1 ) <= 0 ;

                     A113:

                    now

                      assume (t `2 ) < 1;

                      then ( - 1) >= (t `1 ) by A1, A74, A106, A112;

                      then ( - (t `2 )) <= ( - 1) by A104, A106, XXREAL_0: 2;

                      hence (t `2 ) >= 1 by XREAL_1: 24;

                    end;

                     not (t `2 ) = 1 by A1, A76, A104;

                    then

                     A114: (t `2 ) > 1 by A113, XXREAL_0: 1;

                    then (t `2 ) < ((t `2 ) ^2 ) by SQUARE_1: 14;

                    then ((t `2 ) ^2 ) > ( - (t `1 )) by A103, A106, XXREAL_0: 2;

                    then (((t `2 ) ^2 ) / (t `2 )) > (( - (t `1 )) / (t `2 )) by A106, XREAL_1: 74;

                    then (t `2 ) > ( - ((t `1 ) / (t `2 ))) by A106, XCMPLX_1: 89;

                    then ( - (t `2 )) < ( - ( - ((t `1 ) / (t `2 )))) by XREAL_1: 24;

                    then

                     A115: ((( - 1) * (t `2 )) / (t `2 )) < (((t `1 ) / (t `2 )) / (t `2 )) by A106, XREAL_1: 74;

                    ((t `2 ) / (t `2 )) > (1 / (t `2 )) by A114, XREAL_1: 74;

                    hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1 by A106, A112, A115, XCMPLX_1: 60, XCMPLX_1: 89;

                  end;

                end;

                hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1;

              end;

                case

                 A116: (t `2 ) <= 0 ;

                then

                 A117: (t `2 ) < 0 by A103;

                

                 A118: (t `1 ) <= (t `2 ) or (t `1 ) <= ( - (t `2 )) by A103, Th13;

                now

                  per cases ;

                    case

                     A119: (t `1 ) > 0 ;

                    ( - 1) >= (t `2 ) or (t `2 ) >= 1 or ( - 1) >= (t `1 ) or (t `1 ) >= 1 by A1, A74;

                    then (t `2 ) <= ( - 1) or 1 <= ( - (t `2 )) by A104, A116, XXREAL_0: 2;

                    then

                     A120: (t `2 ) <= ( - 1) or ( - 1) >= ( - ( - (t `2 ))) by XREAL_1: 24;

                     not (t `2 ) = ( - 1) by A1, A76, A104;

                    then

                     A121: (t `2 ) < ( - 1) by A120, XXREAL_0: 1;

                    then ((t `2 ) / (t `2 )) > (( - 1) / (t `2 )) by XREAL_1: 75;

                    then

                     A122: ( - ((t `2 ) / (t `2 ))) < ( - (( - 1) / (t `2 ))) by XREAL_1: 24;

                    ( - (t `2 )) < ((t `2 ) ^2 ) by A121, SQUARE_1: 46;

                    then (t `1 ) < ((t `2 ) ^2 ) by A116, A118, XXREAL_0: 2;

                    then ((t `1 ) / (t `2 )) > (((t `2 ) ^2 ) / (t `2 )) by A117, XREAL_1: 75;

                    then ((t `1 ) / (t `2 )) > (t `2 ) by A117, XCMPLX_1: 89;

                    then

                     A123: (((t `1 ) / (t `2 )) / (t `2 )) < ((t `2 ) / (t `2 )) by A117, XREAL_1: 75;

                     0 > ((t `1 ) / (t `2 )) by A117, A119, XREAL_1: 142;

                    then ((( - 1) * (t `2 )) / (t `2 )) < (((t `1 ) / (t `2 )) / (t `2 )) by A117, XREAL_1: 75;

                    hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1 by A117, A122, A123, XCMPLX_1: 60;

                  end;

                    case

                     A124: (t `1 ) <= 0 ;

                    

                     A125: not (t `2 ) = ( - 1) by A1, A76, A104;

                    ( - 1) >= (t `2 ) or ( - 1) >= (t `1 ) by A1, A74, A116, A124;

                    then

                     A126: (t `2 ) < ( - 1) by A103, A116, A125, XXREAL_0: 1, XXREAL_0: 2;

                    then

                     A127: ( - (t `2 )) < ((t `2 ) ^2 ) by SQUARE_1: 46;

                    ( - (t `2 )) >= ( - (t `1 )) by A103, A116, XREAL_1: 24;

                    then ((t `2 ) ^2 ) > ( - (t `1 )) by A127, XXREAL_0: 2;

                    then (((t `2 ) ^2 ) / (t `2 )) < (( - (t `1 )) / (t `2 )) by A117, XREAL_1: 75;

                    then (t `2 ) < ( - ((t `1 ) / (t `2 ))) by A117, XCMPLX_1: 89;

                    then ( - (t `2 )) > ( - ( - ((t `1 ) / (t `2 )))) by XREAL_1: 24;

                    then

                     A128: ((( - 1) * (t `2 )) / (t `2 )) < (((t `1 ) / (t `2 )) / (t `2 )) by A117, XREAL_1: 75;

                    ((t `2 ) / (t `2 )) > (( - 1) / (t `2 )) by A126, XREAL_1: 75;

                    then 1 > (( - 1) / (t `2 )) by A117, XCMPLX_1: 60;

                    then ( - 1) < ( - (( - 1) / (t `2 ))) by XREAL_1: 24;

                    hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1 by A103, A116, A124, A128, XCMPLX_1: 89;

                  end;

                end;

                hence ( - 1) < (1 / (t `2 )) & (1 / (t `2 )) < 1 & ( - 1) < (((t `1 ) / (t `2 )) / (t `2 )) & (((t `1 ) / (t `2 )) / (t `2 )) < 1;

              end;

            end;

            ( Out_In_Sq . t) = |[(((t `1 ) / (t `2 )) / (t `2 )), (1 / (t `2 ))]| by A75, A103, Def1;

            then (p4 `2 ) = (1 / (t `2 )) & (p4 `1 ) = (((t `1 ) / (t `2 )) / (t `2 )) by EUCLID: 52;

            hence thesis by A1, A105;

          end;

        end;

        hence thesis;

      end;

      

       A129: D = ( NonZero ( TOP-REAL 2)) by A1, SUBSET_1:def 4;

      for x1,x2 be object st x1 in ( dom Out_In_Sq ) & x2 in ( dom Out_In_Sq ) & ( Out_In_Sq . x1) = ( Out_In_Sq . x2) holds x1 = x2

      proof

        

         A130: K1a c= D

        proof

          let x be object;

          assume x in K1a;

          then

           A131: ex p8 be Point of ( TOP-REAL 2) st x = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2));

          then not x in {( 0. ( TOP-REAL 2))} by TARSKI:def 1;

          hence thesis by A3, A131, XBOOLE_0:def 5;

        end;

        

         A132: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm1, REVROT_1: 19;

        (( 1.REAL 2) `1 ) <= (( 1.REAL 2) `2 ) & ( - (( 1.REAL 2) `2 )) <= (( 1.REAL 2) `1 ) or (( 1.REAL 2) `1 ) >= (( 1.REAL 2) `2 ) & (( 1.REAL 2) `1 ) <= ( - (( 1.REAL 2) `2 )) by Th5;

        then

         A133: ( 1.REAL 2) in K1a by A132;

        the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

        .= D by PRE_TOPC:def 5;

        then

        reconsider K11 = K1a as non empty Subset of (( TOP-REAL 2) | D) by A133, A130;

        reconsider K01 = K0a as non empty Subset of (( TOP-REAL 2) | D) by A2, Th17;

        let x1,x2 be object;

        assume that

         A134: x1 in ( dom Out_In_Sq ) and

         A135: x2 in ( dom Out_In_Sq ) and

         A136: ( Out_In_Sq . x1) = ( Out_In_Sq . x2);

        ( NonZero ( TOP-REAL 2)) <> {} by Th9;

        then

         A137: ( dom Out_In_Sq ) = ( NonZero ( TOP-REAL 2)) by FUNCT_2:def 1;

        then

         A138: x2 in D by A1, A135, SUBSET_1:def 4;

        reconsider p1 = x1, p2 = x2 as Point of ( TOP-REAL 2) by A134, A135, XBOOLE_0:def 5;

        

         A139: D c= (K01 \/ K11)

        proof

          let x be object;

          assume

           A140: x in D;

          then

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

           not x in {( 0. ( TOP-REAL 2))} by A129, A140, XBOOLE_0:def 5;

          then ((px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 ))) & px <> ( 0. ( TOP-REAL 2)) or ((px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 ))) & px <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1, XREAL_1: 25;

          then x in K01 or x in K11;

          hence thesis by XBOOLE_0:def 3;

        end;

        

         A141: x1 in D by A1, A134, A137, SUBSET_1:def 4;

        now

          per cases by A139, A141, XBOOLE_0:def 3;

            case x1 in K01;

            then

             A142: ex p7 be Point of ( TOP-REAL 2) st p1 = p7 & ((p7 `2 ) <= (p7 `1 ) & ( - (p7 `1 )) <= (p7 `2 ) or (p7 `2 ) >= (p7 `1 ) & (p7 `2 ) <= ( - (p7 `1 ))) & p7 <> ( 0. ( TOP-REAL 2));

            then

             A143: ( Out_In_Sq . p1) = |[(1 / (p1 `1 )), (((p1 `2 ) / (p1 `1 )) / (p1 `1 ))]| by Def1;

            now

              per cases by A139, A138, XBOOLE_0:def 3;

                case x2 in K0a;

                then ex p8 be Point of ( TOP-REAL 2) st p2 = p8 & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2));

                then

                 A144: |[(1 / (p2 `1 )), (((p2 `2 ) / (p2 `1 )) / (p2 `1 ))]| = |[(1 / (p1 `1 )), (((p1 `2 ) / (p1 `1 )) / (p1 `1 ))]| by A136, A143, Def1;

