jgraph_3.miz



    begin

    reserve x for Real;

    

     Lm1: ((x ^2 ) + 1) > 0

    proof

      (x ^2 ) >= 0 by XREAL_1: 63;

      hence thesis;

    end;

    

     Lm2: ( dom proj1 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

    

     Lm3: ( dom proj2 ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

    theorem :: JGRAPH_3:1

    for p be Point of ( TOP-REAL 2) holds |.p.| = ( sqrt (((p `1 ) ^2 ) + ((p `2 ) ^2 ))) & ( |.p.| ^2 ) = (((p `1 ) ^2 ) + ((p `2 ) ^2 )) by JGRAPH_1: 29, JGRAPH_1: 30;

    theorem :: JGRAPH_3:2

    for f be Function, B,C be set holds ((f | B) .: C) = (f .: (C /\ B))

    proof

      let f be Function, B,C be set;

      thus ((f | B) .: C) c= (f .: (C /\ B))

      proof

        let x be object;

        assume x in ((f | B) .: C);

        then

        consider y be object such that

         A1: y in ( dom (f | B)) and

         A2: y in C and

         A3: x = ((f | B) . y) by FUNCT_1:def 6;

        

         A4: ((f | B) . y) = (f . y) by A1, FUNCT_1: 47;

        

         A5: ( dom (f | B)) = (( dom f) /\ B) by RELAT_1: 61;

        then y in B by A1, XBOOLE_0:def 4;

        then

         A6: y in (C /\ B) by A2, XBOOLE_0:def 4;

        y in ( dom f) by A1, A5, XBOOLE_0:def 4;

        hence thesis by A3, A6, A4, FUNCT_1:def 6;

      end;

      let x be object;

      assume x in (f .: (C /\ B));

      then

      consider y be object such that

       A7: y in ( dom f) and

       A8: y in (C /\ B) and

       A9: x = (f . y) by FUNCT_1:def 6;

      

       A10: y in C by A8, XBOOLE_0:def 4;

      y in B by A8, XBOOLE_0:def 4;

      then y in (( dom f) /\ B) by A7, XBOOLE_0:def 4;

      then

       A11: y in ( dom (f | B)) by RELAT_1: 61;

      then ((f | B) . y) = (f . y) by FUNCT_1: 47;

      hence thesis by A9, A10, A11, FUNCT_1:def 6;

    end;

    theorem :: JGRAPH_3:3

    

     Th3: for X,Y be non empty TopSpace, p0 be Point of X, D be non empty Subset of X, E be non empty Subset of Y, f be Function of X, Y st (D ` ) = {p0} & (E ` ) = {(f . p0)} & X is T_2 & Y is T_2 & (for p be Point of (X | D) holds (f . p) <> (f . p0)) & (f | D) is continuous Function of (X | D), (Y | E) & (for V be Subset of Y st (f . p0) in V & V is open holds ex W be Subset of X st p0 in W & W is open & (f .: W) c= V) holds f is continuous

    proof

      let X,Y be non empty TopSpace, p0 be Point of X, D be non empty Subset of X, E be non empty Subset of Y, f be Function of X, Y;

      assume that

       A1: (D ` ) = {p0} and

       A2: (E ` ) = {(f . p0)} and

       A3: X is T_2 and

       A4: Y is T_2 and

       A5: for p be Point of (X | D) holds (f . p) <> (f . p0) and

       A6: (f | D) is continuous Function of (X | D), (Y | E) and

       A7: for V be Subset of Y st (f . p0) in V & V is open holds ex W be Subset of X st p0 in W & W is open & (f .: W) c= V;

      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

        

         A8: the carrier of (X | D) = D by PRE_TOPC: 8;

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

        assume that

         A9: (f . p) in V and

         A10: V is open;

        per cases ;

          suppose p = p0;

          hence thesis by A7, A9, A10;

        end;

          suppose

           A11: p <> p0;

          then not p in (D ` ) by A1, TARSKI:def 1;

          then p in (the carrier of X \ (D ` )) by XBOOLE_0:def 5;

          then

           A12: p in ((D ` ) ` ) by SUBSET_1:def 4;

          then (f . p) <> (f . p0) by A5, A8;

          then

          consider G1,G2 be Subset of Y such that

           A13: G1 is open and G2 is open and

           A14: (f . p) in G1 and (f . p0) in G2 and G1 misses G2 by A4, PRE_TOPC:def 10;

          

           A15: ( [#] (X | D)) = D by PRE_TOPC:def 5;

          then

          reconsider p22 = p as Point of (X | D) by A12;

          consider h be Function of (X | D), (Y | E) such that

           A16: h = (f | D) and

           A17: h is continuous by A6;

          

           A18: (h . p) = (f . p) by A12, A16, FUNCT_1: 49;

          

           A19: ( [#] (Y | E)) = E by PRE_TOPC:def 5;

          then

          reconsider V20 = ((G1 /\ V) /\ E) as Subset of (Y | E) by XBOOLE_1: 17;

          (G1 /\ V) is open by A10, A13, TOPS_1: 11;

          then

           A20: V20 is open by A19, TOPS_2: 24;

          (f . p) <> (f . p0) by A5, A12, A15;

          then not (f . p) in (E ` ) by A2, TARSKI:def 1;

          then not (f . p) in (the carrier of Y \ E) by SUBSET_1:def 4;

          then

           A21: (h . p22) in E by A18, XBOOLE_0:def 5;

          (h . p22) in (G1 /\ V) by A9, A14, A18, XBOOLE_0:def 4;

          then (h . p22) in V20 by A21, XBOOLE_0:def 4;

          then

          consider W2 be Subset of (X | D) such that

           A22: p22 in W2 and

           A23: W2 is open and

           A24: (h .: W2) c= V20 by A17, A20, JGRAPH_2: 10;

          consider W3b be Subset of X such that

           A25: W3b is open and

           A26: W2 = (W3b /\ ( [#] (X | D))) by A23, TOPS_2: 24;

          consider H1,H2 be Subset of X such that

           A27: H1 is open and H2 is open and

           A28: p in H1 and

           A29: p0 in H2 and

           A30: H1 misses H2 by A3, A11, PRE_TOPC:def 10;

          p22 in W3b by A22, A26, XBOOLE_0:def 4;

          then

           A31: p in (H1 /\ W3b) by A28, XBOOLE_0:def 4;

          reconsider W3 = (H1 /\ W3b) as Subset of X;

          

           A32: W3 c= W3b by XBOOLE_1: 17;

          

           A33: (f .: W3) c= (h .: W2)

          proof

            let xx be object;

            assume xx in (f .: W3);

            then

            consider yy be object such that

             A34: yy in ( dom f) and

             A35: yy in W3 and

             A36: xx = (f . yy) by FUNCT_1:def 6;

            H2 c= (H1 ` ) by A30, SUBSET_1: 23;

            then (D ` ) c= (H1 ` ) by A1, A29, ZFMISC_1: 31;

            then W3 c= H1 & H1 c= D by SUBSET_1: 12, XBOOLE_1: 17;

            then

             A37: W3 c= D;

            then

             A38: yy in W2 by A15, A26, A32, A35, XBOOLE_0:def 4;

            ( dom h) = (( dom f) /\ D) by A16, RELAT_1: 61;

            then

             A39: yy in ( dom h) by A34, A35, A37, XBOOLE_0:def 4;

            then (h . yy) = (f . yy) by A16, FUNCT_1: 47;

            hence thesis by A36, A39, A38, FUNCT_1:def 6;

          end;

          ((G1 /\ V) /\ E) c= (G1 /\ V) by XBOOLE_1: 17;

          then (G1 /\ V) c= V & (h .: W2) c= (G1 /\ V) by A24, XBOOLE_1: 17;

          then

           A40: (h .: W2) c= V;

          (H1 /\ W3b) is open by A25, A27, TOPS_1: 11;

          hence thesis by A31, A33, A40, XBOOLE_1: 1;

        end;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    begin

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

    definition

      :: JGRAPH_3:def1

      func Sq_Circ -> Function of the carrier of ( TOP-REAL 2), the carrier of ( TOP-REAL 2) means

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

      existence

      proof

        defpred P[ set, set] means (for p be Point of ( TOP-REAL 2) st p = $1 holds (p = ( 0. ( TOP-REAL 2)) implies $2 = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies $2 = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies $2 = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|));

        set BP = the carrier of ( TOP-REAL 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;

          set q = x;

          per cases ;

            suppose q = ( 0. ( TOP-REAL 2));

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

            hence thesis;

          end;

            suppose

             A2: ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ))) & q <> ( 0. ( TOP-REAL 2));

            set r = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

            for p be Point of ( TOP-REAL 2) st p = x holds (p = ( 0. ( TOP-REAL 2)) implies r = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies r = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies r = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A2;

            hence thesis;

          end;

            suppose

             A3: not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ))) & q <> ( 0. ( TOP-REAL 2));

            set r = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

            for p be Point of ( TOP-REAL 2) st p = x holds (p = ( 0. ( TOP-REAL 2)) implies r = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies r = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies r = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A3;

            hence thesis;

          end;

        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

         A4: for x be Element of BP holds for p be Point of ( TOP-REAL 2) st p = x holds (p = ( 0. ( TOP-REAL 2)) implies (h . x) = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h . x) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h . x) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|);

        for p be Point of ( TOP-REAL 2) holds (p = ( 0. ( TOP-REAL 2)) implies (h . p) = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A4;

        hence thesis;

      end;

      uniqueness

      proof

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

        assume that

         A5: for p be Point of ( TOP-REAL 2) holds (p = ( 0. ( TOP-REAL 2)) implies (h1 . p) = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h1 . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h1 . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) and

         A6: for p be Point of ( TOP-REAL 2) holds (p = ( 0. ( TOP-REAL 2)) implies (h2 . p) = p) & (((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h2 . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) & p <> ( 0. ( TOP-REAL 2)) implies (h2 . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|);

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

        proof

          let x be object;

          assume x in the carrier of ( TOP-REAL 2);

          then

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

          per cases ;

            suppose

             A7: q = ( 0. ( TOP-REAL 2));

            then (h1 . q) = q by A5;

            hence thesis by A6, A7;

          end;

            suppose

             A8: ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ))) & q <> ( 0. ( TOP-REAL 2));

            then (h1 . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A5;

            hence thesis by A6, A8;

          end;

            suppose

             A9: not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ))) & q <> ( 0. ( TOP-REAL 2));

            then (h1 . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A5;

            hence thesis by A6, A9;

          end;

        end;

        hence h1 = h2 by FUNCT_2: 12;

      end;

    end

    theorem :: JGRAPH_3:4

    

     Th4: 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 ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) & ( not ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) implies ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|)

    proof

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

      

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

      assume

       A2: p <> ( 0. ( TOP-REAL 2));

      hereby

        assume

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

        now

          per cases by A3;

            case

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

            now

              assume

               A5: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

               A6:

              now

                per cases by A5;

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

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A4, 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 A4, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A6;

                  case (p `1 ) = (p `2 );

                  hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, Def1;

                end;

                  case

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

                  then (p `1 ) <> 0 & ( - (p `1 )) = (p `2 ) by A2, EUCLID: 53, EUCLID: 54;

                  then

                   A8: ((p `2 ) / (p `1 )) = ( - 1) by XCMPLX_1: 197;

                  (p `2 ) <> 0 by A2, A7, EUCLID: 53, EUCLID: 54;

                  then ((p `1 ) / (p `2 )) = ( - 1) by A7, XCMPLX_1: 197;

                  hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, A8, Def1;

                end;

              end;

              hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

            end;

            hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, Def1;

          end;

            case

             A9: (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ));

            now

              assume

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

               A11:

              now

                per cases by A10;

                  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 A9, XXREAL_0: 1;

                end;

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

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A9, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A11;

                  case (p `1 ) = (p `2 );

                  hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A10, Def1;

                end;

                  case

                   A12: (p `1 ) = ( - (p `2 ));

                  then (p `1 ) <> 0 & ( - (p `1 )) = (p `2 ) by A2, EUCLID: 53, EUCLID: 54;

                  then

                   A13: ((p `2 ) / (p `1 )) = ( - 1) by XCMPLX_1: 197;

                  (p `2 ) <> 0 by A2, A12, EUCLID: 53, EUCLID: 54;

                  then ((p `1 ) / (p `2 )) = ( - 1) by A12, XCMPLX_1: 197;

                  hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A10, A13, Def1;

                end;

              end;

              hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

            end;

            hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, Def1;

          end;

        end;

        hence ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

      end;

      

       A14: ( - (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 thesis by A2, A1, A14, Def1;

    end;

    theorem :: JGRAPH_3:5

    

     Th5: for X be non empty TopSpace, f1 be Function of X, R^1 st f1 is continuous & (for q be Point of X holds ex r be Real st (f1 . q) = r & r >= 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) = ( sqrt 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 ex r be Real st (f1 . q) = r & r >= 0 ;

      defpred P[ set, set] means (for r11 be Real st (f1 . $1) = r11 holds $2 = ( sqrt r11));

      

       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 y = ( sqrt r1) as Element of REAL by XREAL_0:def 1;

        take y;

        thus thesis;

      end;

      ex f be Function of the carrier of X, REAL st for x2 be Element of X holds P[x2, (f . x2)] from FUNCT_2:sch 3( A3);

      then

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

       A4: for x2 be Element of X holds for r11 be Real st (f1 . x2) = r11 holds (f . x2) = ( sqrt r11);

      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 r01 be Real such that

         A5: r01 > 0 and

         A6: ].(r - r01), (r + r01).[ c= V by FRECHET: 8;

        set r0 = ( min (r01,1));

        

         A7: r0 > 0 by A5, XXREAL_0: 21;

        

         A8: r0 > 0 by A5, XXREAL_0: 21;

        r0 <= r01 by XXREAL_0: 17;

        then (r - r01) <= (r - r0) & (r + r0) <= (r + r01) by XREAL_1: 6, XREAL_1: 10;

        then ].(r - r0), (r + r0).[ c= ].(r - r01), (r + r01).[ by XXREAL_1: 46;

        then

         A9: ].(r - r0), (r + r0).[ c= V by A6;

        

         A10: ex r8 be Real st (f1 . p) = r8 & r8 >= 0 by A2;

        

         A11: r = ( sqrt r1) by A4;

        then

         A12: r1 = (r ^2 ) by A10, SQUARE_1:def 2;

        

         A13: r >= 0 by A10, A11, SQUARE_1: 17, SQUARE_1: 26;

        then

         A14: (((2 * r) * r0) + (r0 ^2 )) > ( 0 + 0 ) by A8, SQUARE_1: 12, XREAL_1: 8;

        per cases ;

          suppose

           A15: (r - r0) > 0 ;

          set r4 = (r0 * (r - r0));

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

          

           A16: r1 < (r1 + r4) by A8, A15, XREAL_1: 29, XREAL_1: 129;

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

          then

           A17: (f1 . p) in G1 by A16, XXREAL_1: 4;

          G1 is open by JORDAN6: 35;

          then

          consider W1 be Subset of X such that

           A18: p in W1 & W1 is open and

           A19: (f1 .: W1) c= G1 by A1, A17, JGRAPH_2: 10;

          set W = W1;

          

           A20: ((r - ((1 / 2) * r0)) ^2 ) >= 0 & (r0 ^2 ) >= 0 by XREAL_1: 63;

          now

            assume r1 = 0 ;

            then r = 0 by A4, SQUARE_1: 17;

            hence contradiction by A7, A15;

          end;

          then 0 < r by A10, A11, SQUARE_1: 25;

          then

           A21: (r0 * r) > 0 by A8, XREAL_1: 129;

          then ( 0 + (r * r0)) < ((r * r0) + (r * r0)) by XREAL_1: 8;

          then ((r0 * r) - (r0 * r0)) < (((2 * r) * r0) - (r0 * r0)) by XREAL_1: 14;

          then ( - r4) > ( - (((2 * r) * r0) - (r0 ^2 ))) by XREAL_1: 24;

          then (r1 + ( - r4)) > ((r ^2 ) + ( - (((2 * r) * r0) - (r0 ^2 )))) by A12, XREAL_1: 8;

          then ( sqrt (r1 - r4)) > ( sqrt ((r - r0) ^2 )) by SQUARE_1: 27, XREAL_1: 63;

          then

           A22: ( sqrt (r1 - r4)) > (r - r0) by A15, SQUARE_1: 22;

          ( 0 + (r * r0)) < ((r * r0) + (r * r0)) by A21, XREAL_1: 8;

          then ((r0 * r) + 0 ) < (((2 * r) * r0) + (2 * (r0 * r0))) by A8, XREAL_1: 8;

          then (((r0 * r) - (r0 * r0)) + (r0 * r0)) < ((((2 * r) * r0) + (r0 * r0)) + (r0 * r0));

          then ((r0 * r) - (r0 * r0)) < (((2 * r) * r0) + (r0 * r0)) by XREAL_1: 7;

          then (r1 + r4) < ((r ^2 ) + (((2 * r) * r0) + (r0 ^2 ))) by A12, XREAL_1: 8;

          then ( sqrt (r1 + r4)) < ( sqrt ((r + r0) ^2 )) by A10, A8, A15, SQUARE_1: 27;

          then

           A23: (r + r0) > ( sqrt (r1 + r4)) by A13, A7, SQUARE_1: 22;

          

           A24: (r1 - r4) = ((r ^2 ) - ((r0 * r) - (r0 * r0))) by A10, A11, SQUARE_1:def 2

          .= (((r - ((1 / 2) * r0)) ^2 ) + ((3 / 4) * (r0 ^2 )));

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

          proof

            let x be object;

            assume x in (g0 .: W);

            then

            consider z be object such that

             A25: z in ( dom g0) and

             A26: z in W and

             A27: (g0 . z) = x by FUNCT_1:def 6;

            reconsider pz = z as Point of X by A25;

            reconsider aa1 = (f1 . pz) as Real;