                

                 A145: p1 = |[(p1 `1 ), (p1 `2 )]| by EUCLID: 53;

                set qq = |[(1 / (p2 `1 )), (((p2 `2 ) / (p2 `1 )) / (p2 `1 ))]|;

                

                 A146: ((1 / (p1 `1 )) " ) = (((p1 `1 ) " ) " )

                .= (p1 `1 );

                 A147:

                now

                  assume

                   A148: (p1 `1 ) = 0 ;

                  then (p1 `2 ) = 0 by A142;

                  hence contradiction by A142, A148, EUCLID: 53, EUCLID: 54;

                end;

                (qq `1 ) = (1 / (p2 `1 )) by EUCLID: 52;

                then

                 A149: (1 / (p1 `1 )) = (1 / (p2 `1 )) by A144, EUCLID: 52;

                (qq `2 ) = (((p2 `2 ) / (p2 `1 )) / (p2 `1 )) by EUCLID: 52;

                then ((p1 `2 ) / (p1 `1 )) = ((p2 `2 ) / (p1 `1 )) by A144, A149, A146, A147, EUCLID: 52, XCMPLX_1: 53;

                then (p1 `2 ) = (p2 `2 ) by A147, XCMPLX_1: 53;

                hence thesis by A149, A146, A145, EUCLID: 53;

              end;

                case

                 A150: x2 in K1a & not x2 in K0a;

                 A151:

                now

                  assume

                   A152: (p1 `1 ) = 0 ;

                  then (p1 `2 ) = 0 by A142;

                  hence contradiction by A142, A152, EUCLID: 53, EUCLID: 54;

                end;

                 A153:

                now

                  per cases by A142;

                    case

                     A154: (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 );

                    then ( - (p1 `1 )) <= (p1 `1 ) by XXREAL_0: 2;

                    then (p1 `1 ) >= 0 ;

                    then ((p1 `2 ) / (p1 `1 )) <= ((p1 `1 ) / (p1 `1 )) by A154, XREAL_1: 72;

                    hence ((p1 `2 ) / (p1 `1 )) <= 1 by A151, XCMPLX_1: 60;

                  end;

                    case

                     A155: (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 ));

                    then ( - (p1 `1 )) >= (p1 `1 ) by XXREAL_0: 2;

                    then (p1 `1 ) <= 0 ;

                    then ((p1 `2 ) / (p1 `1 )) <= ((p1 `1 ) / (p1 `1 )) by A155, XREAL_1: 73;

                    hence ((p1 `2 ) / (p1 `1 )) <= 1 by A151, XCMPLX_1: 60;

                  end;

                end;

                 A156:

                now

                  per cases by A142;

                    case

                     A157: (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 );

                    then ( - (p1 `1 )) <= (p1 `1 ) by XXREAL_0: 2;

                    then (p1 `1 ) >= 0 ;

                    then (( - (p1 `1 )) / (p1 `1 )) <= ((p1 `2 ) / (p1 `1 )) by A157, XREAL_1: 72;

                    hence ( - 1) <= ((p1 `2 ) / (p1 `1 )) by A151, XCMPLX_1: 197;

                  end;

                    case

                     A158: (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 ));

                    ( - (p1 `2 )) >= ( - ( - (p1 `1 ))) & (p1 `1 ) <= 0 by XREAL_1: 24, A158;

                    then (( - (p1 `2 )) / ( - (p1 `1 ))) >= ((p1 `1 ) / ( - (p1 `1 ))) by XREAL_1: 72;

                    then (( - (p1 `2 )) / ( - (p1 `1 ))) >= ( - 1) by A151, XCMPLX_1: 198;

                    hence ( - 1) <= ((p1 `2 ) / (p1 `1 )) by XCMPLX_1: 191;

                  end;

                end;

                

                 A159: ex p8 be Point of ( TOP-REAL 2) st p2 = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A150;

                 A160:

                now

                  assume

                   A161: (p2 `2 ) = 0 ;

                  then (p2 `1 ) = 0 by A159;

                  hence contradiction by A159, A161, EUCLID: 53, EUCLID: 54;

                end;

                 not (((p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 ))) & p2 <> ( 0. ( TOP-REAL 2))) by A150;

                then

                 A162: ( Out_In_Sq . p2) = |[(((p2 `1 ) / (p2 `2 )) / (p2 `2 )), (1 / (p2 `2 ))]| by A159, Def1;

                then (((p1 `2 ) / (p1 `1 )) / (p1 `1 )) = (1 / (p2 `2 )) by A136, A143, SPPOL_2: 1;

                

                then

                 A163: ((p1 `2 ) / (p1 `1 )) = ((1 / (p2 `2 )) * (p1 `1 )) by A151, XCMPLX_1: 87

                .= ((p1 `1 ) / (p2 `2 ));

                (1 / (p1 `1 )) = (((p2 `1 ) / (p2 `2 )) / (p2 `2 )) by A136, A143, A162, SPPOL_2: 1;

                

                then

                 A164: ((p2 `1 ) / (p2 `2 )) = ((1 / (p1 `1 )) * (p2 `2 )) by A160, XCMPLX_1: 87

                .= ((p2 `2 ) / (p1 `1 ));

                then

                 A165: (((p2 `1 ) / (p2 `2 )) * ((p1 `2 ) / (p1 `1 ))) = 1 by A160, A151, A163, XCMPLX_1: 112;

                

                 A166: ((((p2 `1 ) / (p2 `2 )) * ((p1 `2 ) / (p1 `1 ))) * (p1 `1 )) = (1 * (p1 `1 )) by A160, A151, A164, A163, XCMPLX_1: 112;

                then

                 A167: (p1 `2 ) <> 0 by A151;

                

                 A168: ex p9 be Point of ( TOP-REAL 2) st p2 = p9 & ((p9 `1 ) <= (p9 `2 ) & ( - (p9 `2 )) <= (p9 `1 ) or (p9 `1 ) >= (p9 `2 ) & (p9 `1 ) <= ( - (p9 `2 ))) & p9 <> ( 0. ( TOP-REAL 2)) by A150;

                 A169:

                now

                  per cases by A168;

                    case

                     A170: (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 );

                    then ( - (p2 `2 )) <= (p2 `2 ) by XXREAL_0: 2;

                    then (p2 `2 ) >= 0 ;

                    then (( - (p2 `2 )) / (p2 `2 )) <= ((p2 `1 ) / (p2 `2 )) by A170, XREAL_1: 72;

                    hence ( - 1) <= ((p2 `1 ) / (p2 `2 )) by A160, XCMPLX_1: 197;

                  end;

                    case

                     A171: (p2 `1 ) >= (p2 `2 ) & (p2 `1 ) <= ( - (p2 `2 ));

                    ( - (p2 `1 )) >= ( - ( - (p2 `2 ))) & (p2 `2 ) <= 0 by XREAL_1: 24, A171;

                    then (( - (p2 `1 )) / ( - (p2 `2 ))) >= ((p2 `2 ) / ( - (p2 `2 ))) by XREAL_1: 72;

                    then (( - (p2 `1 )) / ( - (p2 `2 ))) >= ( - 1) by A160, XCMPLX_1: 198;

                    hence ( - 1) <= ((p2 `1 ) / (p2 `2 )) by XCMPLX_1: 191;

                  end;

                end;

                (((p2 `1 ) / (p2 `2 )) * (((p1 `2 ) / (p1 `1 )) * (p1 `1 ))) = (p1 `1 ) by A166;

                then

                 A172: (((p2 `1 ) / (p2 `2 )) * (p1 `2 )) = (p1 `1 ) by A151, XCMPLX_1: 87;

                then

                 A173: ((p2 `1 ) / (p2 `2 )) = ((p1 `1 ) / (p1 `2 )) by A167, XCMPLX_1: 89;

                 A174:

                now

                  per cases by A168;

                    case

                     A175: (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 );

                    then ( - (p2 `2 )) <= (p2 `2 ) by XXREAL_0: 2;

                    then (p2 `2 ) >= 0 ;

                    then ((p2 `1 ) / (p2 `2 )) <= ((p2 `2 ) / (p2 `2 )) by A175, XREAL_1: 72;

                    hence ((p2 `1 ) / (p2 `2 )) <= 1 by A160, XCMPLX_1: 60;

                  end;

                    case

                     A176: (p2 `1 ) >= (p2 `2 ) & (p2 `1 ) <= ( - (p2 `2 ));

                    then ( - (p2 `2 )) >= (p2 `2 ) by XXREAL_0: 2;

                    then (p2 `2 ) <= 0 ;

                    then ((p2 `1 ) / (p2 `2 )) <= ((p2 `2 ) / (p2 `2 )) by A176, XREAL_1: 73;

                    hence ((p2 `1 ) / (p2 `2 )) <= 1 by A160, XCMPLX_1: 60;

                  end;

                end;

                now

                  per cases ;

                    case 0 <= ((p2 `1 ) / (p2 `2 ));

                    then

                     A177: (p1 `2 ) > 0 & (p1 `1 ) >= 0 or (p1 `2 ) < 0 & (p1 `1 ) <= 0 by A151, A172;

                    now

                      assume ((p1 `2 ) / (p1 `1 )) <> 1;

                      then ((p1 `2 ) / (p1 `1 )) < 1 by A153, XXREAL_0: 1;

                      hence contradiction by A165, A174, A177, XREAL_1: 162;

                    end;

                    then (p1 `2 ) = (1 * (p1 `1 )) by A151, XCMPLX_1: 87;