            

             A28: ex r9 be Real st (f1 . pz) = r9 & r9 >= 0 by A2;

            pz in the carrier of X;

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

            then

             A29: (f1 . pz) in (f1 .: W1) by A26, FUNCT_1:def 6;

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

            then ( sqrt aa1) < ( sqrt (r1 + r4)) by A28, SQUARE_1: 27;

            then

             A30: ( sqrt aa1) < (r + r0) by A23, XXREAL_0: 2;

            

             A31: (r1 - r4) < aa1 by A19, A29, XXREAL_1: 4;

             A32:

            now

              per cases ;

                case 0 <= (r1 - r4);

                then ( sqrt (r1 - r4)) <= ( sqrt aa1) by A31, SQUARE_1: 26;

                hence (r - r0) < ( sqrt aa1) by A22, XXREAL_0: 2;

              end;

                case 0 > (r1 - r4);

                hence contradiction by A24, A20;

              end;

            end;

            x = ( sqrt aa1) by A4, A27;

            hence thesis by A30, A32, XXREAL_1: 4;

          end;

          hence thesis by A9, A18, XBOOLE_1: 1;

        end;

          suppose

           A33: (r - r0) <= 0 ;

          set r4 = ((((2 * r) * r0) + (r0 ^2 )) / 3);

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

          ((((2 * r) * r0) + (r0 ^2 )) / 3) > 0 by A14, XREAL_1: 139;

          then

           A34: r1 < (r1 + r4) by XREAL_1: 29;

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

          then

           A35: (f1 . p) in G1 by A34, XXREAL_1: 4;

          G1 is open by JORDAN6: 35;

          then

          consider W1 be Subset of X such that

           A36: p in W1 & W1 is open and

           A37: (f1 .: W1) c= G1 by A1, A35, JGRAPH_2: 10;

          set W = W1;

          ((((2 * r) * r0) + (r0 ^2 )) / 3) < (((2 * r) * r0) + (r0 ^2 )) by A14, XREAL_1: 221;

          then (r1 + r4) < ((r ^2 ) + (((2 * r) * r0) + (r0 ^2 ))) by A12, XREAL_1: 8;

          then ( sqrt (r1 + r4)) <= ( sqrt ((r + r0) ^2 )) by A10, A13, A8, SQUARE_1: 26;

          then

           A38: (r + r0) >= ( sqrt (r1 + r4)) by A13, A7, SQUARE_1: 22;

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

          proof

            let x be object;

            assume x in (g0 .: W);

            then

            consider z be object such that

             A39: z in ( dom g0) and

             A40: z in W and

             A41: (g0 . z) = x by FUNCT_1:def 6;

            reconsider pz = z as Point of X by A39;

            reconsider aa1 = (f1 . pz) as Real;

            

             A42: ex r9 be Real st (f1 . pz) = r9 & r9 >= 0 by A2;

            pz in the carrier of X;

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

            then

             A43: (f1 . pz) in (f1 .: W1) by A40, FUNCT_1:def 6;

            then aa1 < (r1 + r4) by A37, XXREAL_1: 4;

            then ( sqrt aa1) < ( sqrt (r1 + r4)) by A42, SQUARE_1: 27;

            then

             A44: ( sqrt aa1) < (r + r0) by A38, XXREAL_0: 2;

            

             A45: (r1 - r4) < aa1 by A37, A43, XXREAL_1: 4;

             A46:

            now

              per cases by A33;

                case (r - r0) = 0 ;

                hence (r - r0) < ( sqrt aa1) by A12, A45, SQUARE_1: 17, SQUARE_1: 27;

              end;

                case (r - r0) < 0 ;

                hence (r - r0) < ( sqrt aa1) by A42, SQUARE_1: 17, SQUARE_1: 26;

              end;

            end;

            x = ( sqrt aa1) by A4, A41;

            hence thesis by A44, A46, XXREAL_1: 4;

          end;

          hence thesis by A9, A36, XBOOLE_1: 1;

        end;

      end;

      then

       A47: g0 is continuous by JGRAPH_2: 10;

      for p be Point of X, r11 be Real st (f1 . p) = r11 holds (g0 . p) = ( sqrt r11) by A4;

      hence thesis by A47;

    end;

    theorem :: JGRAPH_3:6

    

     Th6: 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) ^2 )) & g is continuous

    proof

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

      assume f1 is continuous & f2 is continuous & for q be Point of X holds (f2 . q) <> 0 ;

      then

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

       A1: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = (r1 / r2) and

       A2: g2 is continuous by JGRAPH_2: 27;

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

       A3: for p be Point of X, r1 be Real st (g2 . p) = r1 holds (g3 . p) = (r1 * r1) and

       A4: g3 is continuous by A2, JGRAPH_2: 22;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = ((r1 / r2) ^2 )

      proof

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

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

        then (g2 . p) = (r1 / r2) by A1;

        hence thesis by A3;

      end;

      hence thesis by A4;

    end;

    theorem :: JGRAPH_3:7

    

     Th7: 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) = (1 + ((r1 / r2) ^2 ))) & g is continuous

    proof

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

      assume f1 is continuous & f2 is continuous & for q be Point of X holds (f2 . q) <> 0 ;

      then

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

       A1: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = ((r1 / r2) ^2 ) and

       A2: g2 is continuous by Th6;

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

       A3: for p be Point of X, r1 be Real st (g2 . p) = r1 holds (g3 . p) = (r1 + 1) and

       A4: g3 is continuous by A2, JGRAPH_2: 24;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (1 + ((r1 / r2) ^2 ))

      proof

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

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

        then (g2 . p) = ((r1 / r2) ^2 ) by A1;

        hence thesis by A3;

      end;

      hence thesis by A4;

    end;

    theorem :: JGRAPH_3:8

    

     Th8: 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) = ( sqrt (1 + ((r1 / r2) ^2 )))) & g is continuous

    proof

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

      assume f1 is continuous & f2 is continuous & for q be Point of X holds (f2 . q) <> 0 ;

      then

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

       A1: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = (1 + ((r1 / r2) ^2 )) and

       A2: g2 is continuous by Th7;

      for q be Point of X holds ex r be Real st (g2 . q) = r & r >= 0

      proof

        let q be Point of X;

        reconsider r1 = (f1 . q), r2 = (f2 . q) as Real;

        (1 + ((r1 / r2) ^2 )) > 0 by Lm1;

        hence thesis by A1;

      end;

      then

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

       A3: for p be Point of X, r1 be Real st (g2 . p) = r1 holds (g3 . p) = ( sqrt r1) and

       A4: g3 is continuous by A2, Th5;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = ( sqrt (1 + ((r1 / r2) ^2 )))

      proof

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

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

        then (g2 . p) = (1 + ((r1 / r2) ^2 )) by A1;

        hence thesis by A3;

      end;

      hence thesis by A4;

    end;

    theorem :: JGRAPH_3:9

    

     Th9: 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 / ( sqrt (1 + ((r1 / r2) ^2 ))))) & 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) = ( sqrt (1 + ((r1 / r2) ^2 ))) and

       A4: g2 is continuous by A1, A2, Th8;

      for q be Point of X holds (g2 . q) <> 0

      proof

        let q be Point of X;

        reconsider r1 = (f1 . q), r2 = (f2 . q) as Real;

        ( sqrt (1 + ((r1 / r2) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        hence thesis by A3;

      end;

      then

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

       A5: for p be Point of X, r1,r0 be Real st (f1 . p) = r1 & (g2 . p) = r0 holds (g3 . p) = (r1 / r0) and

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

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (r1 / ( sqrt (1 + ((r1 / r2) ^2 ))))

      proof

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

        assume that

         A7: (f1 . p) = r1 and

         A8: (f2 . p) = r2;

        (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) by A3, A7, A8;

        hence thesis by A5, A7;

      end;

      hence thesis by A6;

    end;

    theorem :: JGRAPH_3:10

    

     Th10: 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) = (r2 / ( sqrt (1 + ((r1 / r2) ^2 ))))) & 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 and

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

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

       A4: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) and

       A5: g2 is continuous by A1, A2, A3, Th8;

      for q be Point of X holds (g2 . q) <> 0

      proof

        let q be Point of X;

        reconsider r1 = (f1 . q), r2 = (f2 . q) as Real;

        ( sqrt (1 + ((r1 / r2) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        hence thesis by A4;

      end;

      then

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

       A6: for p be Point of X, r2,r0 be Real st (f2 . p) = r2 & (g2 . p) = r0 holds (g3 . p) = (r2 / r0) and

       A7: g3 is continuous by A2, A5, JGRAPH_2: 27;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (r2 / ( sqrt (1 + ((r1 / r2) ^2 ))))

      proof

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

        assume that

         A8: (f1 . p) = r1 and

         A9: (f2 . p) = r2;

        (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) by A4, A8, A9;

        hence thesis by A6, A9;

      end;

      hence thesis by A7;

    end;

    

     Lm4: for K1 be non empty Subset of ( TOP-REAL 2) holds for q be Point of (( TOP-REAL 2) | K1) holds (( proj2 | K1) . q) = ( proj2 . q)

    proof

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

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

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

      then q in (( dom proj2 ) /\ K1) by Lm3, XBOOLE_0:def 4;

      hence thesis by FUNCT_1: 48;

    end;

    

     Lm5: for K1 be non empty Subset of ( TOP-REAL 2) holds ( proj2 | K1) is continuous Function of (( TOP-REAL 2) | K1), R^1

    proof

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

      reconsider g2 = ( proj2 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      for q be Point of (( TOP-REAL 2) | K1) holds (g2 . q) = ( proj2 . q) by Lm4;

      hence thesis by JGRAPH_2: 30;

    end;

    

     Lm6: for K1 be non empty Subset of ( TOP-REAL 2) holds for q be Point of (( TOP-REAL 2) | K1) holds (( proj1 | K1) . q) = ( proj1 . q)

    proof

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

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

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

      then q in (( dom proj1 ) /\ K1) by Lm2, XBOOLE_0:def 4;

      hence thesis by FUNCT_1: 48;

    end;

    

     Lm7: for K1 be non empty Subset of ( TOP-REAL 2) holds ( proj1 | K1) is continuous Function of (( TOP-REAL 2) | K1), R^1

    proof

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

      reconsider g2 = ( proj1 | K1) as Function of (( TOP-REAL 2) | K1), R^1 by TOPMETR: 17;

      for q be Point of (( TOP-REAL 2) | K1) holds (g2 . q) = ( proj1 . q) by Lm6;

      hence thesis by JGRAPH_2: 29;

    end;

    theorem :: JGRAPH_3:11

    

     Th11: 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 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) & (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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      now

        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 Lm6

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence (g1 . q) <> 0 by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 / ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th10;

      

       A6: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `1 ) / ( sqrt (1 + (((r `2 ) / (r `1 )) ^2 )))) by A1, A7;

        hence thesis by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:12

    

     Th12: 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 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) & (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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      now

        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 Lm6

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence (g1 . q) <> 0 by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 / ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th9;

      

       A6: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `2 ) / ( sqrt (1 + (((r `2 ) / (r `1 )) ^2 )))) by A1, A7;

        hence thesis by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:13

    

     Th13: 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 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      now

        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;

        (g2 . q) = ( proj2 . q) by Lm4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence (g2 . q) <> 0 by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g1 . q) = r1 & (g2 . q) = r2 holds (g3 . q) = (r2 / ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th10;

      

       A6: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `2 ) / ( sqrt (1 + (((r `1 ) / (r `2 )) ^2 )))) by A1, A7;

        hence thesis by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:14

    

     Th14: 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 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      now

        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;

        (g2 . q) = ( proj2 . q) by Lm4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence (g2 . q) <> 0 by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g1 . q) = r1 & (g2 . q) = r2 holds (g3 . q) = (r1 / ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th9;

      

       A6: for x be object st x in ( dom f) holds (f . x) = (g3 . x)

      proof

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `1 ) / ( sqrt (1 + (((r `1 ) / (r `2 )) ^2 )))) by A1, A7;

        hence thesis by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    

     Lm8: ( 0.REAL 2) = ( 0. ( TOP-REAL 2)) by EUCLID: 66;

    

     Lm9: (( 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 JGRAPH_2: 5;

    

     Lm10: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm8, REVROT_1: 19;

    

     Lm11: for K1 be non empty Subset of ( TOP-REAL 2) holds ( dom ( proj2 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1)

    proof

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

      

       A1: ( dom ( Sq_Circ | K1)) c= ( dom ( proj2 * ( Sq_Circ | K1)))

      proof

        let x be object;

        assume

         A2: x in ( dom ( Sq_Circ | K1));

        then x in (( dom Sq_Circ ) /\ K1) by RELAT_1: 61;

        then x in ( dom Sq_Circ ) by XBOOLE_0:def 4;

        then

         A3: ( Sq_Circ . x) in ( rng Sq_Circ ) by FUNCT_1: 3;

        (( Sq_Circ | K1) . x) = ( Sq_Circ . x) by A2, FUNCT_1: 47;

        hence thesis by A2, A3, Lm3, FUNCT_1: 11;

      end;

      ( dom ( proj2 * ( Sq_Circ | K1))) c= ( dom ( Sq_Circ | K1)) by RELAT_1: 25;

      

      hence ( dom ( proj2 * ( Sq_Circ | K1))) = ( dom ( Sq_Circ | K1)) by A1

      .= (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

      .= K1 by XBOOLE_1: 28

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

    end;

    

     Lm12: for K1 be non empty Subset of ( TOP-REAL 2) holds ( dom ( proj1 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1)

    proof

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

      

       A1: ( dom ( Sq_Circ | K1)) c= ( dom ( proj1 * ( Sq_Circ | K1)))

      proof

        let x be object;

        assume

         A2: x in ( dom ( Sq_Circ | K1));

        then x in (( dom Sq_Circ ) /\ K1) by RELAT_1: 61;

        then x in ( dom Sq_Circ ) by XBOOLE_0:def 4;

        then

         A3: ( Sq_Circ . x) in ( rng Sq_Circ ) by FUNCT_1: 3;

        (( Sq_Circ | K1) . x) = ( Sq_Circ . x) by A2, FUNCT_1: 47;

        hence thesis by A2, A3, Lm2, FUNCT_1: 11;

      end;

      ( dom ( proj1 * ( Sq_Circ | K1))) c= ( dom ( Sq_Circ | K1)) by RELAT_1: 25;

      

      hence ( dom ( proj1 * ( Sq_Circ | K1))) = ( dom ( Sq_Circ | K1)) by A1

      .= (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

      .= K1 by XBOOLE_1: 28

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

    end;

    

     Lm13: ( NonZero ( TOP-REAL 2)) <> {} by JGRAPH_2: 9;

    theorem :: JGRAPH_3:15

    

     Th15: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = ( Sq_Circ | 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);

      assume

       A1: f = ( Sq_Circ | 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)) };

      then ( 1.REAL 2) in K0 by Lm9, Lm10;

      then

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

      ( dom ( proj1 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1) & ( rng ( proj1 * ( Sq_Circ | K1))) c= the carrier of R^1 by Lm12, TOPMETR: 17;

      then

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

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

      proof

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

        

         A2: ( dom ( Sq_Circ | K1)) = (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

        .= K1 by XBOOLE_1: 28;

        

         A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A4: 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 A1, A3;

        then

         A5: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Def1;

        (( Sq_Circ | K1) . p) = ( Sq_Circ . p) by A4, A3, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) by A4, A2, A3, A5, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A6: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))));

      ( dom ( proj2 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1) & ( rng ( proj2 * ( Sq_Circ | K1))) c= the carrier of R^1 by Lm11, TOPMETR: 17;

      then

      reconsider g2 = ( proj2 * ( Sq_Circ | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by 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 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))

      proof

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

        

         A7: ( dom ( Sq_Circ | K1)) = (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

        .= K1 by XBOOLE_1: 28;

        

         A8: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A9: 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 A1, A8;

        then

         A10: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Def1;

        (( Sq_Circ | K1) . p) = ( Sq_Circ . p) by A9, A8, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) by A9, A7, A8, A10, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A11: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))));

       A12:

      now

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

        

         A13: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

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

        then

         A14: 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 A1, A13;

        now

          assume

           A15: (q `1 ) = 0 ;

          then (q `2 ) = 0 by A14;

          hence contradiction by A14, A15, EUCLID: 53, EUCLID: 54;

        end;

        hence (q `1 ) <> 0 ;

      end;

      then

       A16: f1 is continuous by A6, Th11;

      

       A17: 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

         A18: |[x, y]| in K1 and

         A19: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A20: 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 A1, A18;

        

         A21: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A22: (f1 . p99) = ((p99 `1 ) / ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 )))) by A6, A18;

        (( Sq_Circ | K0) . |[x, y]|) = ( Sq_Circ . |[x, y]|) by A18, FUNCT_1: 49

        .= |[((p99 `1 ) / ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 )))), ((p99 `2 ) / ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 ))))]| by A20, Def1

        .= |[r, s]| by A11, A18, A19, A21, A22;

        hence thesis by A1;

      end;

      f2 is continuous by A12, A11, Th12;

      hence thesis by A1, A16, A17, Lm13, JGRAPH_2: 35;

    end;

    

     Lm14: (( 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 JGRAPH_2: 5;

    

     Lm15: ( 1.REAL 2) <> ( 0. ( TOP-REAL 2)) by Lm8, REVROT_1: 19;

    theorem :: JGRAPH_3:16

    

     Th16: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = ( Sq_Circ | 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);

      assume

       A1: f = ( Sq_Circ | 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)) };

      then ( 1.REAL 2) in K0 by Lm14, Lm15;

      then

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

      ( dom ( proj2 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1) & ( rng ( proj2 * ( Sq_Circ | K1))) c= the carrier of R^1 by Lm11, TOPMETR: 17;

      then

      reconsider g1 = ( proj2 * ( Sq_Circ | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by FUNCT_2: 2;