                    

                    then (((p2 `1 ) / (p2 `2 )) * (p2 `2 )) = (1 * (p2 `2 )) by A151, A173, XCMPLX_1: 60

                    .= (p2 `2 );

                    then (p2 `1 ) = (p2 `2 ) by A160, XCMPLX_1: 87;

                    hence contradiction by A150, A168;

                  end;

                    case 0 > ((p2 `1 ) / (p2 `2 ));

                    then

                     A178: (p1 `2 ) < 0 & (p1 `1 ) > 0 or (p1 `2 ) > 0 & (p1 `1 ) < 0 by A173, XREAL_1: 143;

                    now

                      assume ((p1 `2 ) / (p1 `1 )) <> ( - 1);

                      then ( - 1) < ((p1 `2 ) / (p1 `1 )) by A156, XXREAL_0: 1;

                      hence contradiction by A165, A169, A178, XREAL_1: 166;

                    end;

                    

                    then (p1 `2 ) = (( - 1) * (p1 `1 )) by A151, XCMPLX_1: 87

                    .= ( - (p1 `1 ));

                    then ( - (p1 `2 )) = (p1 `1 );

                    then ((p2 `1 ) / (p2 `2 )) = ( - 1) by A167, A173, XCMPLX_1: 197;

                    then (p2 `1 ) = (( - 1) * (p2 `2 )) by A160, XCMPLX_1: 87;

                    then ( - (p2 `1 )) = (p2 `2 );

                    hence contradiction by A150, A168;

                  end;

                end;

                hence contradiction;

              end;

            end;

            hence thesis;

          end;

            case x1 in K1a;

            then

             A179: ex p7 be Point of ( TOP-REAL 2) st p1 = p7 & ((p7 `1 ) <= (p7 `2 ) & ( - (p7 `2 )) <= (p7 `1 ) or (p7 `1 ) >= (p7 `2 ) & (p7 `1 ) <= ( - (p7 `2 ))) & p7 <> ( 0. ( TOP-REAL 2));

            then

             A180: ( Out_In_Sq . p1) = |[(((p1 `1 ) / (p1 `2 )) / (p1 `2 )), (1 / (p1 `2 ))]| by Th14;

            now

              per cases by A139, A138, XBOOLE_0:def 3;

                case x2 in K1a;

                then ex p8 be Point of ( TOP-REAL 2) st p2 = p8 & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2));

                then

                 A181: |[(((p2 `1 ) / (p2 `2 )) / (p2 `2 )), (1 / (p2 `2 ))]| = |[(((p1 `1 ) / (p1 `2 )) / (p1 `2 )), (1 / (p1 `2 ))]| by A136, A180, Th14;

                

                 A182: p1 = |[(p1 `1 ), (p1 `2 )]| by EUCLID: 53;

                set qq = |[(((p2 `1 ) / (p2 `2 )) / (p2 `2 )), (1 / (p2 `2 ))]|;

                

                 A183: ((1 / (p1 `2 )) " ) = (((p1 `2 ) " ) " )

                .= (p1 `2 );

                 A184:

                now

                  assume

                   A185: (p1 `2 ) = 0 ;

                  then (p1 `1 ) = 0 by A179;

                  hence contradiction by A179, A185, EUCLID: 53, EUCLID: 54;

                end;

                (qq `2 ) = (1 / (p2 `2 )) by EUCLID: 52;

                then

                 A186: (1 / (p1 `2 )) = (1 / (p2 `2 )) by A181, EUCLID: 52;

                (qq `1 ) = (((p2 `1 ) / (p2 `2 )) / (p2 `2 )) by EUCLID: 52;

                then ((p1 `1 ) / (p1 `2 )) = ((p2 `1 ) / (p1 `2 )) by A181, A186, A183, A184, EUCLID: 52, XCMPLX_1: 53;

                then (p1 `1 ) = (p2 `1 ) by A184, XCMPLX_1: 53;

                hence thesis by A186, A183, A182, EUCLID: 53;

              end;

                case

                 A187: x2 in K0a & not x2 in K1a;

                 A188:

                now

                  assume

                   A189: (p1 `2 ) = 0 ;

                  then (p1 `1 ) = 0 by A179;

                  hence contradiction by A179, A189, EUCLID: 53, EUCLID: 54;

                end;

                 A190:

                now

                  per cases by A179;

                    case

                     A191: (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 );

                    then ( - (p1 `2 )) <= (p1 `2 ) by XXREAL_0: 2;

                    then (p1 `2 ) >= 0 ;

                    then ((p1 `1 ) / (p1 `2 )) <= ((p1 `2 ) / (p1 `2 )) by A191, XREAL_1: 72;

                    hence ((p1 `1 ) / (p1 `2 )) <= 1 by A188, XCMPLX_1: 60;

                  end;

                    case

                     A192: (p1 `1 ) >= (p1 `2 ) & (p1 `1 ) <= ( - (p1 `2 ));

                    then ( - (p1 `2 )) >= (p1 `2 ) by XXREAL_0: 2;

                    then (p1 `2 ) <= 0 ;

                    then ((p1 `1 ) / (p1 `2 )) <= ((p1 `2 ) / (p1 `2 )) by A192, XREAL_1: 73;

                    hence ((p1 `1 ) / (p1 `2 )) <= 1 by A188, XCMPLX_1: 60;

                  end;

                end;

                 A193:

                now

                  per cases by A179;

                    case

                     A194: (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 );

                    then ( - (p1 `2 )) <= (p1 `2 ) by XXREAL_0: 2;

                    then (p1 `2 ) >= 0 ;

                    then (( - (p1 `2 )) / (p1 `2 )) <= ((p1 `1 ) / (p1 `2 )) by A194, XREAL_1: 72;

                    hence ( - 1) <= ((p1 `1 ) / (p1 `2 )) by A188, XCMPLX_1: 197;

                  end;

                    case

                     A195: (p1 `1 ) >= (p1 `2 ) & (p1 `1 ) <= ( - (p1 `2 ));

                    ( - (p1 `1 )) >= ( - ( - (p1 `2 ))) & (p1 `2 ) <= 0 by XREAL_1: 24, A195;

                    then (( - (p1 `1 )) / ( - (p1 `2 ))) >= ((p1 `2 ) / ( - (p1 `2 ))) by XREAL_1: 72;

                    then (( - (p1 `1 )) / ( - (p1 `2 ))) >= ( - 1) by A188, XCMPLX_1: 198;

                    hence ( - 1) <= ((p1 `1 ) / (p1 `2 )) by XCMPLX_1: 191;

                  end;

                end;

                

                 A196: ex p8 be Point of ( TOP-REAL 2) st p2 = p8 & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2)) by A187;

                 A197:

                now

                  assume

                   A198: (p2 `1 ) = 0 ;

                  then (p2 `2 ) = 0 by A196;

                  hence contradiction by A196, A198, EUCLID: 53, EUCLID: 54;

                end;

                

                 A199: ex p9 be Point of ( TOP-REAL 2) st p2 = p9 & ((p9 `2 ) <= (p9 `1 ) & ( - (p9 `1 )) <= (p9 `2 ) or (p9 `2 ) >= (p9 `1 ) & (p9 `2 ) <= ( - (p9 `1 ))) & p9 <> ( 0. ( TOP-REAL 2)) by A187;

                 A200:

                now

                  per cases by A199;

                    case

                     A201: (p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 );

                    then ( - (p2 `1 )) <= (p2 `1 ) by XXREAL_0: 2;

                    then (p2 `1 ) >= 0 ;

                    then (( - (p2 `1 )) / (p2 `1 )) <= ((p2 `2 ) / (p2 `1 )) by A201, XREAL_1: 72;

                    hence ( - 1) <= ((p2 `2 ) / (p2 `1 )) by A197, XCMPLX_1: 197;

                  end;

                    case

                     A202: (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 ));

                    ( - (p2 `2 )) >= ( - ( - (p2 `1 ))) & (p2 `1 ) <= 0 by XREAL_1: 24, A202;

                    then (( - (p2 `2 )) / ( - (p2 `1 ))) >= ((p2 `1 ) / ( - (p2 `1 ))) by XREAL_1: 72;

                    then (( - (p2 `2 )) / ( - (p2 `1 ))) >= ( - 1) by A197, XCMPLX_1: 198;

                    hence ( - 1) <= ((p2 `2 ) / (p2 `1 )) by XCMPLX_1: 191;

                  end;

                end;

                

                 A203: ( Out_In_Sq . p2) = |[(1 / (p2 `1 )), (((p2 `2 ) / (p2 `1 )) / (p2 `1 ))]| by A196, Def1;

                then (1 / (p1 `2 )) = (((p2 `2 ) / (p2 `1 )) / (p2 `1 )) by A136, A180, SPPOL_2: 1;

                

                then

                 A204: ((p2 `2 ) / (p2 `1 )) = ((1 / (p1 `2 )) * (p2 `1 )) by A197, XCMPLX_1: 87

                .= ((p2 `1 ) / (p1 `2 ));

                (((p1 `1 ) / (p1 `2 )) / (p1 `2 )) = (1 / (p2 `1 )) by A136, A180, A203, SPPOL_2: 1;

                

                then ((p1 `1 ) / (p1 `2 )) = ((1 / (p2 `1 )) * (p1 `2 )) by A188, XCMPLX_1: 87

                .= ((p1 `2 ) / (p2 `1 ));

                then

                 A205: (((p2 `2 ) / (p2 `1 )) * ((p1 `1 ) / (p1 `2 ))) = 1 by A197, A188, A204, XCMPLX_1: 112;

                then

                 A206: (p1 `1 ) <> 0 ;