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

      proof

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

        

         A2: ( dom ( Sq_Circ | K1)) = (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

        .= K1 by XBOOLE_1: 28;

        

         A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A4: 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 A1, A3;

        then

         A5: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th4;

        (( Sq_Circ | K1) . p) = ( Sq_Circ . p) by A4, A3, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A4, A2, A3, A5, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A6: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))));

      ( dom ( proj1 * ( Sq_Circ | K1))) = the carrier of (( TOP-REAL 2) | K1) & ( rng ( proj1 * ( Sq_Circ | K1))) c= the carrier of R^1 by Lm12, TOPMETR: 17;

      then

      reconsider g2 = ( proj1 * ( Sq_Circ | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by 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 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))

      proof

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

        

         A7: ( dom ( Sq_Circ | K1)) = (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

        .= K1 by XBOOLE_1: 28;

        

         A8: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A9: 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 A1, A8;

        then

         A10: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th4;

        (( Sq_Circ | K1) . p) = ( Sq_Circ . p) by A9, A8, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A9, A7, A8, A10, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A11: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))));

      

       A12: 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);

        

         A13: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

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

        then

         A14: 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 A1, A13;

        now

          assume

           A15: (q `2 ) = 0 ;

          then (q `1 ) = 0 by A14;

          hence contradiction by A14, A15, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      then

       A16: f1 is continuous by A6, Th13;

       A17:

      now

        let x,y,s,r be Real;

        assume that

         A18: |[x, y]| in K1 and

         A19: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A20: 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 A1, A18;

        

         A21: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A22: (f1 . p99) = ((p99 `2 ) / ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 )))) by A6, A18;

        (( Sq_Circ | K0) . |[x, y]|) = ( Sq_Circ . |[x, y]|) by A18, FUNCT_1: 49

        .= |[((p99 `1 ) / ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 )))), ((p99 `2 ) / ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 ))))]| by A20, Th4

        .= |[s, r]| by A11, A18, A19, A21, A22;

        hence (f . |[x, y]|) = |[s, r]| by A1;

      end;

      f2 is continuous by A12, A11, Th14;

      hence thesis by A1, A16, A17, Lm13, JGRAPH_2: 35;

    end;

    scheme :: JGRAPH_3:sch1

    TopIncl { P[ set] } :

{ p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= ( NonZero ( TOP-REAL 2));

      let x be object;

      assume x in { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      then

       A1: ex p8 be Point of ( TOP-REAL 2) st x = p8 & P[p8] & p8 <> ( 0. ( TOP-REAL 2));

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

      hence thesis by A1, XBOOLE_0:def 5;

    end;

    scheme :: JGRAPH_3:sch2

    TopInter { P[ set] } :

{ p : P[p] & p <> ( 0. ( TOP-REAL 2)) } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ ( NonZero ( TOP-REAL 2)));

      set B0 = ( NonZero ( TOP-REAL 2));

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

      set K0 = { p : P[p] & p <> ( 0. ( TOP-REAL 2)) };

      

       A1: (K1 /\ B0) c= K0

      proof

        let x be object;

        assume

         A2: x in (K1 /\ B0);

        then x in B0 by XBOOLE_0:def 4;

        then not x in {( 0. ( TOP-REAL 2))} by XBOOLE_0:def 5;

        then

         A3: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        x in K1 by A2, XBOOLE_0:def 4;

        then ex p7 be Point of ( TOP-REAL 2) st p7 = x & P[p7];

        hence thesis by A3;

      end;

      K0 c= (K1 /\ B0)

      proof

        let x be object;

        assume x in K0;

        then

         A4: ex p be Point of ( TOP-REAL 2) st x = p & P[p] & p <> ( 0. ( TOP-REAL 2));

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

        then

         A5: x in B0 by A4, XBOOLE_0:def 5;

        x in K1 by A4;

        hence thesis by A5, XBOOLE_0:def 4;

      end;

      hence thesis by A1;

    end;

    theorem :: JGRAPH_3:17

    

     Th17: 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 = ( Sq_Circ | 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 JGRAPH_2: 47;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `1 )) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 47;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      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 )));

      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);

      assume

       A1: f = ( Sq_Circ | 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)) };

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= ( NonZero ( TOP-REAL 2)) from TopIncl;

      then

       A2: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by A1, PRE_TOPC: 7;

      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 )));

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

      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 )));

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } = ({ p7 where p7 be Point of ( TOP-REAL 2) : P[p7] } /\ ( NonZero ( TOP-REAL 2))) from TopInter;

      then

       A3: K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, PRE_TOPC:def 5;

      

       A4: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A5: x in ((K2 /\ K3) \/ (K4 /\ K5));

        per cases by A5, XBOOLE_0:def 3;

          suppose

           A6: x in (K2 /\ K3);

          then x in K3 by XBOOLE_0:def 4;

          then

           A7: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `1 )) <= (p8 `2 );

          x in K2 by A6, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) <= (p7 `1 );

          hence thesis by A7;

        end;

          suppose

           A8: x in (K4 /\ K5);

          then x in K5 by XBOOLE_0:def 4;

          then

           A9: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `2 ) <= ( - (p8 `1 ));

          x in K4 by A8, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) >= (p7 `1 );

          hence thesis by A9;

        end;

      end;

      

       A10: (K2 /\ K3) is closed & (K4 /\ K5) is closed by TOPS_1: 8;

      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 A4;

      then K1 is closed by A10, TOPS_1: 9;

      hence thesis by A1, A2, A3, Th15, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_3:18

    

     Th18: 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 = ( Sq_Circ | 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 JGRAPH_2: 48;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `2 )) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 48;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      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 )));

      set b0 = ( NonZero ( TOP-REAL 2));

      defpred P0[ Point of ( TOP-REAL 2)] means (($1 `1 ) <= ($1 `2 ) & ( - ($1 `2 )) <= ($1 `1 ) or ($1 `1 ) >= ($1 `2 ) & ($1 `1 ) <= ( - ($1 `2 )));

      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);

      assume

       A1: f = ( Sq_Circ | 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)) };

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= ( NonZero ( TOP-REAL 2)) from TopIncl;

      then

       A2: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by A1, PRE_TOPC: 7;

      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 )));

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

      

       A3: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A4: x in ((K2 /\ K3) \/ (K4 /\ K5));

        per cases by A4, XBOOLE_0:def 3;

          suppose

           A5: x in (K2 /\ K3);

          then x in K3 by XBOOLE_0:def 4;

          then

           A6: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `2 )) <= (p8 `1 );

          x in K2 by A5, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `1 ) <= (p7 `2 );

          hence thesis by A6;

        end;

          suppose

           A7: x in (K4 /\ K5);

          then x in K5 by XBOOLE_0:def 4;

          then

           A8: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `1 ) <= ( - (p8 `2 ));

          x in K4 by A7, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `1 ) >= (p7 `2 );

          hence thesis by A8;

        end;

      end;

      set k0 = { p : P0[p] & p <> ( 0. ( TOP-REAL 2)) };

      

       A9: (K2 /\ K3) is closed & (K4 /\ K5) is closed by TOPS_1: 8;

      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 A3;

      then

       A10: K1 is closed by A9, TOPS_1: 9;

      k0 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P0[p7] } /\ b0) from TopInter;

      then K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A1, PRE_TOPC:def 5;

      hence thesis by A1, A2, A10, Th16, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_3:19

    

     Th19: 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 = ( Sq_Circ | D) & h is continuous

    proof

      set Y1 = |[( - 1), 1]|;

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

      

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

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

      

      then

       A2: ( dom ( Sq_Circ | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A1, XBOOLE_1: 28;

      assume

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

      

      then

       A4: D = ( {( 0. ( TOP-REAL 2))} ` )

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

      

       A5: { 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

         A6: 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 A6, XBOOLE_0:def 5;

          then x in (D ` ) by SUBSET_1:def 4;

          hence contradiction by A3, A6, TARSKI:def 1;

        end;

        hence thesis by PRE_TOPC: 8;

      end;

      ( 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 Lm9, Lm10;

      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 A5;

      

       A7: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      

       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 A3, A9, TARSKI:def 1;

        end;

        hence thesis by PRE_TOPC: 8;

      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 JGRAPH_2: 3;

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

      

       A11: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A12: x in D;

        then

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

         not x in {( 0. ( TOP-REAL 2))} by A4, A12, 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;

      

       A13: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= ( NonZero ( TOP-REAL 2)) by A4, PRE_TOPC:def 5;

      

       A14: the carrier of (( TOP-REAL 2) | D) = D by PRE_TOPC: 8;

      

       A15: ( rng ( Sq_Circ | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        reconsider K00 = K0 as Subset of ( TOP-REAL 2) by A14, XBOOLE_1: 1;

        let y be object;

        

         A16: 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);

          

           A17: the carrier of (( TOP-REAL 2) | K00) = K0 by PRE_TOPC: 8;

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

          then

           A18: 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 A17;

          now

            assume

             A19: (q `1 ) = 0 ;

            then (q `2 ) = 0 by A18;

            hence contradiction by A18, A19, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

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

        then

        consider x be object such that

         A20: x in ( dom ( Sq_Circ | K0)) and

         A21: y = (( Sq_Circ | K0) . x) by FUNCT_1:def 3;

        

         A22: x in (( dom Sq_Circ ) /\ K0) by A20, RELAT_1: 61;

        then

         A23: x in K0 by XBOOLE_0:def 4;

        K0 c= the carrier of ( TOP-REAL 2) by A14, XBOOLE_1: 1;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A23;

        K00 = the carrier of (( TOP-REAL 2) | K00) by PRE_TOPC: 8;

        then p in the carrier of (( TOP-REAL 2) | K00) by A22, XBOOLE_0:def 4;

        then

         A24: (p `1 ) <> 0 by A16;

        

         A25: 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 A23;

        then

         A26: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Def1;

        

         A27: ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        then ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & (( - (p `1 )) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) or ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) >= ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= (( - (p `1 )) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A25, XREAL_1: 72;

        then

         A28: ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ( - ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) <= ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) or ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) >= ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ( - ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) by XCMPLX_1: 187;

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

        

         A29: (p9 `1 ) = ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & (p9 `2 ) = ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by EUCLID: 52;

        

         A30: (p9 `1 ) = ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by EUCLID: 52;

         A31:

        now

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

          then ( 0 * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = (((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A30, EUCLID: 52, EUCLID: 54;

          hence contradiction by A24, A27, XCMPLX_1: 87;

        end;

        ( Sq_Circ . p) = y by A21, A23, FUNCT_1: 49;

        then y in K0 by A31, A26, A28, A29;

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

        hence thesis;

      end;

      

       A32: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      ( dom ( Sq_Circ | K0)) = (( dom Sq_Circ ) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by FUNCT_2:def 1

      .= K0 by A32, XBOOLE_1: 28;

      then

      reconsider f = ( Sq_Circ | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A7, A15, FUNCT_2: 2, XBOOLE_1: 1;

      

       A33: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A34: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      

       A35: ( rng ( Sq_Circ | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        reconsider K10 = K1 as Subset of ( TOP-REAL 2) by A34;

        let y be object;

        

         A36: 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);

          

           A37: the carrier of (( TOP-REAL 2) | K10) = K1 by PRE_TOPC: 8;

          assume q in the carrier of (( TOP-REAL 2) | K10);

          then

           A38: 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 A37;

          now

            assume

             A39: (q `2 ) = 0 ;

            then (q `1 ) = 0 by A38;

            hence contradiction by A38, A39, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

        assume y in ( rng ( Sq_Circ | K1));

        then

        consider x be object such that

         A40: x in ( dom ( Sq_Circ | K1)) and

         A41: y = (( Sq_Circ | K1) . x) by FUNCT_1:def 3;

        

         A42: x in (( dom Sq_Circ ) /\ K1) by A40, RELAT_1: 61;

        then

         A43: x in K1 by XBOOLE_0:def 4;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A34;

        K10 = the carrier of (( TOP-REAL 2) | K10) by PRE_TOPC: 8;

        then p in the carrier of (( TOP-REAL 2) | K10) by A42, XBOOLE_0:def 4;

        then

         A44: (p `2 ) <> 0 by A36;

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

        

         A45: (p9 `2 ) = ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & (p9 `1 ) = ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by EUCLID: 52;

        

         A46: 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 A43;

        then

         A47: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th4;

        

         A48: ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        then ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & (( - (p `2 )) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) or ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) >= ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= (( - (p `2 )) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A46, XREAL_1: 72;

        then

         A49: ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ( - ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))) <= ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) or ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) >= ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ( - ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))) by XCMPLX_1: 187;

        

         A50: (p9 `2 ) = ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by EUCLID: 52;

         A51:

        now

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

          then ( 0 * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = (((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A50, EUCLID: 52, EUCLID: 54;

          hence contradiction by A44, A48, XCMPLX_1: 87;

        end;

        ( Sq_Circ . p) = y by A41, A43, FUNCT_1: 49;

        then y in K1 by A51, A47, A49, A45;

        hence thesis by PRE_TOPC: 8;

      end;

      ( dom ( Sq_Circ | K1)) = (( dom Sq_Circ ) /\ K1) by RELAT_1: 61

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

      .= K1 by A34, XBOOLE_1: 28;

      then

      reconsider g = ( Sq_Circ | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A10, A35, FUNCT_2: 2, XBOOLE_1: 1;

      

       A52: ( dom g) = K1 by A10, FUNCT_2:def 1;

      g = ( Sq_Circ | K1);

      then

       A53: K1 is closed by A4, Th18;

      

       A54: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

      

       A55: 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

         A56: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A54, XBOOLE_0:def 4;

        then (f . x) = ( Sq_Circ . x) by FUNCT_1: 49;

        hence thesis by A33, A56, FUNCT_1: 49;

      end;

      f = ( Sq_Circ | K0);

      then

       A57: K0 is closed by A4, Th17;

      

       A58: ( dom f) = K0 by A7, FUNCT_2:def 1;

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

      then

       A59: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A54, A33, A11;

      

       A60: f is continuous & g is continuous by A4, Th17, Th18;

      then

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

       A61: h = (f +* g) and h is continuous by A54, A33, A59, A57, A53, A55, JGRAPH_2: 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A62: f tolerates g by A55, A58, A52, PARTFUN1:def 4;

      

       A63: for x be object st x in ( dom h) holds (h . x) = (( Sq_Circ | D) . x)

      proof

        let x be object;

        assume

         A64: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A13, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A13, A64, XBOOLE_0:def 5;

        then

         A65: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        x in (the carrier of ( TOP-REAL 2) \ (D ` )) by A3, A13, A64;

        then

         A66: x in ((D ` ) ` ) by SUBSET_1:def 4;

        per cases ;

          suppose

           A67: x in K0;

          

           A68: (( Sq_Circ | D) . p) = ( Sq_Circ . p) by A66, FUNCT_1: 49

          .= (f . p) by A67, FUNCT_1: 49;

          (h . p) = ((g +* f) . p) by A61, A62, FUNCT_4: 34

          .= (f . p) by A58, A67, FUNCT_4: 13;

          hence thesis by A68;

        end;

          suppose not x in K0;

          then not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) by A65;

          then (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )) by XREAL_1: 26;

          then

           A69: x in K1 by A65;

          (( Sq_Circ | D) . p) = ( Sq_Circ . p) by A66, FUNCT_1: 49

          .= (g . p) by A69, FUNCT_1: 49;

          hence thesis by A61, A52, A69, FUNCT_4: 13;

        end;

      end;

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

      then (f +* g) = ( Sq_Circ | D) by A61, A2, A63;

      hence thesis by A54, A33, A59, A57, A60, A53, A55, JGRAPH_2: 1;

    end;

    theorem :: JGRAPH_3:20

    

     Th20: for D be non empty Subset of ( TOP-REAL 2) st D = ( NonZero ( TOP-REAL 2)) holds (D ` ) = {( 0. ( TOP-REAL 2))}

    proof

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

      assume

       A1: D = ( NonZero ( TOP-REAL 2));

      

       A2: (D ` ) c= {( 0. ( TOP-REAL 2))}

      proof

        let x be object;

        assume

         A3: x in (D ` );

        then x in (the carrier of ( TOP-REAL 2) \ D) by SUBSET_1:def 4;

        then not x in D by XBOOLE_0:def 5;

        hence thesis by A1, A3, XBOOLE_0:def 5;

      end;

       {( 0. ( TOP-REAL 2))} c= (D ` )

      proof

        let x be object;

        assume

         A4: x in {( 0. ( TOP-REAL 2))};

        then not x in D by A1, XBOOLE_0:def 5;

        then x in (the carrier of ( TOP-REAL 2) \ D) by A4, XBOOLE_0:def 5;

        hence thesis by SUBSET_1:def 4;

      end;

      hence thesis by A2;

    end;

    

     Lm16: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

    theorem :: JGRAPH_3:21

    

     Th21: ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = Sq_Circ & h is continuous

    proof

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      reconsider f = Sq_Circ as Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A1: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

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

        

         A2: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A2, XBOOLE_0:def 5;

        then

         A3: not p = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

           A4: not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

          then

           A5: (q `2 ) <> 0 ;

          set q9 = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

          

           A6: (q9 `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

          

           A7: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

           A8:

          now

            assume q9 = ( 0. ( TOP-REAL 2));

            then ( 0 * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A6, EUCLID: 52, EUCLID: 54;

            hence contradiction by A5, A7, XCMPLX_1: 87;

          end;

          ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A3, A4, Def1;

          hence thesis by A8, Def1;

        end;

          suppose

           A9: (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

           A10:

          now

            assume

             A11: (q `1 ) = 0 ;

            then (q `2 ) = 0 by A9;

            hence contradiction by A3, A11, EUCLID: 53, EUCLID: 54;

          end;

          set q9 = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

          