                ((((p2 `2 ) / (p2 `1 )) * ((p1 `1 ) / (p1 `2 ))) * (p1 `2 )) = (p1 `2 ) by A205;

                then (((p2 `2 ) / (p2 `1 )) * (((p1 `1 ) / (p1 `2 )) * (p1 `2 ))) = (p1 `2 );

                then (((p2 `2 ) / (p2 `1 )) * (p1 `1 )) = (p1 `2 ) by A188, XCMPLX_1: 87;

                then

                 A207: ((p2 `2 ) / (p2 `1 )) = ((p1 `2 ) / (p1 `1 )) by A206, XCMPLX_1: 89;

                 A208:

                now

                  per cases by A199;

                    case

                     A209: (p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 );

                    then ( - (p2 `1 )) <= (p2 `1 ) by XXREAL_0: 2;

                    then (p2 `1 ) >= 0 ;

                    then ((p2 `2 ) / (p2 `1 )) <= ((p2 `1 ) / (p2 `1 )) by A209, XREAL_1: 72;

                    hence ((p2 `2 ) / (p2 `1 )) <= 1 by A197, XCMPLX_1: 60;

                  end;

                    case

                     A210: (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 ));

                    then ( - (p2 `1 )) >= (p2 `1 ) by XXREAL_0: 2;

                    then (p2 `1 ) <= 0 ;

                    then ((p2 `2 ) / (p2 `1 )) <= ((p2 `1 ) / (p2 `1 )) by A210, XREAL_1: 73;

                    hence ((p2 `2 ) / (p2 `1 )) <= 1 by A197, XCMPLX_1: 60;

                  end;

                end;

                now

                  per cases ;

                    case 0 <= ((p2 `2 ) / (p2 `1 ));

                    then

                     A211: (p1 `1 ) > 0 & (p1 `2 ) >= 0 or (p1 `1 ) < 0 & (p1 `2 ) <= 0 by A205, A206;

                    now

                      assume ((p1 `1 ) / (p1 `2 )) <> 1;

                      then ((p1 `1 ) / (p1 `2 )) < 1 by A190, XXREAL_0: 1;

                      hence contradiction by A205, A208, A211, XREAL_1: 162;

                    end;

                    then (p1 `1 ) = (1 * (p1 `2 )) by A188, XCMPLX_1: 87;

                    

                    then (((p2 `2 ) / (p2 `1 )) * (p2 `1 )) = (1 * (p2 `1 )) by A188, A207, XCMPLX_1: 60

                    .= (p2 `1 );

                    then (p2 `2 ) = (p2 `1 ) by A197, XCMPLX_1: 87;

                    hence contradiction by A187, A199;

                  end;

                    case 0 > ((p2 `2 ) / (p2 `1 ));

                    then

                     A212: (p1 `1 ) < 0 & (p1 `2 ) > 0 or (p1 `1 ) > 0 & (p1 `2 ) < 0 by A207, XREAL_1: 143;

                    now

                      assume ((p1 `1 ) / (p1 `2 )) <> ( - 1);

                      then ( - 1) < ((p1 `1 ) / (p1 `2 )) by A193, XXREAL_0: 1;

                      hence contradiction by A205, A200, A212, XREAL_1: 166;

                    end;

                    

                    then (p1 `1 ) = (( - 1) * (p1 `2 )) by A188, XCMPLX_1: 87

                    .= ( - (p1 `2 ));

                    then ( - (p1 `1 )) = (p1 `2 );

                    then ((p2 `2 ) / (p2 `1 )) = ( - 1) by A206, A207, XCMPLX_1: 197;

                    then (p2 `2 ) = (( - 1) * (p2 `1 )) by A197, XCMPLX_1: 87;

                    then ( - (p2 `2 )) = (p2 `1 );

                    hence contradiction by A187, A199;

                  end;

                end;

                hence contradiction;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      then

       A213: Out_In_Sq is one-to-one by FUNCT_1:def 4;

      

       A214: for s be Point of ( TOP-REAL 2) st s in Kb holds ( Out_In_Sq . s) = s

      proof

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

        assume t in Kb;

        then

         A215: ex p4 be Point of ( TOP-REAL 2) st p4 = t & (( - 1) = (p4 `1 ) & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1 or (p4 `1 ) = 1 & ( - 1) <= (p4 `2 ) & (p4 `2 ) <= 1 or ( - 1) = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1 or 1 = (p4 `2 ) & ( - 1) <= (p4 `1 ) & (p4 `1 ) <= 1) by A1;

        then

         A216: t <> ( 0. ( TOP-REAL 2)) by EUCLID: 52, EUCLID: 54;

        

         A217: not t = ( 0. ( TOP-REAL 2)) by A215, EUCLID: 52, EUCLID: 54;

        now

          per cases ;

            case

             A218: (t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 ));

            then

             A219: ( Out_In_Sq . t) = |[(1 / (t `1 )), (((t `2 ) / (t `1 )) / (t `1 ))]| by A217, Def1;

            

             A220: 1 <= (t `1 ) & (t `1 ) >= ( - 1) or 1 >= (t `1 ) & ( - 1) >= ( - ( - (t `1 ))) by A215, A218, XREAL_1: 24;

            now

              per cases by A215, A220, XXREAL_0: 1;

                case (t `1 ) = 1;

                hence thesis by A219, EUCLID: 53;

              end;

                case (t `1 ) = ( - 1);

                hence thesis by A219, EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

            case

             A221: not ((t `2 ) <= (t `1 ) & ( - (t `1 )) <= (t `2 ) or (t `2 ) >= (t `1 ) & (t `2 ) <= ( - (t `1 )));

            then

             A222: ( Out_In_Sq . t) = |[(((t `1 ) / (t `2 )) / (t `2 )), (1 / (t `2 ))]| by A216, Def1;

            now

              per cases by A215, A221;

                case (t `2 ) = 1;

                hence thesis by A222, EUCLID: 53;

              end;

                case (t `2 ) = ( - 1);

                hence thesis by A222, EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = Out_In_Sq & h is continuous by A2, Th40;

      hence thesis by A213, A4, A72, A214;

    end;

    theorem :: JGRAPH_2:42

    

     Th42: for f,g be Function of I[01] , ( TOP-REAL 2), K0 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 & K0 = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & ((f . O) `1 ) = ( - 1) & ((f . I) `1 ) = 1 & ( - 1) <= ((f . O) `2 ) & ((f . O) `2 ) <= 1 & ( - 1) <= ((f . I) `2 ) & ((f . I) `2 ) <= 1 & ((g . O) `2 ) = ( - 1) & ((g . I) `2 ) = 1 & ( - 1) <= ((g . O) `1 ) & ((g . O) `1 ) <= 1 & ( - 1) <= ((g . I) `1 ) & ((g . I) `1 ) <= 1 & ( rng f) misses K0 & ( rng g) misses K0 holds ( rng f) meets ( rng g)

    proof

      reconsider B = {( 0. ( TOP-REAL 2))} as Subset of ( TOP-REAL 2);

      

       A1: (B ` ) <> {} by Th9;

      reconsider W = (B ` ) as non empty Subset of ( TOP-REAL 2) by Th9;

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

      

       A2: the carrier of (( TOP-REAL 2) | (B ` )) = ( [#] (( TOP-REAL 2) | (B ` )))

      .= (B ` ) by PRE_TOPC:def 5;

      reconsider Kb = { q : P[q] } as Subset of ( TOP-REAL 2) from TopSubset;

      let f,g be Function of I[01] , ( TOP-REAL 2), K0 be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

      

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

      assume

       A4: O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & ((f . O) `1 ) = ( - 1) & ((f . I) `1 ) = 1 & ( - 1) <= ((f . O) `2 ) & ((f . O) `2 ) <= 1 & ( - 1) <= ((f . I) `2 ) & ((f . I) `2 ) <= 1 & ((g . O) `2 ) = ( - 1) & ((g . I) `2 ) = 1 & ( - 1) <= ((g . O) `1 ) & ((g . O) `1 ) <= 1 & ( - 1) <= ((g . I) `1 ) & ((g . I) `1 ) <= 1 & (( rng f) /\ K0) = {} & (( rng g) /\ K0) = {} ;

      then

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

       A5: h is continuous and

       A6: h is one-to-one and for t be Point of ( TOP-REAL 2) st t in K0 & t <> ( 0. ( TOP-REAL 2)) holds not (h . t) in (K0 \/ Kb) and

       A7: for r be Point of ( TOP-REAL 2) st not r in (K0 \/ Kb) holds (h . r) in K0 and

       A8: for s be Point of ( TOP-REAL 2) st s in Kb holds (h . s) = s by Th41;

      ( rng f) c= (B ` )

      proof

        let x be object;

        assume

         A9: x in ( rng f);

        now

          assume x in B;

          then

           A10: x = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

          (( 0. ( TOP-REAL 2)) `1 ) = 0 & (( 0. ( TOP-REAL 2)) `2 ) = 0 by EUCLID: 52, EUCLID: 54;

          then ( 0. ( TOP-REAL 2)) in K0 by A4;

          hence contradiction by A4, A9, A10, XBOOLE_0:def 4;

        end;

        then x in (the carrier of ( TOP-REAL 2) \ B) by A9, XBOOLE_0:def 5;

        hence thesis by SUBSET_1:def 4;

      end;

      then

       A11: ex w be Function of I[01] , ( TOP-REAL 2) st w is continuous & w = (h * f) by A4, A5, A1, Th12;

      then

      reconsider d1 = (h * f) as Function of I[01] , ( TOP-REAL 2);

      the carrier of (( TOP-REAL 2) | W) <> {} ;

      then

       A12: ( dom h) = the carrier of (( TOP-REAL 2) | (B ` )) by FUNCT_2:def 1;