           A12: (q9 `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          

           A13: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

           A14:

          now

            assume q9 = ( 0. ( TOP-REAL 2));

            then ( 0 * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A12, EUCLID: 52, EUCLID: 54;

            hence contradiction by A10, A13, XCMPLX_1: 87;

          end;

          ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A3, A9, Def1;

          hence thesis by A14, Def1;

        end;

      end;

      

       A15: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Def1;

      

       A16: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

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

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm16;

        assume that

         A17: (f . ( 0. ( TOP-REAL 2))) in V and

         A18: V is open;

        VV is open by A18, Lm16, PRE_TOPC: 30;

        then

        consider r be Real such that

         A19: r > 0 and

         A20: ( Ball (u0,r)) c= V by A15, A17, TOPMETR: 15;

        reconsider r as Real;

        reconsider W1 = ( Ball (u0,r)) as Subset of ( TOP-REAL 2) by EUCLID: 67;

        

         A21: W1 is open by GOBOARD6: 3;

        

         A22: (f .: W1) c= W1

        proof

          let z be object;

          assume z in (f .: W1);

          then

          consider y be object such that

           A23: y in ( dom f) and

           A24: y in W1 and

           A25: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A23, A25, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          reconsider q = y as Point of ( TOP-REAL 2) by A23;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          ( dist (u0,qy)) < r by A24, METRIC_1: 11;

          then |.(( 0. ( TOP-REAL 2)) - q).| < r by JGRAPH_1: 28;

          then ( sqrt ((((( 0. ( TOP-REAL 2)) - q) `1 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - q) `2 ) ^2 ))) < r by JGRAPH_1: 30;

          then ( sqrt ((((( 0. ( TOP-REAL 2)) `1 ) - (q `1 )) ^2 ) + (((( 0. ( TOP-REAL 2)) - q) `2 ) ^2 ))) < r by TOPREAL3: 3;

          then

           A26: ( sqrt ((((( 0. ( TOP-REAL 2)) `1 ) - (q `1 )) ^2 ) + (((( 0. ( TOP-REAL 2)) `2 ) - (q `2 )) ^2 ))) < r by TOPREAL3: 3;

          per cases ;

            suppose q = ( 0. ( TOP-REAL 2));

            hence thesis by A24, A25, Def1;

          end;

            suppose

             A27: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

            

             A28: ((q `2 ) ^2 ) >= 0 by XREAL_1: 63;

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

            then (1 + (((q `2 ) / (q `1 )) ^2 )) >= (1 + 0 ) by XREAL_1: 7;

            then

             A29: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            then (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ) >= ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) by XREAL_1: 151;

            then

             A30: 1 <= (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ) by A29, XXREAL_0: 2;

            

             A31: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A27, Def1;

            

            then ((qz `2 ) ^2 ) = (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by A25, EUCLID: 52

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

            then

             A32: ((qz `2 ) ^2 ) <= (((q `2 ) ^2 ) / 1) by A30, A28, XREAL_1: 118;

            

             A33: ((q `1 ) ^2 ) >= 0 by XREAL_1: 63;

            ((qz `1 ) ^2 ) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by A25, A31, EUCLID: 52

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

            then ((qz `1 ) ^2 ) <= (((q `1 ) ^2 ) / 1) by A30, A33, XREAL_1: 118;

            then

             A34: (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A32, XREAL_1: 7;

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

            then

             A35: ( sqrt (((qz `1 ) ^2 ) + ((qz `2 ) ^2 ))) <= ( sqrt (((q `1 ) ^2 ) + ((q `2 ) ^2 ))) by A34, SQUARE_1: 26;

            

             A36: ((( 0. ( TOP-REAL 2)) - qz) `2 ) = ((( 0. ( TOP-REAL 2)) `2 ) - (qz `2 )) by TOPREAL3: 3

            .= ( - (qz `2 )) by JGRAPH_2: 3;

            ((( 0. ( TOP-REAL 2)) - qz) `1 ) = ((( 0. ( TOP-REAL 2)) `1 ) - (qz `1 )) by TOPREAL3: 3

            .= ( - (qz `1 )) by JGRAPH_2: 3;

            then ( sqrt ((((( 0. ( TOP-REAL 2)) - qz) `1 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - qz) `2 ) ^2 ))) < r by A26, A36, A35, JGRAPH_2: 3, XXREAL_0: 2;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by JGRAPH_1: 30;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

            suppose

             A37: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

            

             A38: ((q `2 ) ^2 ) >= 0 by XREAL_1: 63;

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

            then (1 + (((q `1 ) / (q `2 )) ^2 )) >= (1 + 0 ) by XREAL_1: 7;

            then

             A39: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            then (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ) >= ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) by XREAL_1: 151;

            then

             A40: 1 <= (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ) by A39, XXREAL_0: 2;

            

             A41: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A37, Def1;

            

            then ((qz `2 ) ^2 ) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by A25, EUCLID: 52

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

            then

             A42: ((qz `2 ) ^2 ) <= (((q `2 ) ^2 ) / 1) by A40, A38, XREAL_1: 118;

            

             A43: ((q `1 ) ^2 ) >= 0 by XREAL_1: 63;

            ((qz `1 ) ^2 ) = (((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by A25, A41, EUCLID: 52

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

            then ((qz `1 ) ^2 ) <= (((q `1 ) ^2 ) / 1) by A40, A43, XREAL_1: 118;

            then

             A44: (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) <= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by A42, XREAL_1: 7;

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

            then

             A45: ( sqrt (((qz `1 ) ^2 ) + ((qz `2 ) ^2 ))) <= ( sqrt (((q `1 ) ^2 ) + ((q `2 ) ^2 ))) by A44, SQUARE_1: 26;

            

             A46: ((( 0. ( TOP-REAL 2)) - qz) `2 ) = ((( 0. ( TOP-REAL 2)) `2 ) - (qz `2 )) by TOPREAL3: 3

            .= ( - (qz `2 )) by JGRAPH_2: 3;

            ((( 0. ( TOP-REAL 2)) - qz) `1 ) = ((( 0. ( TOP-REAL 2)) `1 ) - (qz `1 )) by TOPREAL3: 3

            .= ( - (qz `1 )) by JGRAPH_2: 3;

            then ( sqrt ((((( 0. ( TOP-REAL 2)) - qz) `1 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - qz) `2 ) ^2 ))) < r by A26, A46, A45, JGRAPH_2: 3, XXREAL_0: 2;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by JGRAPH_1: 30;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

        end;

        u0 in W1 by A19, GOBOARD6: 1;

        hence thesis by A20, A21, A22, XBOOLE_1: 1;

      end;

      

       A47: (D ` ) = {( 0. ( TOP-REAL 2))} by Th20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = ( Sq_Circ | D) & h is continuous by Th19;

      hence thesis by A15, A47, A1, A16, Th3;

    end;

    theorem :: JGRAPH_3:22

    

     Th22: Sq_Circ is one-to-one

    proof

      let x1,x2 be object;

      assume that

       A1: x1 in ( dom Sq_Circ ) and

       A2: x2 in ( dom Sq_Circ ) and

       A3: ( Sq_Circ . x1) = ( Sq_Circ . x2);

      reconsider p2 = x2 as Point of ( TOP-REAL 2) by A2;

      reconsider p1 = x1 as Point of ( TOP-REAL 2) by A1;

      set q = p1, p = p2;

      per cases ;

        suppose

         A4: q = ( 0. ( TOP-REAL 2));

        then

         A5: ( Sq_Circ . q) = ( 0. ( TOP-REAL 2)) by Def1;

        now

          per cases ;

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

            hence thesis by A4;

          end;

            case

             A6: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

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

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

            then

             A7: ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            

             A8: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A6, Def1;

            then ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = 0 by A3, A5, EUCLID: 52, JGRAPH_2: 3;

            

            then

             A9: (p `2 ) = ( 0 * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A7, XCMPLX_1: 87

            .= 0 ;

            ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = 0 by A3, A5, A8, EUCLID: 52, JGRAPH_2: 3;

            

            then (p `1 ) = ( 0 * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A7, XCMPLX_1: 87

            .= 0 ;

            hence contradiction by A6, A9, EUCLID: 53, EUCLID: 54;

          end;

            case

             A10: p <> ( 0. ( TOP-REAL 2)) & not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

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

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

            then

             A11: ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A10, Def1;

            then ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = 0 by A3, A5, EUCLID: 52, JGRAPH_2: 3;

            

            then (p `2 ) = ( 0 * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A11, XCMPLX_1: 87

            .= 0 ;

            hence contradiction by A10;

          end;

        end;

        hence thesis;

      end;

        suppose

         A12: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

        

         A13: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A14: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A12, Def1;

        

         A15: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        

         A16: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

        

         A17: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        now

          per cases ;

            case

             A18: p = ( 0. ( TOP-REAL 2));

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

            then (1 + (((q `2 ) / (q `1 )) ^2 )) >= (1 + 0 ) by XREAL_1: 7;

            then

             A19: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            

             A20: ( Sq_Circ . p) = ( 0. ( TOP-REAL 2)) by A18, Def1;

            then ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A3, A14, EUCLID: 52, JGRAPH_2: 3;

            

            then

             A21: (q `2 ) = ( 0 * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A19, XCMPLX_1: 87

            .= 0 ;

            ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A3, A14, A20, EUCLID: 52, JGRAPH_2: 3;

            

            then (q `1 ) = ( 0 * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A19, XCMPLX_1: 87

            .= 0 ;

            hence contradiction by A12, A21, EUCLID: 53, EUCLID: 54;

          end;

            case

             A22: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

            now

              assume

               A23: (p `1 ) = 0 ;

              then (p `2 ) = 0 by A22;

              hence contradiction by A22, A23, EUCLID: 53, EUCLID: 54;

            end;

            then

             A24: ((p `1 ) ^2 ) > 0 by SQUARE_1: 12;

            

             A25: ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            

             A26: (1 + (((p `2 ) / (p `1 )) ^2 )) > 0 by Lm1;

            

             A27: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A22, Def1;

            then

             A28: ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A3, A14, A15, EUCLID: 52;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by A26, SQUARE_1:def 2;

            then

             A29: (((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A16, SQUARE_1:def 2;

            

             A30: ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A3, A14, A17, A27, EUCLID: 52;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by A26, SQUARE_1:def 2;

            then (((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A16, SQUARE_1:def 2;

            then ((((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) / ((p `1 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

            then ((((p `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

            then (1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A24, XCMPLX_1: 60;

            then

             A31: ((1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (((q `1 ) ^2 ) / ((p `1 ) ^2 )) by A16, XCMPLX_1: 87;

            now

              assume

               A32: (q `1 ) = 0 ;

              then (q `2 ) = 0 by A12;

              hence contradiction by A12, A32, EUCLID: 53, EUCLID: 54;

            end;

            then

             A33: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

            now

              per cases ;

                case

                 A34: (p `2 ) = 0 ;

                then ((q `2 ) ^2 ) = 0 by A16, A29, XCMPLX_1: 50;

                then

                 A35: (q `2 ) = 0 by XCMPLX_1: 6;

                then p = |[(q `1 ), 0 ]| by A3, A14, A27, A34, EUCLID: 53, SQUARE_1: 18;

                hence thesis by A35, EUCLID: 53;

              end;

                case (p `2 ) <> 0 ;

                then

                 A36: ((p `2 ) ^2 ) > 0 by SQUARE_1: 12;

                ((((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) / ((p `2 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A29, XCMPLX_1: 48;

                then ((((p `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

                then (1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A36, XCMPLX_1: 60;

                then ((1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (((q `2 ) ^2 ) / ((p `2 ) ^2 )) by A16, XCMPLX_1: 87;

                then ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) / ((p `1 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / ((q `1 ) ^2 )) by A31, XCMPLX_1: 48;

                then (1 / ((p `1 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / ((q `1 ) ^2 )) by A33, XCMPLX_1: 60;

                then ((1 / ((p `1 ) ^2 )) * ((p `2 ) ^2 )) = ((((p `2 ) ^2 ) * (((q `2 ) ^2 ) / ((p `2 ) ^2 ))) / ((q `1 ) ^2 )) by XCMPLX_1: 74;

                then ((1 / ((p `1 ) ^2 )) * ((p `2 ) ^2 )) = (((q `2 ) ^2 ) / ((q `1 ) ^2 )) by A36, XCMPLX_1: 87;

                then (((p `2 ) ^2 ) / ((p `1 ) ^2 )) = (((q `2 ) ^2 ) / ((q `1 ) ^2 )) by XCMPLX_1: 99;

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

                then

                 A37: (1 + (((p `2 ) / (p `1 )) ^2 )) = (1 + (((q `2 ) / (q `1 )) ^2 )) by XCMPLX_1: 76;

                then (p `2 ) = (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A28, A25, XCMPLX_1: 87;

                then

                 A38: (p `2 ) = (q `2 ) by A13, XCMPLX_1: 87;

                (p `1 ) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A30, A25, A37, XCMPLX_1: 87;

                then (p `1 ) = (q `1 ) by A13, XCMPLX_1: 87;

                then p = |[(q `1 ), (q `2 )]| by A38, EUCLID: 53;

                hence thesis by EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

            case

             A39: p <> ( 0. ( TOP-REAL 2)) & not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

            

             A40: (1 + (((p `1 ) / (p `2 )) ^2 )) > 0 by Lm1;

            

             A41: p <> ( 0. ( TOP-REAL 2)) & (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )) by A39, JGRAPH_2: 13;

            (p `2 ) <> 0 by A39;

            then

             A42: ((p `2 ) ^2 ) > 0 by SQUARE_1: 12;

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

            then

             A43: ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A3, A14, A15, A39, Def1;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by A40, SQUARE_1:def 2;

            then (((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A16, SQUARE_1:def 2;

            then ((((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) / ((p `2 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

            then ((((p `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

            then (1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A42, XCMPLX_1: 60;

            then

             A44: ((1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (((q `2 ) ^2 ) / ((p `2 ) ^2 )) by A16, XCMPLX_1: 87;

            

             A45: ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

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

            then

             A46: ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A3, A14, A17, A39, Def1;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by A40, SQUARE_1:def 2;

            then

             A47: (((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A16, SQUARE_1:def 2;

             A48:

            now

              assume

               A49: (q `1 ) = 0 ;

              then (q `2 ) = 0 by A12;

              hence contradiction by A12, A49, EUCLID: 53, EUCLID: 54;

            end;

            then

             A50: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

            now

              per cases ;

                case (p `1 ) = 0 ;

                then ((q `1 ) ^2 ) = 0 by A16, A47, XCMPLX_1: 50;

                then

                 A51: (q `1 ) = 0 by XCMPLX_1: 6;

                then (q `2 ) = 0 by A12;

                hence contradiction by A12, A51, EUCLID: 53, EUCLID: 54;

              end;

                case

                 A52: (p `1 ) <> 0 ;

                set a = ((q `2 ) / (q `1 ));

                ((((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) / ((p `1 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A47, XCMPLX_1: 48;

                then

                 A53: ((((p `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 48;

                

                 A54: ((q `1 ) * a) <= (q `1 ) & ( - (q `1 )) <= ((q `1 ) * a) or ((q `1 ) * a) >= (q `1 ) & ((q `1 ) * a) <= ( - (q `1 )) by A12, A48, XCMPLX_1: 87;

                 A55:

                now

                  per cases by A48;

                    case

                     A56: (q `1 ) > 0 ;

                    then ((a * (q `1 )) / (q `1 )) <= ((q `1 ) / (q `1 )) & (( - (q `1 )) / (q `1 )) <= ((a * (q `1 )) / (q `1 )) or ((a * (q `1 )) / (q `1 )) >= ((q `1 ) / (q `1 )) & ((a * (q `1 )) / (q `1 )) <= (( - (q `1 )) / (q `1 )) by A54, XREAL_1: 72;

                    then

                     A57: a <= ((q `1 ) / (q `1 )) & (( - (q `1 )) / (q `1 )) <= a or a >= ((q `1 ) / (q `1 )) & a <= (( - (q `1 )) / (q `1 )) by A56, XCMPLX_1: 89;

                    ((q `1 ) / (q `1 )) = 1 by A56, XCMPLX_1: 60;

                    hence a <= 1 & ( - 1) <= a or a >= 1 & a <= ( - 1) by A57, XCMPLX_1: 187;

                  end;

                    case

                     A58: (q `1 ) < 0 ;

                    then

                     A59: ((q `1 ) / (q `1 )) = 1 & (( - (q `1 )) / (q `1 )) = ( - 1) by XCMPLX_1: 60, XCMPLX_1: 197;

                    ((a * (q `1 )) / (q `1 )) >= ((q `1 ) / (q `1 )) & (( - (q `1 )) / (q `1 )) >= ((a * (q `1 )) / (q `1 )) or ((a * (q `1 )) / (q `1 )) <= ((q `1 ) / (q `1 )) & ((a * (q `1 )) / (q `1 )) >= (( - (q `1 )) / (q `1 )) by A54, A58, XREAL_1: 73;

                    hence a <= 1 & ( - 1) <= a or a >= 1 & a <= ( - 1) by A58, A59, XCMPLX_1: 89;

                  end;

                end;

                ((p `1 ) ^2 ) > 0 by A52, SQUARE_1: 12;

                then (1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A53, XCMPLX_1: 60;

                then ((1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (((q `1 ) ^2 ) / ((p `1 ) ^2 )) by A16, XCMPLX_1: 87;

                then ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) / ((p `1 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / ((q `1 ) ^2 )) by A44, XCMPLX_1: 48;

                then (1 / ((p `1 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / ((q `1 ) ^2 )) by A50, XCMPLX_1: 60;

                then ((1 / ((p `1 ) ^2 )) * ((p `2 ) ^2 )) = ((((p `2 ) ^2 ) * (((q `2 ) ^2 ) / ((p `2 ) ^2 ))) / ((q `1 ) ^2 )) by XCMPLX_1: 74;