      ( rng g) c= (B ` )

      proof

        let x be object;

        assume

         A13: x in ( rng g);

        now

          assume x in B;

          then

           A14: x = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

          ( 0. ( TOP-REAL 2)) in K0 by A4, Th3;

          hence contradiction by A4, A13, A14, XBOOLE_0:def 4;

        end;

        then x in (the carrier of ( TOP-REAL 2) \ B) by A13, XBOOLE_0:def 5;

        hence thesis by SUBSET_1:def 4;

      end;

      then

       A15: ex w2 be Function of I[01] , ( TOP-REAL 2) st w2 is continuous & w2 = (h * g) by A4, A5, A1, Th12;

      then

      reconsider d2 = (h * g) as Function of I[01] , ( TOP-REAL 2);

      

       A16: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

       A17: for r be Point of I[01] holds ( - 1) <= ((d1 . r) `1 ) & ((d1 . r) `1 ) <= 1 & ( - 1) <= ((d2 . r) `1 ) & ((d2 . r) `1 ) <= 1 & ( - 1) <= ((d1 . r) `2 ) & ((d1 . r) `2 ) <= 1 & ( - 1) <= ((d2 . r) `2 ) & ((d2 . r) `2 ) <= 1

      proof

        let r be Point of I[01] ;

        

         A18: (g . r) in Kb implies (d2 . r) in (K0 \/ Kb)

        proof

          

           A19: (d2 . r) = (h . (g . r)) by A16, FUNCT_1: 13;

          assume

           A20: (g . r) in Kb;

          then (h . (g . r)) = (g . r) by A8;

          hence thesis by A20, A19, XBOOLE_0:def 3;

        end;

        (f . r) in ( rng f) by A3, FUNCT_1: 3;

        then

         A21: not (f . r) in K0 by A4, XBOOLE_0:def 4;

        

         A22: not (f . r) in Kb implies (d1 . r) in (K0 \/ Kb)

        proof

          assume not (f . r) in Kb;

          then not (f . r) in (K0 \/ Kb) by A21, XBOOLE_0:def 3;

          then

           A23: (h . (f . r)) in K0 by A7;

          (d1 . r) = (h . (f . r)) by A3, FUNCT_1: 13;

          hence thesis by A23, XBOOLE_0:def 3;

        end;

        (g . r) in ( rng g) by A16, FUNCT_1: 3;

        then

         A24: not (g . r) in K0 by A4, XBOOLE_0:def 4;

        

         A25: not (g . r) in Kb implies (d2 . r) in (K0 \/ Kb)

        proof

          assume not (g . r) in Kb;

          then not (g . r) in (K0 \/ Kb) by A24, XBOOLE_0:def 3;

          then

           A26: (h . (g . r)) in K0 by A7;

          (d2 . r) = (h . (g . r)) by A16, FUNCT_1: 13;

          hence thesis by A26, XBOOLE_0:def 3;

        end;

        

         A27: (f . r) in Kb implies (d1 . r) in (K0 \/ Kb)

        proof

          

           A28: (d1 . r) = (h . (f . r)) by A3, FUNCT_1: 13;

          assume

           A29: (f . r) in Kb;

          then (h . (f . r)) = (f . r) by A8;

          hence thesis by A29, A28, XBOOLE_0:def 3;

        end;

        now

          per cases by A22, A27, A25, A18, XBOOLE_0:def 3;

            case (d1 . r) in K0 & (d2 . r) in K0;

            then (ex p be Point of ( TOP-REAL 2) st p = (d1 . r) & ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1) & ex q be Point of ( TOP-REAL 2) st q = (d2 . r) & ( - 1) < (q `1 ) & (q `1 ) < 1 & ( - 1) < (q `2 ) & (q `2 ) < 1 by A4;

            hence thesis;

          end;

            case (d1 . r) in K0 & (d2 . r) in Kb;

            then (ex p be Point of ( TOP-REAL 2) st p = (d1 . r) & ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1) & ex q be Point of ( TOP-REAL 2) st q = (d2 . r) & (( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1) by A4;

            hence thesis;

          end;

            case (d1 . r) in Kb & (d2 . r) in K0;

            then (ex p be Point of ( TOP-REAL 2) st p = (d2 . r) & ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1) & ex q be Point of ( TOP-REAL 2) st q = (d1 . r) & (( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1) by A4;

            hence thesis;

          end;

            case (d1 . r) in Kb & (d2 . r) in Kb;

            then (ex p be Point of ( TOP-REAL 2) st p = (d2 . r) & (( - 1) = (p `1 ) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1)) & ex q be Point of ( TOP-REAL 2) st q = (d1 . r) & (( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1);

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      (f . I) in Kb by A4;

      then (h . (f . I)) = (f . I) by A8;

      then

       A30: ((d1 . I) `1 ) = 1 by A4, A3, FUNCT_1: 13;

      (f . O) in Kb by A4;

      then (h . (f . O)) = (f . O) by A8;

      then

       A31: ((d1 . O) `1 ) = ( - 1) by A4, A3, FUNCT_1: 13;

      (g . I) in Kb by A4;

      then (h . (g . I)) = (g . I) by A8;

      then

       A32: ((d2 . I) `2 ) = 1 by A4, A16, FUNCT_1: 13;

      (g . O) in Kb by A4;

      then (h . (g . O)) = (g . O) by A8;

      then

       A33: ((d2 . O) `2 ) = ( - 1) by A4, A16, FUNCT_1: 13;

      set s = the Element of (( rng d1) /\ ( rng d2));

      d1 is one-to-one & d2 is one-to-one by A4, A6, FUNCT_1: 24;

      then ( rng d1) meets ( rng d2) by A4, A11, A15, A31, A30, A33, A32, A17, JGRAPH_1: 47;

      then

       A34: (( rng d1) /\ ( rng d2)) <> {} ;

      then s in ( rng d1) by XBOOLE_0:def 4;

      then

      consider t1 be object such that

       A35: t1 in ( dom d1) and

       A36: s = (d1 . t1) by FUNCT_1:def 3;

      

       A37: (f . t1) in ( rng f) by A3, A35, FUNCT_1: 3;

      s in ( rng d2) by A34, XBOOLE_0:def 4;

      then

      consider t2 be object such that

       A38: t2 in ( dom d2) and

       A39: s = (d2 . t2) by FUNCT_1:def 3;

      (h . (f . t1)) = (d1 . t1) by A35, FUNCT_1: 12;

      then

       A40: (h . (f . t1)) = (h . (g . t2)) by A36, A38, A39, FUNCT_1: 12;

      ( rng g) c= (the carrier of ( TOP-REAL 2) \ B)

      proof

        let e be object;

        assume

         A41: e in ( rng g);

        now

          assume e in B;

          then

           A42: e = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

          ( 0. ( TOP-REAL 2)) in { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } by Th3;

          hence contradiction by A4, A41, A42, XBOOLE_0:def 4;

        end;

        hence thesis by A41, XBOOLE_0:def 5;

      end;

      then

       A43: ( rng g) c= the carrier of (( TOP-REAL 2) | (B ` )) by A2, SUBSET_1:def 4;

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

      then

       A44: (g . t2) in ( rng g) by A38, FUNCT_1: 3;

      ( rng f) c= (the carrier of ( TOP-REAL 2) \ B)

      proof

        let e be object;

        assume

         A45: e in ( rng f);

        now

          assume e in B;

          then

           A46: e = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

          ( 0. ( TOP-REAL 2)) in { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } by Th3;

          hence contradiction by A4, A45, A46, XBOOLE_0:def 4;

        end;

        hence thesis by A45, XBOOLE_0:def 5;

      end;

      then ( rng f) c= the carrier of (( TOP-REAL 2) | (B ` )) by A2, SUBSET_1:def 4;

      then (f . t1) = (g . t2) by A6, A43, A40, A12, A37, A44, FUNCT_1:def 4;

      then (( rng f) /\ ( rng g)) <> {} by A37, A44, XBOOLE_0:def 4;

      hence thesis;

    end;

    theorem :: JGRAPH_2:43

    

     Th43: for A,B,C,D be Real, f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st (for t be Point of ( TOP-REAL 2) holds (f . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|) holds f is continuous

    proof

      reconsider h11 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

      set K0 = ( [#] ( TOP-REAL 2));

      let A,B,C,D be Real, f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A1: (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) = the TopStruct of ( TOP-REAL 2) by TSEP_1: 93;

      then

      reconsider h1 = h11 as Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 ;

      h11 is continuous by JORDAN5A: 27;

      then h1 is continuous by A1, PRE_TOPC: 32;

      then

      consider g1 be Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 such that

       A2: for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), r1 be Real st (h1 . p) = r1 holds (g1 . p) = (A * r1) and

       A3: g1 is continuous by Th23;

      reconsider f1 = ( proj1 * f) as Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 by A1, TOPMETR: 17;

      consider g11 be Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 such that

       A4: for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), r1 be Real st (g1 . p) = r1 holds (g11 . p) = (r1 + B) and

       A5: g11 is continuous by A3, Th24;

      reconsider f2 = ( proj2 * f) as Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 by A1, TOPMETR: 17;

      reconsider h11 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

      reconsider h1 = h11 as Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 by A1;

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

      then

       A6: ( dom f1) = ( dom g11) by A1, FUNCT_2:def 1;

      assume

       A7: for t be Point of ( TOP-REAL 2) holds (f . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

      

       A8: for x be object st x in ( dom f1) holds (f1 . x) = (g11 . x)

      proof

        let x be object;

        assume

         A9: x in ( dom f1);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by FUNCT_2:def 1;

        (f1 . x) = ( proj1 . (f . x)) by A9, FUNCT_1: 12;

        

        then

         A10: (f1 . x) = ( proj1 . |[((A * (p `1 )) + B), ((C * (p `2 )) + D)]|) by A7

        .= ((A * (p `1 )) + B) by PSCOMP_1: 65

        .= ((A * ( proj1 . p)) + B) by PSCOMP_1:def 5;

        (A * ( proj1 . p)) = (g1 . p) by A1, A2;

        hence thesis by A1, A4, A10;

      end;

      h11 is continuous by JORDAN5A: 27;

      then h1 is continuous by A1, PRE_TOPC: 32;

      then

      consider g1 be Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 such that

       A11: for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), r1 be Real st (h1 . p) = r1 holds (g1 . p) = (C * r1) and

       A12: g1 is continuous by Th23;

      consider g11 be Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), R^1 such that

       A13: for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), r1 be Real st (g1 . p) = r1 holds (g11 . p) = (r1 + D) and

       A14: g11 is continuous by A12, Th24;

      

       A15: for x be object st x in ( dom f2) holds (f2 . x) = (g11 . x)

      proof

        let x be object;

        assume

         A16: x in ( dom f2);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by FUNCT_2:def 1;

        (f2 . x) = ( proj2 . (f . x)) by A16, FUNCT_1: 12;

        

        then

         A17: (f2 . x) = ( proj2 . |[((A * (p `1 )) + B), ((C * (p `2 )) + D)]|) by A7

        .= ((C * (p `2 )) + D) by PSCOMP_1: 65

        .= ((C * ( proj2 . p)) + D) by PSCOMP_1:def 6;

        (C * ( proj2 . p)) = (g1 . p) by A1, A11;

        hence thesis by A1, A13, A17;

      end;

      reconsider f0 = f as Function of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))), (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by A1;

      

       A18: for x,y,r,s be Real st |[x, y]| in K0 & r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|) holds (f0 . |[x, y]|) = |[r, s]|

      proof

        let x,y,r,s be Real;

        assume that |[x, y]| in K0 and

         A19: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        

         A20: (f . |[x, y]|) is Point of ( TOP-REAL 2);

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

        then ( proj1 . (f0 . |[x, y]|)) = r & ( proj2 . (f0 . |[x, y]|)) = s by A19, FUNCT_1: 13;

        hence thesis by A20, Th8;

      end;

      ( dom f2) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      then ( dom f2) = ( dom g11) by A1, FUNCT_2:def 1;

      then

       A21: f2 is continuous by A14, A15, FUNCT_1: 2;

      f1 is continuous by A5, A6, A8, FUNCT_1: 2;

      then f0 is continuous by A21, A18, Th35;

      hence f is continuous by A1, PRE_TOPC: 34;

    end;

    definition

      let A,B,C,D be Real;

      :: JGRAPH_2:def2

      func AffineMap (A,B,C,D) -> Function of ( TOP-REAL 2), ( TOP-REAL 2) means

      : Def2: for t be Point of ( TOP-REAL 2) holds (it . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

      existence

      proof

        defpred P[ object, object] means for t be Point of ( TOP-REAL 2) st t = $1 holds $2 = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

        

         A1: for x be object st x in the carrier of ( TOP-REAL 2) holds ex y be object st P[x, y]

        proof

          let x be object;

          assume x in the carrier of ( TOP-REAL 2);

          then

          reconsider t2 = x as Point of ( TOP-REAL 2);

          reconsider y2 = |[((A * (t2 `1 )) + B), ((C * (t2 `2 )) + D)]| as set;

          for t be Point of ( TOP-REAL 2) st t = x holds y2 = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

          hence thesis;

        end;

        ex ff be Function st ( dom ff) = the carrier of ( TOP-REAL 2) & for x be object st x in the carrier of ( TOP-REAL 2) holds P[x, (ff . x)] from CLASSES1:sch 1( A1);

        then

        consider ff be Function such that

         A2: ( dom ff) = the carrier of ( TOP-REAL 2) and

         A3: for x be object st x in the carrier of ( TOP-REAL 2) holds for t be Point of ( TOP-REAL 2) st t = x holds (ff . x) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

        for x be object st x in the carrier of ( TOP-REAL 2) holds (ff . x) in the carrier of ( TOP-REAL 2)

        proof

          let x be object;

          assume x in the carrier of ( TOP-REAL 2);

          then

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

          (ff . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by A3;

          hence thesis;

        end;

        then

        reconsider ff as Function of ( TOP-REAL 2), ( TOP-REAL 2) by A2, FUNCT_2: 3;

        take ff;

        thus thesis by A3;

      end;

      uniqueness

      proof

        let m1,m2 be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

         A4: for t be Point of ( TOP-REAL 2) holds (m1 . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| and

         A5: for t be Point of ( TOP-REAL 2) holds (m2 . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]|;

        for x be Point of ( TOP-REAL 2) holds (m1 . x) = (m2 . x)

        proof

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

          

          thus (m1 . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by A4

          .= (m2 . t) by A5;

        end;

        hence m1 = m2 by FUNCT_2: 63;

      end;

    end

    registration

      let a,b,c,d be Real;

      cluster ( AffineMap (a,b,c,d)) -> continuous;

      coherence

      proof

        for t be Point of ( TOP-REAL 2) holds (( AffineMap (a,b,c,d)) . t) = |[((a * (t `1 )) + b), ((c * (t `2 )) + d)]| by Def2;

        hence thesis by Th43;

      end;

    end

    theorem :: JGRAPH_2:44

    

     Th44: for A,B,C,D be Real st A > 0 & C > 0 holds ( AffineMap (A,B,C,D)) is one-to-one

    proof

      let A,B,C,D be Real such that

       A1: A > 0 and

       A2: C > 0 ;

      set ff = ( AffineMap (A,B,C,D));

      for x1,x2 be object st x1 in ( dom ff) & x2 in ( dom ff) & (ff . x1) = (ff . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A3: x1 in ( dom ff) and

         A4: x2 in ( dom ff) and

         A5: (ff . x1) = (ff . x2);

        reconsider p2 = x2 as Point of ( TOP-REAL 2) by A4;

        reconsider p1 = x1 as Point of ( TOP-REAL 2) by A3;

        

         A6: (ff . x1) = |[((A * (p1 `1 )) + B), ((C * (p1 `2 )) + D)]| & (ff . x2) = |[((A * (p2 `1 )) + B), ((C * (p2 `2 )) + D)]| by Def2;

        then ((A * (p1 `1 )) + B) = ((A * (p2 `1 )) + B) by A5, SPPOL_2: 1;

        then (p1 `1 ) = ((A * (p2 `1 )) / A) by A1, XCMPLX_1: 89;

        then

         A7: (p1 `1 ) = (p2 `1 ) by A1, XCMPLX_1: 89;

        ((C * (p1 `2 )) + D) = ((C * (p2 `2 )) + D) by A5, A6, SPPOL_2: 1;

        then (p1 `2 ) = ((C * (p2 `2 )) / C) by A2, XCMPLX_1: 89;

        hence thesis by A2, A7, TOPREAL3: 6, XCMPLX_1: 89;

      end;

      hence thesis by FUNCT_1:def 4;

    end;

    theorem :: JGRAPH_2:45

    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 & ((f . O) `1 ) = a & ((f . I) `1 ) = b & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & c <= ((f . I) `2 ) & ((f . I) `2 ) <= d & ((g . O) `2 ) = c & ((g . I) `2 ) = d & a <= ((g . O) `1 ) & ((g . O) `1 ) <= b & a <= ((g . I) `1 ) & ((g . I) `1 ) <= b & a < b & c < d & not (ex r be Point of I[01] st a < ((f . r) `1 ) & ((f . r) `1 ) < b & c < ((f . r) `2 ) & ((f . r) `2 ) < d) & not (ex r be Point of I[01] st a < ((g . r) `1 ) & ((g . r) `1 ) < b & c < ((g . r) `2 ) & ((g . r) `2 ) < d) holds ( rng f) meets ( rng g)

    proof

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

      reconsider K0 = { p : P[p] } as Subset of ( TOP-REAL 2) from TopSubset;

      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 and

       A2: f is continuous one-to-one & g is continuous one-to-one and

       A3: ((f . O) `1 ) = a and

       A4: ((f . I) `1 ) = b and

       A5: c <= ((f . O) `2 ) and

       A6: ((f . O) `2 ) <= d and

       A7: c <= ((f . I) `2 ) and

       A8: ((f . I) `2 ) <= d and

       A9: ((g . O) `2 ) = c and

       A10: ((g . I) `2 ) = d and

       A11: a <= ((g . O) `1 ) and

       A12: ((g . O) `1 ) <= b and

       A13: a <= ((g . I) `1 ) and

       A14: ((g . I) `1 ) <= b and

       A15: a < b and

       A16: c < d and

       A17: not (ex r be Point of I[01] st a < ((f . r) `1 ) & ((f . r) `1 ) < b & c < ((f . r) `2 ) & ((f . r) `2 ) < d) and

       A18: not (ex r be Point of I[01] st a < ((g . r) `1 ) & ((g . r) `1 ) < b & c < ((g . r) `2 ) & ((g . r) `2 ) < d);

      set A = (2 / (b - a)), B = (1 - ((2 * b) / (b - a))), C = (2 / (d - c)), D = (1 - ((2 * d) / (d - c)));

      set ff = ( AffineMap (A,B,C,D));

      reconsider f2 = (ff * f), g2 = (ff * g) as Function of I[01] , ( TOP-REAL 2);

      

       A19: (d - c) > 0 by A16, XREAL_1: 50;

      then

       A20: C > 0 by XREAL_1: 139;

      

       A21: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      

      then

       A22: (g2 . I) = (ff . (g . I)) by FUNCT_1: 13

      .= |[((A * ((g . I) `1 )) + B), ((C * d) + D)]| by A10, Def2;