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

                then (((p `2 ) ^2 ) / ((p `1 ) ^2 )) = (((q `2 ) ^2 ) / ((q `1 ) ^2 )) by XCMPLX_1: 99;

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

                then

                 A60: (((p `2 ) / (p `1 )) ^2 ) = (((q `2 ) / (q `1 )) ^2 ) by XCMPLX_1: 76;

                then

                 A61: (((p `2 ) / (p `1 )) * (p `1 )) = (a * (p `1 )) or (((p `2 ) / (p `1 )) * (p `1 )) = (( - a) * (p `1 )) by SQUARE_1: 40;

                 A62:

                now

                  per cases by A52, A61, XCMPLX_1: 87;

                    case

                     A63: (p `2 ) = (a * (p `1 ));

                    now

                      per cases by A52;

                        case (p `1 ) > 0 ;

                        then ((p `1 ) / (p `1 )) <= ((a * (p `1 )) / (p `1 )) & (( - (a * (p `1 ))) / (p `1 )) <= ((p `1 ) / (p `1 )) or ((p `1 ) / (p `1 )) >= ((a * (p `1 )) / (p `1 )) & ((p `1 ) / (p `1 )) <= (( - (a * (p `1 ))) / (p `1 )) by A41, A63, XREAL_1: 72;

                        then

                         A64: 1 <= ((a * (p `1 )) / (p `1 )) & (( - (a * (p `1 ))) / (p `1 )) <= 1 or 1 >= ((a * (p `1 )) / (p `1 )) & 1 <= (( - (a * (p `1 ))) / (p `1 )) by A52, XCMPLX_1: 60;

                        ((a * (p `1 )) / (p `1 )) = a by A52, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A64, XCMPLX_1: 187;

                      end;

                        case (p `1 ) < 0 ;

                        then ((p `1 ) / (p `1 )) >= ((a * (p `1 )) / (p `1 )) & (( - (a * (p `1 ))) / (p `1 )) >= ((p `1 ) / (p `1 )) or ((p `1 ) / (p `1 )) <= ((a * (p `1 )) / (p `1 )) & ((p `1 ) / (p `1 )) >= (( - (a * (p `1 ))) / (p `1 )) by A41, A63, XREAL_1: 73;

                        then

                         A65: 1 >= ((a * (p `1 )) / (p `1 )) & (( - (a * (p `1 ))) / (p `1 )) >= 1 or 1 <= ((a * (p `1 )) / (p `1 )) & 1 >= (( - (a * (p `1 ))) / (p `1 )) by A52, XCMPLX_1: 60;

                        ((a * (p `1 )) / (p `1 )) = a by A52, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A65, XCMPLX_1: 187;

                      end;

                    end;

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

                    hence 1 <= a or ( - 1) >= a;

                  end;

                    case

                     A66: (p `2 ) = (( - a) * (p `1 ));

                    now

                      per cases by A52;

                        case (p `1 ) > 0 ;

                        then ((p `1 ) / (p `1 )) <= ((( - a) * (p `1 )) / (p `1 )) & (( - (( - a) * (p `1 ))) / (p `1 )) <= ((p `1 ) / (p `1 )) or ((p `1 ) / (p `1 )) >= ((( - a) * (p `1 )) / (p `1 )) & ((p `1 ) / (p `1 )) <= (( - (( - a) * (p `1 ))) / (p `1 )) by A41, A66, XREAL_1: 72;

                        then 1 <= ((( - a) * (p `1 )) / (p `1 )) & (( - (( - a) * (p `1 ))) / (p `1 )) <= 1 or 1 >= ((( - a) * (p `1 )) / (p `1 )) & 1 <= (( - (( - a) * (p `1 ))) / (p `1 )) by A52, XCMPLX_1: 60;

                        then

                         A67: 1 <= ( - a) & ( - ((( - a) * (p `1 )) / (p `1 ))) <= 1 or 1 >= ( - a) & 1 <= ( - ((( - a) * (p `1 )) / (p `1 ))) by A52, XCMPLX_1: 89, XCMPLX_1: 187;

                        ((( - a) * (p `1 )) / (p `1 )) = ( - a) by A52, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A67;

                      end;

                        case (p `1 ) < 0 ;

                        then ((p `1 ) / (p `1 )) >= ((( - a) * (p `1 )) / (p `1 )) & (( - (( - a) * (p `1 ))) / (p `1 )) >= ((p `1 ) / (p `1 )) or ((p `1 ) / (p `1 )) <= ((( - a) * (p `1 )) / (p `1 )) & ((p `1 ) / (p `1 )) >= (( - (( - a) * (p `1 ))) / (p `1 )) by A41, A66, XREAL_1: 73;

                        then 1 >= ((( - a) * (p `1 )) / (p `1 )) & (( - (( - a) * (p `1 ))) / (p `1 )) >= 1 or 1 <= ((( - a) * (p `1 )) / (p `1 )) & 1 >= (( - (( - a) * (p `1 ))) / (p `1 )) by A52, XCMPLX_1: 60;

                        then

                         A68: 1 >= ( - a) & ( - ((( - a) * (p `1 )) / (p `1 ))) >= 1 or 1 <= ( - a) & 1 >= ( - ((( - a) * (p `1 )) / (p `1 ))) by A52, XCMPLX_1: 89, XCMPLX_1: 187;

                        ((( - a) * (p `1 )) / (p `1 )) = ( - a) by A52, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A68;

                      end;

                    end;

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

                    hence 1 <= a or ( - 1) >= a;

                  end;

                end;

                 A69:

                now

                  per cases by A62, A55, XXREAL_0: 1;

                    case a = 1;

                    then (((p `2 ) ^2 ) / ((p `1 ) ^2 )) = 1 by A60, XCMPLX_1: 76;

                    then

                     A70: ((p `2 ) ^2 ) = ((p `1 ) ^2 ) by XCMPLX_1: 58;

                    (((p `1 ) / (p `2 )) ^2 ) = (((p `1 ) ^2 ) / ((p `2 ) ^2 )) by XCMPLX_1: 76;

                    hence (((p `1 ) / (p `2 )) ^2 ) = (((q `2 ) / (q `1 )) ^2 ) by A60, A70, XCMPLX_1: 76;

                  end;

                    case a = ( - 1);

                    then (((p `2 ) ^2 ) / ((p `1 ) ^2 )) = 1 by A60, XCMPLX_1: 76;

                    then

                     A71: ((p `2 ) ^2 ) = ((p `1 ) ^2 ) by XCMPLX_1: 58;

                    (((p `1 ) / (p `2 )) ^2 ) = (((p `1 ) ^2 ) / ((p `2 ) ^2 )) by XCMPLX_1: 76;

                    hence (((p `1 ) / (p `2 )) ^2 ) = (((q `2 ) / (q `1 )) ^2 ) by A60, A71, XCMPLX_1: 76;

                  end;

                end;

                then (p `2 ) = (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A43, A45, XCMPLX_1: 87;

                then

                 A72: (p `2 ) = (q `2 ) by A13, XCMPLX_1: 87;

                (p `1 ) = (((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A46, A45, A69, XCMPLX_1: 87;

                then (p `1 ) = (q `1 ) by A13, XCMPLX_1: 87;

                then p = |[(q `1 ), (q `2 )]| by A72, EUCLID: 53;

                hence thesis by EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

        suppose

         A73: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

        

         A74: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

        

         A75: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

        

         A76: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by Lm1;

        

         A77: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A78: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A73, Def1;

        

         A79: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A73, JGRAPH_2: 13;

        now

          per cases ;

            case

             A80: p = ( 0. ( TOP-REAL 2));

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

            then (1 + (((q `1 ) / (q `2 )) ^2 )) >= (1 + 0 ) by XREAL_1: 7;

            then

             A81: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

            ( Sq_Circ . p) = ( 0. ( TOP-REAL 2)) by A80, Def1;

            then ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A3, A78, EUCLID: 52, JGRAPH_2: 3;

            

            then (q `2 ) = ( 0 * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A81, XCMPLX_1: 87

            .= 0 ;

            hence contradiction by A73;

          end;

            case

             A82: p <> ( 0. ( TOP-REAL 2)) & ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

            now

              assume

               A83: (p `1 ) = 0 ;

              then (p `2 ) = 0 by A82;

              hence contradiction by A82, A83, EUCLID: 53, EUCLID: 54;

            end;

            then

             A84: ((p `1 ) ^2 ) > 0 by SQUARE_1: 12;

            

             A85: (1 + (((p `2 ) / (p `1 )) ^2 )) > 0 by Lm1;

            

             A86: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A82, Def1;

            then

             A87: ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A3, A78, A75, EUCLID: 52;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by A85, SQUARE_1:def 2;

            then (((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A76, SQUARE_1:def 2;

            then ((((p `1 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) / ((p `1 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

            then ((((p `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

            then (1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A84, XCMPLX_1: 60;

            then

             A88: ((1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) * (1 + (((q `1 ) / (q `2 )) ^2 ))) = (((q `1 ) ^2 ) / ((p `1 ) ^2 )) by A76, XCMPLX_1: 87;

            

             A89: ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A3, A78, A74, A86, EUCLID: 52;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) ^2 )) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by A85, SQUARE_1:def 2;

            then

             A90: (((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A76, SQUARE_1:def 2;

            

             A91: ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            

             A92: (q `2 ) <> 0 by A73;

            then

             A93: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

            now

              per cases ;

                case (p `2 ) = 0 ;

                then ((q `2 ) ^2 ) = 0 by A76, A90, XCMPLX_1: 50;

                then (q `2 ) = 0 by XCMPLX_1: 6;

                hence contradiction by A73;

              end;

                case

                 A94: (p `2 ) <> 0 ;

                set a = ((q `1 ) / (q `2 ));

                ((((p `2 ) ^2 ) / (1 + (((p `2 ) / (p `1 )) ^2 ))) / ((p `2 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A90, XCMPLX_1: 48;

                then

                 A95: ((((p `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

                

                 A96: ((q `2 ) * a) <= (q `2 ) & ( - (q `2 )) <= ((q `2 ) * a) or ((q `2 ) * a) >= (q `2 ) & ((q `2 ) * a) <= ( - (q `2 )) by A79, A92, XCMPLX_1: 87;

                 A97:

                now

                  per cases by A73;

                    case

                     A98: (q `2 ) > 0 ;

                    then

                     A99: ((q `2 ) / (q `2 )) = 1 & (( - (q `2 )) / (q `2 )) = ( - 1) by XCMPLX_1: 60, XCMPLX_1: 197;

                    ((a * (q `2 )) / (q `2 )) <= ((q `2 ) / (q `2 )) & (( - (q `2 )) / (q `2 )) <= ((a * (q `2 )) / (q `2 )) or ((a * (q `2 )) / (q `2 )) >= ((q `2 ) / (q `2 )) & ((a * (q `2 )) / (q `2 )) <= (( - (q `2 )) / (q `2 )) by A96, A98, XREAL_1: 72;

                    hence a <= 1 & ( - 1) <= a or a >= 1 & a <= ( - 1) by A98, A99, XCMPLX_1: 89;

                  end;

                    case

                     A100: (q `2 ) < 0 ;

                    then ((a * (q `2 )) / (q `2 )) >= ((q `2 ) / (q `2 )) & (( - (q `2 )) / (q `2 )) >= ((a * (q `2 )) / (q `2 )) or ((a * (q `2 )) / (q `2 )) <= ((q `2 ) / (q `2 )) & ((a * (q `2 )) / (q `2 )) >= (( - (q `2 )) / (q `2 )) by A96, XREAL_1: 73;

                    then a >= ((q `2 ) / (q `2 )) & (( - (q `2 )) / (q `2 )) >= a or a <= ((q `2 ) / (q `2 )) & a >= (( - (q `2 )) / (q `2 )) by A100, XCMPLX_1: 89;

                    hence a <= 1 & ( - 1) <= a or a >= 1 & a <= ( - 1) by A100, XCMPLX_1: 60, XCMPLX_1: 197;

                  end;

                end;

                ((p `2 ) ^2 ) > 0 by A94, SQUARE_1: 12;

                then (1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A95, XCMPLX_1: 60;

                then ((1 / (1 + (((p `2 ) / (p `1 )) ^2 ))) * (1 + (((q `1 ) / (q `2 )) ^2 ))) = (((q `2 ) ^2 ) / ((p `2 ) ^2 )) by A76, XCMPLX_1: 87;

                then ((((q `2 ) ^2 ) / ((q `2 ) ^2 )) / ((p `2 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / ((q `2 ) ^2 )) by A88, XCMPLX_1: 48;

                then (1 / ((p `2 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / ((q `2 ) ^2 )) by A93, XCMPLX_1: 60;

                then ((1 / ((p `2 ) ^2 )) * ((p `1 ) ^2 )) = ((((p `1 ) ^2 ) * (((q `1 ) ^2 ) / ((p `1 ) ^2 ))) / ((q `2 ) ^2 )) by XCMPLX_1: 74;

                then ((1 / ((p `2 ) ^2 )) * ((p `1 ) ^2 )) = (((q `1 ) ^2 ) / ((q `2 ) ^2 )) by A84, XCMPLX_1: 87;

                then (((p `1 ) ^2 ) / ((p `2 ) ^2 )) = (((q `1 ) ^2 ) / ((q `2 ) ^2 )) by XCMPLX_1: 99;

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

                then

                 A101: (((p `1 ) / (p `2 )) ^2 ) = (((q `1 ) / (q `2 )) ^2 ) by XCMPLX_1: 76;

                then

                 A102: ((p `1 ) / (p `2 )) = ((q `1 ) / (q `2 )) or ((p `1 ) / (p `2 )) = ( - ((q `1 ) / (q `2 ))) by SQUARE_1: 40;

                 A103:

                now

                  per cases by A94, A102, XCMPLX_1: 87;

                    case

                     A104: (p `1 ) = (a * (p `2 ));

                    now

                      per cases by A94;

                        case (p `2 ) > 0 ;

                        then ((p `2 ) / (p `2 )) <= ((a * (p `2 )) / (p `2 )) & (( - (a * (p `2 ))) / (p `2 )) <= ((p `2 ) / (p `2 )) or ((p `2 ) / (p `2 )) >= ((a * (p `2 )) / (p `2 )) & ((p `2 ) / (p `2 )) <= (( - (a * (p `2 ))) / (p `2 )) by A82, A104, XREAL_1: 72;

                        then

                         A105: 1 <= ((a * (p `2 )) / (p `2 )) & (( - (a * (p `2 ))) / (p `2 )) <= 1 or 1 >= ((a * (p `2 )) / (p `2 )) & 1 <= (( - (a * (p `2 ))) / (p `2 )) by A94, XCMPLX_1: 60;

                        ((a * (p `2 )) / (p `2 )) = a by A94, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A105, XCMPLX_1: 187;

                      end;

                        case (p `2 ) < 0 ;

                        then ((p `2 ) / (p `2 )) >= ((a * (p `2 )) / (p `2 )) & (( - (a * (p `2 ))) / (p `2 )) >= ((p `2 ) / (p `2 )) or ((p `2 ) / (p `2 )) <= ((a * (p `2 )) / (p `2 )) & ((p `2 ) / (p `2 )) >= (( - (a * (p `2 ))) / (p `2 )) by A82, A104, XREAL_1: 73;

                        then

                         A106: 1 >= ((a * (p `2 )) / (p `2 )) & (( - (a * (p `2 ))) / (p `2 )) >= 1 or 1 <= ((a * (p `2 )) / (p `2 )) & 1 >= (( - (a * (p `2 ))) / (p `2 )) by A94, XCMPLX_1: 60;

                        ((a * (p `2 )) / (p `2 )) = a by A94, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A106, XCMPLX_1: 187;

                      end;

                    end;

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

                    hence 1 <= a or ( - 1) >= a;

                  end;

                    case

                     A107: (p `1 ) = (( - a) * (p `2 ));

                    now

                      per cases by A94;

                        case (p `2 ) > 0 ;

                        then ((p `2 ) / (p `2 )) <= ((( - a) * (p `2 )) / (p `2 )) & (( - (( - a) * (p `2 ))) / (p `2 )) <= ((p `2 ) / (p `2 )) or ((p `2 ) / (p `2 )) >= ((( - a) * (p `2 )) / (p `2 )) & ((p `2 ) / (p `2 )) <= (( - (( - a) * (p `2 ))) / (p `2 )) by A82, A107, XREAL_1: 72;

                        then 1 <= ((( - a) * (p `2 )) / (p `2 )) & (( - (( - a) * (p `2 ))) / (p `2 )) <= 1 or 1 >= ((( - a) * (p `2 )) / (p `2 )) & 1 <= (( - (( - a) * (p `2 ))) / (p `2 )) by A94, XCMPLX_1: 60;

                        then

                         A108: 1 <= ( - a) & ( - ((( - a) * (p `2 )) / (p `2 ))) <= 1 or 1 >= ( - a) & 1 <= ( - ((( - a) * (p `2 )) / (p `2 ))) by A94, XCMPLX_1: 89, XCMPLX_1: 187;

                        ((( - a) * (p `2 )) / (p `2 )) = ( - a) by A94, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A108;

                      end;

                        case (p `2 ) < 0 ;

                        then ((p `2 ) / (p `2 )) >= ((( - a) * (p `2 )) / (p `2 )) & (( - (( - a) * (p `2 ))) / (p `2 )) >= ((p `2 ) / (p `2 )) or ((p `2 ) / (p `2 )) <= ((( - a) * (p `2 )) / (p `2 )) & ((p `2 ) / (p `2 )) >= (( - (( - a) * (p `2 ))) / (p `2 )) by A82, A107, XREAL_1: 73;