      

      then

       A23: ((g2 . I) `2 ) = ((C * d) + D) by EUCLID: 52

      .= (((d * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

      .= 1;

      

       A24: (g2 . O) = (ff . (g . O)) by A21, FUNCT_1: 13

      .= |[((A * ((g . O) `1 )) + B), ((C * c) + D)]| by A9, Def2;

      

      then

       A25: ((g2 . O) `2 ) = (((2 / (d - c)) * c) + (1 - ((2 * d) / (d - c)))) by EUCLID: 52

      .= (((c * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

      .= (((c * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c)))) by A19, XCMPLX_1: 60

      .= (((c * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)))

      .= (((c * 2) + ((d - c) - (2 * d))) / (d - c))

      .= (( - (d - c)) / (d - c))

      .= ( - ((d - c) / (d - c)))

      .= ( - 1) by A19, XCMPLX_1: 60;

      

       A26: (b - a) > 0 by A15, XREAL_1: 50;

      

       A27: ( - 1) <= ((g2 . O) `1 ) & ((g2 . O) `1 ) <= 1 & ( - 1) <= ((g2 . I) `1 ) & ((g2 . I) `1 ) <= 1

      proof

        reconsider s1 = ((g . I) `1 ) as Real;

        reconsider s0 = ((g . O) `1 ) as Real;

        

         A28: ((a - b) / (b - a)) = (( - (b - a)) / (b - a))

        .= ( - ((b - a) / (b - a)))

        .= ( - 1) by A26, XCMPLX_1: 60;

        

         A29: ((g2 . I) `1 ) = ((A * s1) + B) by A22, EUCLID: 52

        .= (((s1 * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

        .= (((s1 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a)))) by A26, XCMPLX_1: 60

        .= (((s1 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)))

        .= (((s1 * 2) + ((b - a) - (2 * b))) / (b - a))

        .= ((((s1 - b) + (s1 - b)) - (a - b)) / (b - a));

        (b - b) >= (s0 - b) by A12, XREAL_1: 9;

        then (( 0 + (b - b)) - (a - b)) >= (((s0 - b) + (s0 - b)) - (a - b)) by XREAL_1: 9;

        then

         A30: ((b - a) / (b - a)) >= ((((s0 - b) + (s0 - b)) - (a - b)) / (b - a)) by A26, XREAL_1: 72;

        (b - b) >= (s1 - b) by A14, XREAL_1: 9;

        then

         A31: (( 0 + (b - b)) - (a - b)) >= (((s1 - b) + (s1 - b)) - (a - b)) by XREAL_1: 9;

        (a - b) <= (s1 - b) by A13, XREAL_1: 9;

        then ((a - b) + (a - b)) <= ((s1 - b) + (s1 - b)) by XREAL_1: 7;

        then

         A32: (((a - b) + (a - b)) - (a - b)) <= (((s1 - b) + (s1 - b)) - (a - b)) by XREAL_1: 9;

        (a - b) <= (s0 - b) by A11, XREAL_1: 9;

        then ((a - b) + (a - b)) <= ((s0 - b) + (s0 - b)) by XREAL_1: 7;

        then

         A33: (((a - b) + (a - b)) - (a - b)) <= (((s0 - b) + (s0 - b)) - (a - b)) by XREAL_1: 9;

        ((g2 . O) `1 ) = ((A * s0) + B) by A24, EUCLID: 52

        .= (((s0 * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

        .= (((s0 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a)))) by A26, XCMPLX_1: 60

        .= (((s0 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)))

        .= (((s0 * 2) + ((b - a) - (2 * b))) / (b - a))

        .= ((((s0 - b) + (s0 - b)) - (a - b)) / (b - a));

        hence thesis by A26, A33, A28, A30, A29, A32, A31, XREAL_1: 72;

      end;

       A34:

      now

        assume ( rng f2) meets K0;

        then

        consider x be object such that

         A35: x in ( rng f2) and

         A36: x in K0 by XBOOLE_0: 3;

        reconsider q = x as Point of ( TOP-REAL 2) by A35;

        consider p such that

         A37: p = q and

         A38: ( - 1) < (p `1 ) and

         A39: (p `1 ) < 1 and

         A40: ( - 1) < (p `2 ) and

         A41: (p `2 ) < 1 by A36;

        consider z be object such that

         A42: z in ( dom f2) and

         A43: x = (f2 . z) by A35, FUNCT_1:def 3;

        reconsider u = z as Point of I[01] by A42;

        reconsider t = (f . u) as Point of ( TOP-REAL 2);

        

         A44: ((A * (t `1 )) + B) = ((((t `1 ) * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

        .= ((((t `1 ) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a)))) by A26, XCMPLX_1: 60

        .= ((((t `1 ) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)))

        .= ((((t `1 ) * 2) + ((b - a) - (2 * b))) / (b - a))

        .= (((2 * ((t `1 ) - b)) - (a - b)) / (b - a));

        

         A45: (ff . t) = p by A37, A42, A43, FUNCT_1: 12;

        

         A46: ((C * (t `2 )) + D) = ((((t `2 ) * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

        .= ((((t `2 ) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c)))) by A19, XCMPLX_1: 60

        .= ((((t `2 ) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)))

        .= ((((t `2 ) * 2) + ((d - c) - (2 * d))) / (d - c))

        .= (((2 * ((t `2 ) - d)) - (c - d)) / (d - c));

        

         A47: (ff . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by Def2;

        then ( - 1) < ((C * (t `2 )) + D) by A40, A45, EUCLID: 52;

        then (( - 1) * (d - c)) < ((((2 * ((t `2 ) - d)) - (c - d)) / (d - c)) * (d - c)) by A19, A46, XREAL_1: 68;

        then (( - 1) * (d - c)) < ((2 * ((t `2 ) - d)) - (c - d)) by A19, XCMPLX_1: 87;

        then ((( - 1) * (d - c)) + (c - d)) < (((2 * ((t `2 ) - d)) - (c - d)) + (c - d)) by XREAL_1: 8;

        then ((2 * (c - d)) / 2) < ((2 * ((t `2 ) - d)) / 2) by XREAL_1: 74;

        then

         A48: c < (t `2 ) by XREAL_1: 9;

        ((C * (t `2 )) + D) < 1 by A41, A47, A45, EUCLID: 52;

        then (1 * (d - c)) > ((((2 * ((t `2 ) - d)) - (c - d)) / (d - c)) * (d - c)) by A19, A46, XREAL_1: 68;

        then (1 * (d - c)) > ((2 * ((t `2 ) - d)) - (c - d)) by A19, XCMPLX_1: 87;

        then ((1 * (d - c)) + (c - d)) > (((2 * ((t `2 ) - d)) - (c - d)) + (c - d)) by XREAL_1: 8;

        then ( 0 / 2) > ((((t `2 ) - d) * 2) / 2);

        then

         A49: ( 0 + d) > (t `2 ) by XREAL_1: 19;

        ((A * (t `1 )) + B) < 1 by A39, A47, A45, EUCLID: 52;

        then (1 * (b - a)) > ((((2 * ((t `1 ) - b)) - (a - b)) / (b - a)) * (b - a)) by A26, A44, XREAL_1: 68;

        then (1 * (b - a)) > ((2 * ((t `1 ) - b)) - (a - b)) by A26, XCMPLX_1: 87;

        then ((1 * (b - a)) + (a - b)) > (((2 * ((t `1 ) - b)) - (a - b)) + (a - b)) by XREAL_1: 8;

        then ( 0 / 2) > ((((t `1 ) - b) * 2) / 2);

        then

         A50: ( 0 + b) > (t `1 ) by XREAL_1: 19;

        ( - 1) < ((A * (t `1 )) + B) by A38, A47, A45, EUCLID: 52;

        then (( - 1) * (b - a)) < ((((2 * ((t `1 ) - b)) - (a - b)) / (b - a)) * (b - a)) by A26, A44, XREAL_1: 68;

        then (( - 1) * (b - a)) < ((2 * ((t `1 ) - b)) - (a - b)) by A26, XCMPLX_1: 87;

        then ((( - 1) * (b - a)) + (a - b)) < (((2 * ((t `1 ) - b)) - (a - b)) + (a - b)) by XREAL_1: 8;

        then ((2 * (a - b)) / 2) < ((2 * ((t `1 ) - b)) / 2) by XREAL_1: 74;

        then a < (t `1 ) by XREAL_1: 9;

        hence contradiction by A17, A50, A48, A49;

      end;

      

       A51: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      

      then

       A52: (f2 . I) = (ff . (f . I)) by FUNCT_1: 13

      .= |[((A * b) + B), ((C * ((f . I) `2 )) + D)]| by A4, Def2;

      

      then

       A53: ((f2 . I) `1 ) = ((A * b) + B) by EUCLID: 52

      .= (((b * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

      .= 1;

      

       A54: (f2 . O) = (ff . (f . O)) by A51, FUNCT_1: 13

      .= |[((A * a) + B), ((C * ((f . O) `2 )) + D)]| by A3, Def2;

      

      then

       A55: ((f2 . O) `1 ) = ((A * a) + B) by EUCLID: 52

      .= (((a * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

      .= (((a * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a)))) by A26, XCMPLX_1: 60

      .= (((a * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)))

      .= (((a * 2) + ((b - a) - (2 * b))) / (b - a))

      .= (( - (b - a)) / (b - a))

      .= ( - ((b - a) / (b - a)))

      .= ( - 1) by A26, XCMPLX_1: 60;