                        then 1 >= ((( - a) * (p `2 )) / (p `2 )) & (( - (( - a) * (p `2 ))) / (p `2 )) >= 1 or 1 <= ((( - a) * (p `2 )) / (p `2 )) & 1 >= (( - (( - a) * (p `2 ))) / (p `2 )) by A94, XCMPLX_1: 60;

                        then

                         A109: 1 >= ( - a) & ( - ((( - a) * (p `2 )) / (p `2 ))) >= 1 or 1 <= ( - a) & 1 >= ( - ((( - a) * (p `2 )) / (p `2 ))) by A94, XCMPLX_1: 89, XCMPLX_1: 187;

                        ((( - a) * (p `2 )) / (p `2 )) = ( - a) by A94, XCMPLX_1: 89;

                        hence 1 <= a & ( - a) <= 1 or 1 >= a & 1 <= ( - a) by A109;

                      end;

                    end;

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

                    hence 1 <= a or ( - 1) >= a;

                  end;

                end;

                 A110:

                now

                  per cases by A103, A97, XXREAL_0: 1;

                    case a = 1;

                    then (((p `1 ) ^2 ) / ((p `2 ) ^2 )) = 1 by A101, XCMPLX_1: 76;

                    then

                     A111: ((p `1 ) ^2 ) = ((p `2 ) ^2 ) by XCMPLX_1: 58;

                    (((p `2 ) / (p `1 )) ^2 ) = (((p `2 ) ^2 ) / ((p `1 ) ^2 )) by XCMPLX_1: 76;

                    hence (((p `2 ) / (p `1 )) ^2 ) = (((q `1 ) / (q `2 )) ^2 ) by A101, A111, XCMPLX_1: 76;

                  end;

                    case a = ( - 1);

                    then (((p `1 ) ^2 ) / ((p `2 ) ^2 )) = 1 by A101, XCMPLX_1: 76;

                    then

                     A112: ((p `1 ) ^2 ) = ((p `2 ) ^2 ) by XCMPLX_1: 58;

                    (((p `2 ) / (p `1 )) ^2 ) = (((p `2 ) ^2 ) / ((p `1 ) ^2 )) by XCMPLX_1: 76;

                    hence (((p `2 ) / (p `1 )) ^2 ) = (((q `1 ) / (q `2 )) ^2 ) by A101, A112, XCMPLX_1: 76;

                  end;

                end;

                then (p `1 ) = (((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A87, A91, XCMPLX_1: 87;

                then

                 A113: (p `1 ) = (q `1 ) by A77, XCMPLX_1: 87;

                (p `2 ) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A89, A91, A110, XCMPLX_1: 87;

                then (p `2 ) = (q `2 ) by A77, XCMPLX_1: 87;

                then p = |[(q `1 ), (q `2 )]| by A113, EUCLID: 53;

                hence thesis by EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

            case

             A114: p <> ( 0. ( TOP-REAL 2)) & not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

            then (p `2 ) <> 0 ;

            then

             A115: ((p `2 ) ^2 ) > 0 by SQUARE_1: 12;

            

             A116: ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            

             A117: (1 + (((p `1 ) / (p `2 )) ^2 )) > 0 by Lm1;

            

             A118: ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A114, Def1;

            then

             A119: ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A3, A78, A75, EUCLID: 52;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by A117, SQUARE_1:def 2;

            then

             A120: (((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A76, SQUARE_1:def 2;

            

             A121: ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A3, A78, A74, A118, EUCLID: 52;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ^2 )) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by XCMPLX_1: 76;

            then (((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by A117, SQUARE_1:def 2;

            then (((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A76, SQUARE_1:def 2;

            then ((((p `2 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) / ((p `2 ) ^2 )) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

            then ((((p `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

            then (1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `2 ) ^2 ) / ((p `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A115, XCMPLX_1: 60;

            then

             A122: ((1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) * (1 + (((q `1 ) / (q `2 )) ^2 ))) = (((q `2 ) ^2 ) / ((p `2 ) ^2 )) by A76, XCMPLX_1: 87;

            (q `2 ) <> 0 by A73;

            then

             A123: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

            now

              per cases ;

                case

                 A124: (p `1 ) = 0 ;

                then ((q `1 ) ^2 ) = 0 by A76, A120, XCMPLX_1: 50;

                then

                 A125: (q `1 ) = 0 by XCMPLX_1: 6;

                then p = |[ 0 , (q `2 )]| by A3, A78, A118, A124, EUCLID: 53, SQUARE_1: 18;

                hence thesis by A125, EUCLID: 53;

              end;

                case (p `1 ) <> 0 ;

                then

                 A126: ((p `1 ) ^2 ) > 0 by SQUARE_1: 12;

                ((((p `1 ) ^2 ) / (1 + (((p `1 ) / (p `2 )) ^2 ))) / ((p `1 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A120, XCMPLX_1: 48;

                then ((((p `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 48;

                then (1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A126, XCMPLX_1: 60;

                then ((1 / (1 + (((p `1 ) / (p `2 )) ^2 ))) * (1 + (((q `1 ) / (q `2 )) ^2 ))) = (((q `1 ) ^2 ) / ((p `1 ) ^2 )) by A76, XCMPLX_1: 87;

                then ((((q `2 ) ^2 ) / ((q `2 ) ^2 )) / ((p `2 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / ((q `2 ) ^2 )) by A122, XCMPLX_1: 48;

                then (1 / ((p `2 ) ^2 )) = ((((q `1 ) ^2 ) / ((p `1 ) ^2 )) / ((q `2 ) ^2 )) by A123, XCMPLX_1: 60;

                then ((1 / ((p `2 ) ^2 )) * ((p `1 ) ^2 )) = ((((p `1 ) ^2 ) * (((q `1 ) ^2 ) / ((p `1 ) ^2 ))) / ((q `2 ) ^2 )) by XCMPLX_1: 74;

                then ((1 / ((p `2 ) ^2 )) * ((p `1 ) ^2 )) = (((q `1 ) ^2 ) / ((q `2 ) ^2 )) by A126, XCMPLX_1: 87;

                then (((p `1 ) ^2 ) / ((p `2 ) ^2 )) = (((q `1 ) ^2 ) / ((q `2 ) ^2 )) by XCMPLX_1: 99;

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

                then

                 A127: (1 + (((p `1 ) / (p `2 )) ^2 )) = (1 + (((q `1 ) / (q `2 )) ^2 )) by XCMPLX_1: 76;

                then (p `1 ) = (((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A119, A116, XCMPLX_1: 87;

                then

                 A128: (p `1 ) = (q `1 ) by A77, XCMPLX_1: 87;

                (p `2 ) = (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A121, A116, A127, XCMPLX_1: 87;

                then (p `2 ) = (q `2 ) by A77, XCMPLX_1: 87;

                then p = |[(q `1 ), (q `2 )]| by A128, EUCLID: 53;

                hence thesis by EUCLID: 53;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

    end;

    registration

      cluster Sq_Circ -> one-to-one;

      coherence by Th22;

    end

    theorem :: JGRAPH_3:23

    

     Th23: for Kb,Cb be Subset of ( TOP-REAL 2) st 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 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| = 1 } holds ( Sq_Circ .: Kb) = Cb

    proof

      let Kb,Cb be Subset of ( TOP-REAL 2);

      assume

       A1: 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 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| = 1 };

      thus ( Sq_Circ .: Kb) c= Cb

      proof

        let y be object;

        assume y in ( Sq_Circ .: Kb);

        then

        consider x be object such that x in ( dom Sq_Circ ) and

         A2: x in Kb and

         A3: y = ( Sq_Circ . x) by FUNCT_1:def 6;

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

         A4: q = x and

         A5: ( - 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 A1, A2;

        now

          per cases ;

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

            hence contradiction by A5, JGRAPH_2: 3;

          end;

            case

             A6: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

            

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

            

             A8: (1 + ((q `2 ) ^2 )) > 0 by Lm1;

            

             A9: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A6, Def1;

            now

              per cases by A5;

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

                

                then ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = (((( - 1) / ( sqrt (1 + (((q `2 ) / ( - 1)) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / ( - 1)) ^2 )))) ^2 )) by A7, JGRAPH_1: 29

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

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

                .= ((1 / (1 + ((q `2 ) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + ((q `2 ) ^2 ))) ^2 ))) by A8, SQUARE_1:def 2

                .= ((1 / (1 + ((q `2 ) ^2 ))) + (((q `2 ) ^2 ) / (1 + ((q `2 ) ^2 )))) by A8, SQUARE_1:def 2

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

                .= 1 by A8, XCMPLX_1: 60;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A9;

              end;

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

                

                then ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = (((1 / ( sqrt (1 + (((q `2 ) / 1) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / 1) ^2 )))) ^2 )) by A7, JGRAPH_1: 29

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

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

                .= ((1 / (1 + ((q `2 ) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + ((q `2 ) ^2 ))) ^2 ))) by A8, SQUARE_1:def 2

                .= ((1 / (1 + ((q `2 ) ^2 ))) + (((q `2 ) ^2 ) / (1 + ((q `2 ) ^2 )))) by A8, SQUARE_1:def 2

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

                .= 1 by A8, XCMPLX_1: 60;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A9;

              end;

                case

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

                then ( - 1) <= (q `1 ) & (q `1 ) >= 1 or ( - 1) >= (q `1 ) & 1 >= (q `1 ) by A6, XREAL_1: 24;

                then

                 A11: (q `1 ) = 1 or (q `1 ) = ( - 1) by A10, XXREAL_0: 1;

                ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + ((( - 1) / (q `1 )) ^2 )))) ^2 ) + ((( - 1) / ( sqrt (1 + ((( - 1) / (q `1 )) ^2 )))) ^2 )) by A7, A10, JGRAPH_1: 29

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

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

                .= ((1 / 2) + (1 / (( sqrt 2) ^2 ))) by A11, SQUARE_1:def 2

                .= ((1 / 2) + (1 / 2)) by SQUARE_1:def 2

                .= 1;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A9;

              end;

                case

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

                then 1 <= (q `1 ) & (q `1 ) >= ( - 1) or 1 >= (q `1 ) & ( - 1) >= (q `1 ) by A6, XREAL_1: 25;

                then

                 A13: (q `1 ) = 1 or (q `1 ) = ( - 1) by A12, XXREAL_0: 1;

                ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + ((1 / (q `1 )) ^2 )))) ^2 ) + ((1 / ( sqrt (1 + ((1 / (q `1 )) ^2 )))) ^2 )) by A7, A12, JGRAPH_1: 29

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

                .= ((1 / (( sqrt (1 + (1 / 1))) ^2 )) + (1 / (( sqrt (1 + (1 / 1))) ^2 ))) by A13, XCMPLX_1: 76

                .= ((1 / 2) + (1 / (( sqrt 2) ^2 ))) by SQUARE_1:def 2

                .= ((1 / 2) + (1 / 2)) by SQUARE_1:def 2

                .= 1;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A9;

              end;

            end;

            hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1;

          end;

            case

             A14: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

            

             A15: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) & ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

            

             A16: (1 + ((q `1 ) ^2 )) > 0 by Lm1;

            

             A17: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A14, Def1;

            now

              per cases by A5;

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

                

                then ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / ( - 1)) ^2 )))) ^2 ) + ((( - 1) / ( sqrt (1 + (((q `1 ) / ( - 1)) ^2 )))) ^2 )) by A15, JGRAPH_1: 29

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

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

                .= ((1 / (1 + ((q `1 ) ^2 ))) + (((q `1 ) ^2 ) / (( sqrt (1 + ((q `1 ) ^2 ))) ^2 ))) by A16, SQUARE_1:def 2

                .= ((1 / (1 + ((q `1 ) ^2 ))) + (((q `1 ) ^2 ) / (1 + ((q `1 ) ^2 )))) by A16, SQUARE_1:def 2

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

                .= 1 by A16, XCMPLX_1: 60;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A17;

              end;

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

                

                then ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = (((1 / ( sqrt (1 + (((q `1 ) / 1) ^2 )))) ^2 ) + (((q `1 ) / ( sqrt (1 + (((q `1 ) / 1) ^2 )))) ^2 )) by A15, JGRAPH_1: 29

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

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

                .= ((1 / (1 + ((q `1 ) ^2 ))) + (((q `1 ) ^2 ) / (( sqrt (1 + ((q `1 ) ^2 ))) ^2 ))) by A16, SQUARE_1:def 2

                .= ((1 / (1 + ((q `1 ) ^2 ))) + (((q `1 ) ^2 ) / (1 + ((q `1 ) ^2 )))) by A16, SQUARE_1:def 2

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

                .= 1 by A16, XCMPLX_1: 60;

                then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| = 1 by SQUARE_1: 18, SQUARE_1: 22;

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A3, A4, A17;

              end;

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

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A14;

              end;

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

                hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1 by A14;

              end;

            end;

            hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| = 1;

          end;

        end;

        hence thesis by A1;

      end;

      let y be object;

      assume y in Cb;

      then

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

       A18: p2 = y and

       A19: |.p2.| = 1 by A1;

      set q = p2;

      now

        per cases ;

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

          hence contradiction by A19, TOPRNS_1: 23;

        end;

          case

           A20: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

          

           A21: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_1: 29;

          set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

          

           A22: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          

           A23: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

          

          then

           A24: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by XCMPLX_1: 89

          .= ((px `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          

           A25: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          then

           A26: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A22, A23, XCMPLX_1: 91;

          then

           A27: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A25, A23, XCMPLX_1: 89;

          (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A20, A23, XREAL_1: 64;

          then

           A28: (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A22, A25, A23, XREAL_1: 64;

          

           A29: (1 + (((px `2 ) / (px `1 )) ^2 )) > 0 by Lm1;

          (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A23, XCMPLX_1: 89

          .= ((px `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          then ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) ^2 )) = 1 by A19, A26, A24, A21, XCMPLX_1: 76;

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

          then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) = 1 by A29, SQUARE_1:def 2;

          

          then (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) = ((1 + (((px `2 ) / (px `1 )) ^2 )) * ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))))) by A29, SQUARE_1:def 2

          .= (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))));

          then (((px `1 ) ^2 ) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) = (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A29, XCMPLX_1: 87;

          

          then

           A30: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) = (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A29, XCMPLX_1: 87

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

           A31:

          now

            assume that

             A32: (px `1 ) = 0 and

             A33: (px `2 ) = 0 ;

            ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A33, EUCLID: 52;

            then

             A34: (q `2 ) = 0 by A23, XCMPLX_1: 6;

            ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A32, EUCLID: 52;

            then (q `1 ) = 0 by A23, XCMPLX_1: 6;

            hence contradiction by A20, A34, EUCLID: 53, EUCLID: 54;

          end;

          then not (px `1 ) = 0 by A22, A25, A23, A28, XREAL_1: 64;

          then ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) = ((px `2 ) ^2 ) by A30, XCMPLX_1: 6, XCMPLX_1: 87;

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

          then

           A35: (((px `1 ) ^2 ) - 1) = 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) = 0 by XCMPLX_1: 6;

          now

            per cases by A31, A35, COMPLEX1: 1, SQUARE_1: 41;

              case (px `1 ) = 1;

              hence ( - 1) = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or (px `1 ) = 1 & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or ( - 1) = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or 1 = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 by A22, A25, A23, A28, XREAL_1: 64;

            end;

              case (px `1 ) = ( - 1);

              hence ( - 1) = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or (px `1 ) = 1 & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or ( - 1) = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or 1 = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 by A22, A23, A28, XREAL_1: 64;

            end;

          end;

          then

           A36: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) & px in Kb by A1, FUNCT_2:def 1;

          (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A22, A25, A23, A28, XREAL_1: 64;

          then

           A37: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A31, Def1, JGRAPH_2: 3;

          ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A22, A23, A26, XCMPLX_1: 89;

          hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A18, A37, A27, A36, EUCLID: 53;

        end;

          case

           A38: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

          

           A39: ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ((q `1 ) ^2 )) by JGRAPH_1: 29;

          set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

          

           A40: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

          

           A41: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

          then

           A42: (q `1 ) = ((px `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A40, XCMPLX_1: 89;

          

           A43: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

          then

           A44: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A41, A40, XCMPLX_1: 91;

          then

           A45: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A41, A40, XCMPLX_1: 89;

          (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A38, JGRAPH_2: 13;

          then (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A40, XREAL_1: 64;

          then

           A46: (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A43, A41, A40, XREAL_1: 64;

          

           A47: (1 + (((px `1 ) / (px `2 )) ^2 )) > 0 by Lm1;

          (q `2 ) = ((px `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A43, A40, XCMPLX_1: 89;

          then ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) ^2 )) = 1 by A19, A44, A42, A39, XCMPLX_1: 76;

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

          then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) = 1 by A47, SQUARE_1:def 2;

          

          then (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) = ((1 + (((px `1 ) / (px `2 )) ^2 )) * ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))))) by A47, SQUARE_1:def 2

          .= (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))));

          then (((px `2 ) ^2 ) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) = (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A47, XCMPLX_1: 87;

          then (((px `2 ) ^2 ) + ((px `1 ) ^2 )) = (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A47, XCMPLX_1: 87;

          then

           A48: ((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) = (((px `1 ) ^2 ) / ((px `2 ) ^2 )) by XCMPLX_1: 76;

           A49:

          now

            assume that

             A50: (px `2 ) = 0 and (px `1 ) = 0 ;

            (q `2 ) = 0 by A43, A40, A50, XCMPLX_1: 6;

            hence contradiction by A38;

          end;

          then (px `2 ) <> 0 by A43, A41, A40, A46, XREAL_1: 64;

          then ((((px `2 ) ^2 ) + (((px `1 ) ^2 ) - 1)) * ((px `2 ) ^2 )) = ((px `1 ) ^2 ) by A48, XCMPLX_1: 6, XCMPLX_1: 87;

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

          then

           A51: (((px `2 ) ^2 ) - 1) = 0 or (((px `2 ) ^2 ) + ((px `1 ) ^2 )) = 0 by XCMPLX_1: 6;

          now

            per cases by A49, A51, COMPLEX1: 1, SQUARE_1: 41;

              case (px `2 ) = 1;

              hence ( - 1) = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or (px `2 ) = 1 & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or ( - 1) = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or 1 = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 by A43, A41, A40, A46, XREAL_1: 64;

            end;

              case (px `2 ) = ( - 1);

              hence ( - 1) = (px `2 ) & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or (px `2 ) = 1 & ( - 1) <= (px `1 ) & (px `1 ) <= 1 or ( - 1) = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 or 1 = (px `1 ) & ( - 1) <= (px `2 ) & (px `2 ) <= 1 by A43, A40, A46, XREAL_1: 64;

            end;

          end;

          then

           A52: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) & px in Kb by A1, FUNCT_2:def 1;

          (px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A43, A41, A40, A46, XREAL_1: 64;

          then

           A53: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A49, Th4, JGRAPH_2: 3;

          ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A43, A40, A44, XCMPLX_1: 89;

          hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A18, A53, A45, A52, EUCLID: 53;

        end;

      end;

      hence thesis by FUNCT_1:def 6;

    end;

    theorem :: JGRAPH_3:24

    

     Th24: for P,Kb be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | P) st 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 } & f is being_homeomorphism holds P is being_simple_closed_curve

    proof

      set X = (( TOP-REAL 2) | R^2-unit_square );

      set b = 1, a = 0 ;

      set v = |[1, 0 ]|;

      let P,Kb be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | P);

      assume

       A1: 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 } & f is being_homeomorphism;

      (v `1 ) = 1 & (v `2 ) = 0 by EUCLID: 52;

      then

       A2: |[1, 0 ]| in { 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 Kbb = Kb as non empty Subset of ( TOP-REAL 2) by A1;

      set A = (2 / (b - a)), B = (1 - ((2 * b) / (b - a))), C = (2 / (b - a)), D = (1 - ((2 * b) / (b - a)));

      reconsider Kbd = Kbb as non empty Subset of ( TOP-REAL 2);

      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)]|);

      

       A3: 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( A3);

      then

      consider ff be Function such that

       A4: ( dom ff) = the carrier of ( TOP-REAL 2) and

       A5: 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)]|;

      

       A6: for t be Point of ( TOP-REAL 2) holds (ff . t) = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by A5;

      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 A5;

        hence thesis;

      end;

      then

      reconsider ff as Function of ( TOP-REAL 2), ( TOP-REAL 2) by A4, FUNCT_2: 3;

      reconsider f11 = (ff | R^2-unit_square ) as Function of (( TOP-REAL 2) | R^2-unit_square ), ( TOP-REAL 2) by PRE_TOPC: 9;

      

       A7: f11 is continuous by A6, JGRAPH_2: 43, TOPMETR: 7;

      ff is one-to-one

      proof

        let x1,x2 be object;

        assume that

         A8: x1 in ( dom ff) & x2 in ( dom ff) and

         A9: (ff . x1) = (ff . x2);

        reconsider p1 = x1, p2 = x2 as Point of ( TOP-REAL 2) by A8;

        

         A10: (ff . x1) = |[((A * (p1 `1 )) + B), ((C * (p1 `2 )) + D)]| & (ff . x2) = |[((A * (p2 `1 )) + B), ((C * (p2 `2 )) + D)]| by A5;

        then (((A * (p1 `1 )) + B) - B) = (((A * (p2 `1 )) + B) - B) by A9, SPPOL_2: 1;

        then ((A * (p1 `1 )) / A) = (p2 `1 ) by XCMPLX_1: 89;

        then

         A11: (p1 `1 ) = (p2 `1 ) by XCMPLX_1: 89;

        (((C * (p1 `2 )) + D) - D) = (((C * (p2 `2 )) + D) - D) by A9, A10, SPPOL_2: 1;

        then ((C * (p1 `2 )) / C) = (p2 `2 ) by XCMPLX_1: 89;

        hence thesis by A11, TOPREAL3: 6, XCMPLX_1: 89;

      end;

      then

       A12: f11 is one-to-one by FUNCT_1: 52;

      

       A13: ( dom f11) = (( dom ff) /\ R^2-unit_square ) by RELAT_1: 61

      .= R^2-unit_square by A4, XBOOLE_1: 28;

      

       A14: Kbd c= ( rng f11)

      proof

        let y be object;

        assume

         A15: y in Kbd;

        then

        reconsider py = y as Point of ( TOP-REAL 2);

        set t = |[(((py `1 ) - B) / 2), (((py `2 ) - D) / 2)]|;

        

         A16: ex q st py = 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) by A1, A15;

        now

          per cases by A16;

            case

             A17: ( - 1) = (py `1 ) & ( - 1) <= (py `2 ) & (py `2 ) <= 1;

            then (2 - 1) >= (py `2 );

            then 2 >= ((py `2 ) + 1) by XREAL_1: 19;

            then

             A18: (2 / 2) >= (((py `2 ) - D) / 2) by XREAL_1: 72;

            ( 0 - 1) <= (py `2 ) by A17;

            then 0 <= ((py `2 ) + 1) by XREAL_1: 20;

            hence (t `1 ) = 0 & (t `2 ) <= 1 & (t `2 ) >= 0 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 1 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 0 or (t `1 ) = 1 & (t `2 ) <= 1 & (t `2 ) >= 0 by A17, A18, EUCLID: 52;

          end;

            case

             A19: (py `1 ) = 1 & ( - 1) <= (py `2 ) & (py `2 ) <= 1;

            then (2 - 1) >= (py `2 );

            then 2 >= ((py `2 ) + 1) by XREAL_1: 19;

            then

             A20: (2 / 2) >= (((py `2 ) - D) / 2) by XREAL_1: 72;

            ( 0 - 1) <= (py `2 ) by A19;

            then 0 <= ((py `2 ) + 1) by XREAL_1: 20;

            hence (t `1 ) = 0 & (t `2 ) <= 1 & (t `2 ) >= 0 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 1 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 0 or (t `1 ) = 1 & (t `2 ) <= 1 & (t `2 ) >= 0 by A19, A20, EUCLID: 52;

          end;

            case

             A21: ( - 1) = (py `2 ) & ( - 1) <= (py `1 ) & (py `1 ) <= 1;

            then (2 - 1) >= (py `1 );

            then 2 >= ((py `1 ) + 1) by XREAL_1: 19;

            then

             A22: (2 / 2) >= (((py `1 ) - B) / 2) by XREAL_1: 72;

            ( 0 - 1) <= (py `1 ) by A21;

            then 0 <= ((py `1 ) + 1) by XREAL_1: 20;

            hence (t `1 ) = 0 & (t `2 ) <= 1 & (t `2 ) >= 0 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 1 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 0 or (t `1 ) = 1 & (t `2 ) <= 1 & (t `2 ) >= 0 by A21, A22, EUCLID: 52;

          end;

            case

             A23: 1 = (py `2 ) & ( - 1) <= (py `1 ) & (py `1 ) <= 1;

            then (2 - 1) >= (py `1 );

            then 2 >= ((py `1 ) + 1) by XREAL_1: 19;

            then

             A24: (2 / 2) >= (((py `1 ) - B) / 2) by XREAL_1: 72;

            ( 0 - 1) <= (py `1 ) by A23;

            then 0 <= ((py `1 ) + 1) by XREAL_1: 20;

            hence (t `1 ) = 0 & (t `2 ) <= 1 & (t `2 ) >= 0 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 1 or (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 0 or (t `1 ) = 1 & (t `2 ) <= 1 & (t `2 ) >= 0 by A23, A24, EUCLID: 52;

          end;

        end;

        then

         A25: t in R^2-unit_square by TOPREAL1: 14;

        (t `1 ) = (((py `1 ) - B) / 2) & (t `2 ) = (((py `2 ) - D) / 2) by EUCLID: 52;

        then py = |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by EUCLID: 53;

        

        then py = (ff . t) by A5

        .= (f11 . t) by A25, FUNCT_1: 49;

        hence thesis by A13, A25, FUNCT_1:def 3;

      end;

      ( rng f11) c= Kbd

      proof

        let y be object;

        assume y in ( rng f11);

        then

        consider x be object such that

         A26: x in ( dom f11) and

         A27: y = (f11 . x) by FUNCT_1:def 3;

        reconsider t = x as Point of ( TOP-REAL 2) by A13, A26;

        

         A28: y = (ff . t) by A13, A26, A27, FUNCT_1: 49

        .= |[((A * (t `1 )) + B), ((C * (t `2 )) + D)]| by A5;

        then

        reconsider qy = y as Point of ( TOP-REAL 2);

        

         A29: ex p st t = p & ((p `1 ) = 0 & (p `2 ) <= 1 & (p `2 ) >= 0 or (p `1 ) <= 1 & (p `1 ) >= 0 & (p `2 ) = 1 or (p `1 ) <= 1 & (p `1 ) >= 0 & (p `2 ) = 0 or (p `1 ) = 1 & (p `2 ) <= 1 & (p `2 ) >= 0 ) by A13, A26, TOPREAL1: 14;

        now

          per cases by A29;

            suppose

             A30: (t `1 ) = 0 & (t `2 ) <= 1 & (t `2 ) >= 0 ;

            

             A31: (qy `2 ) = ((2 * (t `2 )) - 1) by A28, EUCLID: 52;

            (2 * 1) >= (2 * (t `2 )) by A30, XREAL_1: 64;

            then

             A32: ((1 + 1) - 1) >= (((qy `2 ) + 1) - 1) by A31, XREAL_1: 9;

            ( 0 - 1) <= (((qy `2 ) + 1) - 1) by A30, A31, XREAL_1: 9;

            hence ( - 1) = (qy `1 ) & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or (qy `1 ) = 1 & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or ( - 1) = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 or 1 = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 by A28, A30, A32, EUCLID: 52;

          end;

            suppose

             A33: (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 1;

            

             A34: (qy `1 ) = ((2 * (t `1 )) - 1) by A28, EUCLID: 52;

            (2 * 1) >= (2 * (t `1 )) by A33, XREAL_1: 64;

            then

             A35: ((1 + 1) - 1) >= (((qy `1 ) + 1) - 1) by A34, XREAL_1: 9;

            ( 0 - 1) <= (((qy `1 ) + 1) - 1) by A33, A34, XREAL_1: 9;

            hence ( - 1) = (qy `1 ) & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or (qy `1 ) = 1 & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or ( - 1) = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 or 1 = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 by A28, A33, A35, EUCLID: 52;

          end;

            suppose

             A36: (t `1 ) <= 1 & (t `1 ) >= 0 & (t `2 ) = 0 ;

            

             A37: (qy `1 ) = ((2 * (t `1 )) - 1) by A28, EUCLID: 52;

            (2 * 1) >= (2 * (t `1 )) by A36, XREAL_1: 64;

            then

             A38: ((1 + 1) - 1) >= (((qy `1 ) + 1) - 1) by A37, XREAL_1: 9;

            ( 0 - 1) <= (((qy `1 ) + 1) - 1) by A36, A37, XREAL_1: 9;

            hence ( - 1) = (qy `1 ) & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or (qy `1 ) = 1 & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or ( - 1) = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 or 1 = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 by A28, A36, A38, EUCLID: 52;

          end;

            suppose

             A39: (t `1 ) = 1 & (t `2 ) <= 1 & (t `2 ) >= 0 ;

            

             A40: (qy `2 ) = ((2 * (t `2 )) - 1) by A28, EUCLID: 52;

            (2 * 1) >= (2 * (t `2 )) by A39, XREAL_1: 64;

            then

             A41: ((1 + 1) - 1) >= (((qy `2 ) + 1) - 1) by A40, XREAL_1: 9;

            ( 0 - 1) <= (((qy `2 ) + 1) - 1) by A39, A40, XREAL_1: 9;

            hence ( - 1) = (qy `1 ) & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or (qy `1 ) = 1 & ( - 1) <= (qy `2 ) & (qy `2 ) <= 1 or ( - 1) = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 or 1 = (qy `2 ) & ( - 1) <= (qy `1 ) & (qy `1 ) <= 1 by A28, A39, A41, EUCLID: 52;

          end;

        end;

        hence thesis by A1;

      end;

      then Kbd = ( rng f11) by A14;

      then

      consider f1 be Function of X, (( TOP-REAL 2) | Kbd) such that f11 = f1 and

       A42: f1 is being_homeomorphism by A7, A12, JGRAPH_1: 46;

      ( dom f) = ( [#] (( TOP-REAL 2) | Kb)) by A1, TOPS_2:def 5

      .= Kb by PRE_TOPC:def 5;

      then (f . |[1, 0 ]|) in ( rng f) by A1, A2, FUNCT_1: 3;

      then

      reconsider PP = P as non empty Subset of ( TOP-REAL 2);

      reconsider g = f as Function of (( TOP-REAL 2) | Kbb), (( TOP-REAL 2) | PP);

      reconsider g as Function of (( TOP-REAL 2) | Kbb), (( TOP-REAL 2) | PP);

      reconsider f22 = f1 as Function of X, (( TOP-REAL 2) | Kbb);

      reconsider h = (g * f22) as Function of (( TOP-REAL 2) | R^2-unit_square ), (( TOP-REAL 2) | PP);

      h is being_homeomorphism by A1, A42, TOPS_2: 57;

      hence thesis by TOPREAL2:def 1;

    end;

    theorem :: JGRAPH_3:25

    

     Th25: for Kb be Subset of ( TOP-REAL 2) st 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 Kb is being_simple_closed_curve & Kb is compact

    proof

      set v = |[1, 0 ]|;

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

      assume

       A1: 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 };

      (v `1 ) = 1 & (v `2 ) = 0 by EUCLID: 52;

      then |[1, 0 ]| in { 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 Kbd = Kb as non empty Subset of ( TOP-REAL 2) by A1;

      set P = Kb;

      ( id (( TOP-REAL 2) | Kbd)) is being_homeomorphism;

      hence Kb is being_simple_closed_curve by A1, Th24;

      then

      consider f be Function of (( TOP-REAL 2) | R^2-unit_square ), (( TOP-REAL 2) | P) such that

       A2: f is being_homeomorphism by TOPREAL2:def 1;

      per cases ;

        suppose

         A3: P is empty;

        Kbd <> {} ;

        hence thesis by A3;

      end;

        suppose P is non empty;

        then

        reconsider R = P as non empty Subset of ( TOP-REAL 2);

        f is continuous & ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A2, TOPS_2:def 5;

        then (( TOP-REAL 2) | R) is compact by COMPTS_1: 14;

        hence thesis by COMPTS_1: 3;

      end;

    end;

    theorem :: JGRAPH_3:26

    for Cb be Subset of ( TOP-REAL 2) st Cb = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds Cb is being_simple_closed_curve

    proof

      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;

      

       A1: ( |[1, 0 ]| `1 ) = 1 & ( |[1, 0 ]| `2 ) = 0 by EUCLID: 52;

      

       A2: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      set v = |[1, 0 ]|;

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

      assume

       A3: Cb = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 };

      (v `1 ) = 1 & (v `2 ) = 0 by EUCLID: 52;

      then

       A4: |[1, 0 ]| in { 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 };

      { q where q be Element of ( TOP-REAL 2) : P[q] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

      then

      reconsider 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 } as non empty Subset of ( TOP-REAL 2) by A4;

       |. |[1, 0 ]|.| = ( sqrt ((( |[1, 0 ]| `1 ) ^2 ) + (( |[1, 0 ]| `2 ) ^2 ))) by JGRAPH_1: 30

      .= 1 by A1, SQUARE_1: 18;

      then |[1, 0 ]| in Cb by A3;

      then

      reconsider Cbb = Cb as non empty Subset of ( TOP-REAL 2);

      

       A5: the carrier of (( TOP-REAL 2) | Kb) = Kb by PRE_TOPC: 8;

      

       A6: ( dom ( Sq_Circ | Kb)) = (( dom Sq_Circ ) /\ Kb) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | Kb) by A5, A2, XBOOLE_1: 28;

      

       A7: ( rng ( Sq_Circ | Kb)) c= (( Sq_Circ | Kb) .: the carrier of (( TOP-REAL 2) | Kb))

      proof

        let u be object;

        assume u in ( rng ( Sq_Circ | Kb));

        then ex z be object st z in ( dom ( Sq_Circ | Kb)) & u = (( Sq_Circ | Kb) . z) by FUNCT_1:def 3;

        hence thesis by A6, FUNCT_1:def 6;

      end;

      (( Sq_Circ | Kb) .: the carrier of (( TOP-REAL 2) | Kb)) = ( Sq_Circ .: Kb) by A5, RELAT_1: 129

      .= Cb by A3, Th23

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

      then

      reconsider f0 = ( Sq_Circ | Kb) as Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | Cbb) by A6, A7, FUNCT_2: 2;

      ( rng ( Sq_Circ | Kb)) c= the carrier of ( TOP-REAL 2);

      then

      reconsider f00 = f0 as Function of (( TOP-REAL 2) | Kb), ( TOP-REAL 2) by A6, FUNCT_2: 2;

      

       A8: f0 is one-to-one & Kb is compact by Th25, FUNCT_1: 52;

      ( rng f0) = (( Sq_Circ | Kb) .: the carrier of (( TOP-REAL 2) | Kb)) by RELSET_1: 22