       A56:

      now

        assume ( rng g2) meets K0;

        then

        consider x be object such that

         A57: x in ( rng g2) and

         A58: x in K0 by XBOOLE_0: 3;

        reconsider q = x as Point of ( TOP-REAL 2) by A57;

        consider p such that

         A59: p = q and

         A60: ( - 1) < (p `1 ) and

         A61: (p `1 ) < 1 and

         A62: ( - 1) < (p `2 ) and

         A63: (p `2 ) < 1 by A58;

        consider z be object such that

         A64: z in ( dom g2) and

         A65: x = (g2 . z) by A57, FUNCT_1:def 3;

        reconsider u = z as Point of I[01] by A64;

        reconsider t = (g . u) as Point of ( TOP-REAL 2);

        

         A66: ((A * (t `1 )) + B) = ((((t `1 ) * 2) / (b - a)) + (1 - ((2 * b) / (b - a))))

        .= ((((t `1 ) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a)))) by A26, XCMPLX_1: 60

        .= ((((t `1 ) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)))

        .= ((((t `1 ) * 2) + ((b - a) - (2 * b))) / (b - a))

        .= (((2 * ((t `1 ) - b)) - (a - b)) / (b - a));

        

         A67: (ff . t) = p by A59, A64, A65, FUNCT_1: 12;

        

         A68: ((C * (t `2 )) + D) = ((((t `2 ) * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

        .= ((((t `2 ) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c)))) by A19, XCMPLX_1: 60

        .= ((((t `2 ) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)))

        .= ((((t `2 ) * 2) + ((d - c) - (2 * d))) / (d - c))

        .= (((2 * ((t `2 ) - d)) - (c - d)) / (d - c));

        

         A69: (ff . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by Def2;

        then ( - 1) < ((C * (t `2 )) + D) by A62, A67, EUCLID: 52;

        then (( - 1) * (d - c)) < ((((2 * ((t `2 ) - d)) - (c - d)) / (d - c)) * (d - c)) by A19, A68, XREAL_1: 68;

        then (( - 1) * (d - c)) < ((2 * ((t `2 ) - d)) - (c - d)) by A19, XCMPLX_1: 87;

        then ((( - 1) * (d - c)) + (c - d)) < (((2 * ((t `2 ) - d)) - (c - d)) + (c - d)) by XREAL_1: 8;

        then ((2 * (c - d)) / 2) < ((2 * ((t `2 ) - d)) / 2) by XREAL_1: 74;

        then

         A70: c < (t `2 ) by XREAL_1: 9;

        ((C * (t `2 )) + D) < 1 by A63, A69, A67, EUCLID: 52;

        then (1 * (d - c)) > ((((2 * ((t `2 ) - d)) - (c - d)) / (d - c)) * (d - c)) by A19, A68, XREAL_1: 68;

        then (1 * (d - c)) > ((2 * ((t `2 ) - d)) - (c - d)) by A19, XCMPLX_1: 87;

        then ((1 * (d - c)) + (c - d)) > (((2 * ((t `2 ) - d)) - (c - d)) + (c - d)) by XREAL_1: 8;

        then ( 0 / 2) > ((((t `2 ) - d) * 2) / 2);

        then

         A71: ( 0 + d) > (t `2 ) by XREAL_1: 19;

        ((A * (t `1 )) + B) < 1 by A61, A69, A67, EUCLID: 52;

        then (1 * (b - a)) > ((((2 * ((t `1 ) - b)) - (a - b)) / (b - a)) * (b - a)) by A26, A66, XREAL_1: 68;

        then (1 * (b - a)) > ((2 * ((t `1 ) - b)) - (a - b)) by A26, XCMPLX_1: 87;

        then ((1 * (b - a)) + (a - b)) > (((2 * ((t `1 ) - b)) - (a - b)) + (a - b)) by XREAL_1: 8;

        then ( 0 / 2) > ((((t `1 ) - b) * 2) / 2);

        then

         A72: ( 0 + b) > (t `1 ) by XREAL_1: 19;

        ( - 1) < ((A * (t `1 )) + B) by A60, A69, A67, EUCLID: 52;

        then (( - 1) * (b - a)) < ((((2 * ((t `1 ) - b)) - (a - b)) / (b - a)) * (b - a)) by A26, A66, XREAL_1: 68;

        then (( - 1) * (b - a)) < ((2 * ((t `1 ) - b)) - (a - b)) by A26, XCMPLX_1: 87;

        then ((( - 1) * (b - a)) + (a - b)) < (((2 * ((t `1 ) - b)) - (a - b)) + (a - b)) by XREAL_1: 8;

        then ((2 * (a - b)) / 2) < ((2 * ((t `1 ) - b)) / 2) by XREAL_1: 74;

        then a < (t `1 ) by XREAL_1: 9;

        hence contradiction by A18, A72, A70, A71;

      end;

      

       A73: ( - 1) <= ((f2 . O) `2 ) & ((f2 . O) `2 ) <= 1 & ( - 1) <= ((f2 . I) `2 ) & ((f2 . I) `2 ) <= 1

      proof

        reconsider s1 = ((f . I) `2 ) as Real;

        reconsider s0 = ((f . O) `2 ) as Real;

        

         A74: ((c - d) / (d - c)) = (( - (d - c)) / (d - c))

        .= ( - ((d - c) / (d - c)))

        .= ( - 1) by A19, XCMPLX_1: 60;

        

         A75: ((f2 . I) `2 ) = ((C * s1) + D) by A52, EUCLID: 52

        .= (((s1 * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

        .= (((s1 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c)))) by A19, XCMPLX_1: 60

        .= (((s1 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)))

        .= (((s1 * 2) + ((d - c) - (2 * d))) / (d - c))

        .= ((((s1 - d) + (s1 - d)) - (c - d)) / (d - c));

        (d - d) >= (s0 - d) by A6, XREAL_1: 9;

        then (( 0 + (d - d)) - (c - d)) >= (((s0 - d) + (s0 - d)) - (c - d)) by XREAL_1: 9;

        then

         A76: ((d - c) / (d - c)) >= ((((s0 - d) + (s0 - d)) - (c - d)) / (d - c)) by A19, XREAL_1: 72;

        (d - d) >= (s1 - d) by A8, XREAL_1: 9;

        then

         A77: (( 0 + (d - d)) - (c - d)) >= (((s1 - d) + (s1 - d)) - (c - d)) by XREAL_1: 9;

        (c - d) <= (s1 - d) by A7, XREAL_1: 9;

        then ((c - d) + (c - d)) <= ((s1 - d) + (s1 - d)) by XREAL_1: 7;

        then

         A78: (((c - d) + (c - d)) - (c - d)) <= (((s1 - d) + (s1 - d)) - (c - d)) by XREAL_1: 9;

        (c - d) <= (s0 - d) by A5, XREAL_1: 9;

        then ((c - d) + (c - d)) <= ((s0 - d) + (s0 - d)) by XREAL_1: 7;

        then

         A79: (((c - d) + (c - d)) - (c - d)) <= (((s0 - d) + (s0 - d)) - (c - d)) by XREAL_1: 9;

        ((f2 . O) `2 ) = ((C * s0) + D) by A54, EUCLID: 52

        .= (((s0 * 2) / (d - c)) + (1 - ((2 * d) / (d - c))))

        .= (((s0 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c)))) by A19, XCMPLX_1: 60

        .= (((s0 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)))

        .= (((s0 * 2) + ((d - c) - (2 * d))) / (d - c))

        .= ((((s0 - d) + (s0 - d)) - (c - d)) / (d - c));

        hence thesis by A19, A79, A74, A76, A75, A78, A77, XREAL_1: 72;

      end;

      set y = the Element of (( rng f2) /\ ( rng g2));

      A > 0 by A26, XREAL_1: 139;

      then

       A80: ff is one-to-one by A20, Th44;

      then f2 is one-to-one & g2 is one-to-one by A2, FUNCT_1: 24;

      then ( rng f2) meets ( rng g2) by A1, A2, A55, A53, A25, A23, A73, A27, A34, A56, Th42;

      then

       A81: (( rng f2) /\ ( rng g2)) <> {} ;

      then y in ( rng f2) by XBOOLE_0:def 4;

      then

      consider x be object such that

       A82: x in ( dom f2) and

       A83: y = (f2 . x) by FUNCT_1:def 3;

      ( dom f2) c= ( dom f) by RELAT_1: 25;

      then

       A84: (f . x) in ( rng f) by A82, FUNCT_1: 3;

      y in ( rng g2) by A81, XBOOLE_0:def 4;

      then

      consider x2 be object such that

       A85: x2 in ( dom g2) and

       A86: y = (g2 . x2) by FUNCT_1:def 3;

      

       A87: y = (ff . (g . x2)) by A85, A86, FUNCT_1: 12;

      ( dom g2) c= ( dom g) by RELAT_1: 25;

      then

       A88: (g . x2) in ( rng g) by A85, FUNCT_1: 3;

      ( dom ff) = the carrier of ( TOP-REAL 2) & y = (ff . (f . x)) by A82, A83, FUNCT_1: 12, FUNCT_2:def 1;

      then (f . x) = (g . x2) by A80, A87, A84, A88, FUNCT_1:def 4;

      then (( rng f) /\ ( rng g)) <> {} by A84, A88, XBOOLE_0:def 4;

      hence thesis;

    end;

    theorem :: JGRAPH_2:46

    { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } is closed Subset of ( TOP-REAL 2) & { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } is closed Subset of ( TOP-REAL 2) by Lm5, Lm8;

    theorem :: JGRAPH_2:47

    { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `1 )) <= (p7 `2 ) } is closed Subset of ( TOP-REAL 2) & { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= ( - (p7 `1 )) } is closed Subset of ( TOP-REAL 2) by Lm11, Lm14;

    theorem :: JGRAPH_2:48

    { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `2 )) <= (p7 `1 ) } is closed Subset of ( TOP-REAL 2) & { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= ( - (p7 `2 )) } is closed Subset of ( TOP-REAL 2) by Lm17, Lm20;