      .= ( Sq_Circ .: Kb) by A5, RELAT_1: 129

      .= Cb by A3, Th23;

      then ex f1 be Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | Cbb) st f00 = f1 & f1 is being_homeomorphism by A8, Th21, JGRAPH_1: 46, TOPMETR: 7;

      hence thesis by Th24;

    end;

    begin

    theorem :: JGRAPH_3:27

    for K0,C0 be Subset of ( TOP-REAL 2) st K0 = { p : ( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 } & C0 = { p1 where p1 be Point of ( TOP-REAL 2) : |.p1.| <= 1 } holds ( Sq_Circ " C0) c= K0

    proof

      let K0,C0 be Subset of ( TOP-REAL 2);

      assume

       A1: K0 = { p : ( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 } & C0 = { p1 where p1 be Point of ( TOP-REAL 2) : |.p1.| <= 1 };

      let x be object;

      assume

       A2: x in ( Sq_Circ " C0);

      then

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

      set q = px;

      

       A3: ( Sq_Circ . x) in C0 by A2, FUNCT_1:def 7;

      now

        per cases ;

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

          hence ( - 1) <= (px `1 ) & (px `1 ) <= 1 & ( - 1) <= (px `2 ) & (px `2 ) <= 1 by JGRAPH_2: 3;

        end;

          case

           A4: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

           A5:

          now

            assume (((px `1 ) ^2 ) + ((px `2 ) ^2 )) = 0 ;

            then (px `1 ) = 0 & (px `2 ) = 0 by COMPLEX1: 1;

            hence contradiction by A4, EUCLID: 53, EUCLID: 54;

          end;

          

           A6: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

           A7:

          now

            assume

             A8: (px `1 ) = 0 ;

            then (px `2 ) = 0 by A4;

            hence contradiction by A4, A8, EUCLID: 53, EUCLID: 54;

          end;

          

           A9: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

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

           A10: p1 = ( Sq_Circ . q) and

           A11: |.p1.| <= 1 by A1, A3;

          ( |.p1.| ^2 ) <= |.p1.| by A11, SQUARE_1: 42;

          then

           A12: ( |.p1.| ^2 ) <= 1 by A11, XXREAL_0: 2;

          

           A13: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

          ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A4, Def1;

          

          then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A9, A10, JGRAPH_1: 29

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

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

          .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A13, SQUARE_1:def 2

          .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A13, SQUARE_1:def 2

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

          then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) <= (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A13, A12, XREAL_1: 64;

          then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) <= (1 + (((q `2 ) / (q `1 )) ^2 )) by A13, XCMPLX_1: 87;

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

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

          then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A6, XREAL_1: 64;

          then ((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + ((((px `2 ) ^2 ) - 1) * ((px `1 ) ^2 ))) <= ((px `2 ) ^2 ) by A7, XCMPLX_1: 6, XCMPLX_1: 87;

          then ((((((px `1 ) ^2 ) * ((px `1 ) ^2 )) - (((px `1 ) ^2 ) * 1)) + (((px `1 ) ^2 ) * ((px `2 ) ^2 ))) - (1 * ((px `2 ) ^2 ))) <= 0 by XREAL_1: 47;

          then

           A14: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) <= 0 ;

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

          then

           A15: (((px `1 ) ^2 ) - 1) <= 0 by A6, A14, A5, XREAL_1: 129;

          then

           A16: (px `1 ) <= 1 by SQUARE_1: 43;

          

           A17: ( - 1) <= (px `1 ) by A15, SQUARE_1: 43;

          then (q `2 ) <= 1 & ( - ( - (q `1 ))) >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - ( - (q `1 ))) by A4, A16, XREAL_1: 24, XXREAL_0: 2;

          then (q `2 ) <= 1 & 1 >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= (q `1 ) by A16, XXREAL_0: 2;

          then (q `2 ) <= 1 & ( - 1) <= ( - ( - (q `2 ))) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - 1) by A17, XREAL_1: 24, XXREAL_0: 2;

          hence ( - 1) <= (px `1 ) & (px `1 ) <= 1 & ( - 1) <= (px `2 ) & (px `2 ) <= 1 by A15, SQUARE_1: 43, XREAL_1: 24;

        end;

          case

           A18: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

           A19:

          now

            assume (((px `2 ) ^2 ) + ((px `1 ) ^2 )) = 0 ;

            then (px `2 ) = 0 by COMPLEX1: 1;

            hence contradiction by A18;

          end;

          

           A20: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

          

           A21: (px `2 ) <> 0 by A18;

          

           A22: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) & ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

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

           A23: p1 = ( Sq_Circ . q) and

           A24: |.p1.| <= 1 by A1, A3;

          ( |.p1.| ^2 ) <= |.p1.| by A24, SQUARE_1: 42;

          then

           A25: ( |.p1.| ^2 ) <= 1 by A24, XXREAL_0: 2;

          

           A26: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by Lm1;

          ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A18, Def1;

          

          then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by A22, A23, JGRAPH_1: 29

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

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

          .= ((((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by A26, SQUARE_1:def 2

          .= ((((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 )))) by A26, SQUARE_1:def 2

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

          then (((((q `2 ) ^2 ) + ((q `1 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) * (1 + (((q `1 ) / (q `2 )) ^2 ))) <= (1 * (1 + (((q `1 ) / (q `2 )) ^2 ))) by A26, A25, XREAL_1: 64;

          then (((q `2 ) ^2 ) + ((q `1 ) ^2 )) <= (1 + (((q `1 ) / (q `2 )) ^2 )) by A26, XCMPLX_1: 87;

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

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

          then (((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) * ((px `2 ) ^2 )) <= ((((px `1 ) ^2 ) / ((px `2 ) ^2 )) * ((px `2 ) ^2 )) by A20, XREAL_1: 64;

          then ((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + ((((px `1 ) ^2 ) - 1) * ((px `2 ) ^2 ))) <= ((px `1 ) ^2 ) by A21, XCMPLX_1: 6, XCMPLX_1: 87;

          then ((((((px `2 ) ^2 ) * ((px `2 ) ^2 )) - (((px `2 ) ^2 ) * 1)) + (((px `2 ) ^2 ) * ((px `1 ) ^2 ))) - (1 * ((px `1 ) ^2 ))) <= 0 by XREAL_1: 47;

          then

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

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

          then

           A28: (((px `2 ) ^2 ) - 1) <= 0 by A20, A27, A19, XREAL_1: 129;

          then ( - 1) <= (px `2 ) & (px `2 ) <= 1 by SQUARE_1: 43;

          then (q `1 ) <= 1 & 1 >= ( - (q `1 )) or (q `1 ) >= ( - 1) & ( - (q `1 )) >= ( - 1) by A18, XXREAL_0: 2;

          then (q `1 ) <= 1 & ( - 1) <= ( - ( - (q `1 ))) or (q `1 ) >= ( - 1) & (q `1 ) <= 1 by XREAL_1: 24;

          hence ( - 1) <= (px `1 ) & (px `1 ) <= 1 & ( - 1) <= (px `2 ) & (px `2 ) <= 1 by A28, SQUARE_1: 43;

        end;

      end;

      hence thesis by A1;

    end;

    theorem :: JGRAPH_3:28

    

     Th28: for p holds (p = ( 0. ( TOP-REAL 2)) implies (( Sq_Circ " ) . p) = ( 0. ( 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)) implies (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) & ( not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) implies (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|)

    proof

      let p;

      set q = p;

      set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

      

       A1: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

      

       A2: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      hereby

        assume

         A3: p = ( 0. ( TOP-REAL 2));

        then ( Sq_Circ . p) = p by Def1;

        hence (( Sq_Circ " ) . p) = ( 0. ( TOP-REAL 2)) by A2, A3, FUNCT_1: 34;

      end;

      hereby

        

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

        set q = p;

        assume that

         A5: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )) and

         A6: p <> ( 0. ( TOP-REAL 2));

        set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

        

         A7: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        

         A8: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A9: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        then

         A10: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A7, A8, XCMPLX_1: 91;

        then

         A11: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A9, A8, XCMPLX_1: 89;

         A12:

        now

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

          then (q `1 ) = 0 & (q `2 ) = 0 by A7, A9, A8, XCMPLX_1: 6;

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

        end;

        (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A5, A8, XREAL_1: 64;

        then (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A7, A9, A8, XREAL_1: 64;

        then ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A7, A8, EUCLID: 52, XREAL_1: 64;

        then

         A13: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A7, A9, A12, Def1, JGRAPH_2: 3;

        ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A7, A8, A10, XCMPLX_1: 89;

        then q = ( Sq_Circ . px) by A13, A11, EUCLID: 53;

        hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A4, FUNCT_1: 34;

      end;

      

       A14: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      

       A15: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

      

       A16: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

      then

       A17: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A1, A15, XCMPLX_1: 91;

      then

       A18: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A16, A15, XCMPLX_1: 89;

      assume

       A19: not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

       A20:

      now

        assume that

         A21: (px `2 ) = 0 and (px `1 ) = 0 ;

        (q `2 ) = 0 by A1, A15, A21, XCMPLX_1: 6;

        hence contradiction by A19;

      end;

      (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )) by A19, JGRAPH_2: 13;

      then (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A15, XREAL_1: 64;

      then (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A1, A16, A15, XREAL_1: 64;

      then ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A1, A15, EUCLID: 52, XREAL_1: 64;

      then

       A22: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A1, A16, A20, Th4, JGRAPH_2: 3;

      ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A1, A15, A17, XCMPLX_1: 89;

      then q = ( Sq_Circ . px) by A22, A18, EUCLID: 53;

      hence thesis by A14, FUNCT_1: 34;

    end;

    theorem :: JGRAPH_3:29

    

     Th29: ( Sq_Circ " ) is Function of ( TOP-REAL 2), ( TOP-REAL 2)

    proof

      

       A1: the carrier of ( TOP-REAL 2) c= ( rng Sq_Circ )

      proof

        let y be object;

        assume y in the carrier of ( TOP-REAL 2);

        then

        reconsider py = y as Point of ( TOP-REAL 2);

        

         A2: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        now

          per cases ;

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

            then ( Sq_Circ . py) = py by Def1;

            hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A2;

          end;

            case

             A3: ((py `2 ) <= (py `1 ) & ( - (py `1 )) <= (py `2 ) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 ))) & py <> ( 0. ( TOP-REAL 2));

            set q = py;

            set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

            

             A4: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

             A5:

            now

              assume that

               A6: (px `1 ) = 0 and

               A7: (px `2 ) = 0 ;

              ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A7, EUCLID: 52;

              then

               A8: (q `2 ) = 0 by A4, XCMPLX_1: 6;

              ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A6, EUCLID: 52;

              then (q `1 ) = 0 by A4, XCMPLX_1: 6;

              hence contradiction by A3, A8, EUCLID: 53, EUCLID: 54;

            end;

            

             A9: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

            

             A10: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            

             A11: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            then

             A12: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A10, A4, XCMPLX_1: 91;

            then

             A13: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A11, A4, XCMPLX_1: 89;

            (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A3, A4, XREAL_1: 64;

            then (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A10, A11, A4, XREAL_1: 64;

            then ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A10, A4, EUCLID: 52, XREAL_1: 64;

            then

             A14: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A10, A11, A5, Def1, JGRAPH_2: 3;

            ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A10, A4, A12, XCMPLX_1: 89;

            hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A14, A13, A9, EUCLID: 53;

          end;

            case

             A15: not ((py `2 ) <= (py `1 ) & ( - (py `1 )) <= (py `2 ) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 ))) & py <> ( 0. ( TOP-REAL 2));

            set q = py;

            set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

            

             A16: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            

             A17: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

             A18:

            now

              assume that

               A19: (px `2 ) = 0 and (px `1 ) = 0 ;

              (q `2 ) = 0 by A17, A16, A19, XCMPLX_1: 6;

              hence contradiction by A15;

            end;

            

             A20: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

            then

             A21: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A17, A16, XCMPLX_1: 91;

            then

             A22: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A20, A16, XCMPLX_1: 89;

            (py `1 ) <= (py `2 ) & ( - (py `2 )) <= (py `1 ) or (py `1 ) >= (py `2 ) & (py `1 ) <= ( - (py `2 )) by A15, JGRAPH_2: 13;

            then (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A16, XREAL_1: 64;

            then (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A17, A20, A16, XREAL_1: 64;

            then ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A17, A16, EUCLID: 52, XREAL_1: 64;

            then

             A23: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A17, A20, A18, Th4, JGRAPH_2: 3;

            

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

            ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A17, A16, A21, XCMPLX_1: 89;

            hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A23, A22, A24, EUCLID: 53;

          end;

        end;

        hence thesis by FUNCT_1:def 3;

      end;

      

       A25: ( rng ( Sq_Circ " )) = ( dom Sq_Circ ) by FUNCT_1: 33

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

      ( dom ( Sq_Circ " )) = ( rng Sq_Circ ) by FUNCT_1: 33;

      then ( dom ( Sq_Circ " )) = the carrier of ( TOP-REAL 2) by A1;

      hence thesis by A25, FUNCT_2: 1;

    end;

    theorem :: JGRAPH_3:30

    

     Th30: 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 (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) & ( not ((p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ))) implies (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|)

    proof

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

      

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

      assume

       A2: p <> ( 0. ( TOP-REAL 2));

      hereby

        assume

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

        now

          per cases by A3;

            case

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

            now

              assume

               A5: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

               A6:

              now

                per cases by A5;

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

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A4, 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 A4, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A6;

                  case (p `1 ) = (p `2 );

                  hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, Th28;

                end;

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

                  then (p `1 ) <> 0 & ( - (p `1 )) = (p `2 ) by A2, EUCLID: 53, EUCLID: 54;

                  then ((p `1 ) / (p `2 )) = ( - 1) & ((p `2 ) / (p `1 )) = ( - 1) by XCMPLX_1: 197, XCMPLX_1: 198;

                  hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, Th28;

                end;

              end;

              hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

            end;

            hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th28;

          end;

            case

             A7: (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 ));

            now

              assume

               A8: (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

               A9:

              now

                per cases by A8;

                  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 A7, XXREAL_0: 1;

                end;

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

                  hence (p `1 ) = (p `2 ) or (p `1 ) = ( - (p `2 )) by A7, XXREAL_0: 1;

                end;

              end;

              now

                per cases by A9;

                  case (p `1 ) = (p `2 );

                  hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A8, Th28;

                end;

                  case

                   A10: (p `1 ) = ( - (p `2 ));

                  then (p `1 ) <> 0 & ( - (p `1 )) = (p `2 ) by A2, EUCLID: 53, EUCLID: 54;

                  then

                   A11: ((p `2 ) / (p `1 )) = ( - 1) by XCMPLX_1: 197;

                  (p `2 ) <> 0 by A2, A10, EUCLID: 53, EUCLID: 54;

                  then ((p `1 ) / (p `2 )) = ( - 1) by A10, XCMPLX_1: 197;

                  hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A8, A11, Th28;

                end;

              end;

              hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

            end;

            hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th28;

          end;

        end;

        hence (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

      end;

      

       A12: ( - (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 thesis by A2, A1, A12, Th28;

    end;

    theorem :: JGRAPH_3:31

    

     Th31: 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 * ( sqrt (1 + ((r1 / r2) ^2 ))))) & 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) = ( sqrt (1 + ((r1 / r2) ^2 ))) and

       A4: g2 is continuous by A1, A2, Th8;

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

       A5: for p be Point of X, r1,r0 be Real st (f1 . p) = r1 & (g2 . p) = r0 holds (g3 . p) = (r1 * r0) and

       A6: g3 is continuous by A1, A4, JGRAPH_2: 25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (r1 * ( sqrt (1 + ((r1 / r2) ^2 ))))

      proof

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

        assume that

         A7: (f1 . p) = r1 and

         A8: (f2 . p) = r2;

        (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) by A3, A7, A8;

        hence thesis by A5, A7;

      end;

      hence thesis by A6;

    end;

    theorem :: JGRAPH_3:32

    

     Th32: 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) = (r2 * ( sqrt (1 + ((r1 / r2) ^2 ))))) & 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 and

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

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

       A4: for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) and

       A5: g2 is continuous by A1, A2, A3, Th8;

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

       A6: for p be Point of X, r2,r0 be Real st (f2 . p) = r2 & (g2 . p) = r0 holds (g3 . p) = (r2 * r0) and

       A7: g3 is continuous by A2, A5, JGRAPH_2: 25;

      for p be Point of X, r1,r2 be Real st (f1 . p) = r1 & (f2 . p) = r2 holds (g3 . p) = (r2 * ( sqrt (1 + ((r1 / r2) ^2 ))))

      proof

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

        assume that

         A8: (f1 . p) = r1 and

         A9: (f2 . p) = r2;

        (g2 . p) = ( sqrt (1 + ((r1 / r2) ^2 ))) by A4, A8, A9;

        hence thesis by A6, A9;

      end;

      hence thesis by A7;

    end;

    theorem :: JGRAPH_3:33

    

     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 `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) & (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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      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 Lm6

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence thesis by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r2 * ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th32;

       A6:

      now

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `1 ) * ( sqrt (1 + (((r `2 ) / (r `1 )) ^2 )))) by A1, A7;

        hence (f . x) = (g3 . x) by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:34

    

     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 `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) & (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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) and

       A2: for q be Point of ( TOP-REAL 2) st q in the carrier of (( TOP-REAL 2) | K1) holds (q `1 ) <> 0 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      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 Lm6

        .= (q2 `1 ) by PSCOMP_1:def 5;

        hence thesis by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g2 . q) = r1 & (g1 . q) = r2 holds (g3 . q) = (r1 * ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th31;

       A6:

      now

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `2 ) * ( sqrt (1 + (((r `2 ) / (r `1 )) ^2 )))) by A1, A7;

        hence (f . x) = (g3 . x) by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:35

    

     Th35: 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 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      for q be Point of (( TOP-REAL 2) | K1) holds (g2 . 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;

        (g2 . q) = ( proj2 . q) by Lm4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence thesis by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g1 . q) = r1 & (g2 . q) = r2 holds (g3 . q) = (r2 * ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th32;

       A6:

      now

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `2 ) * ( sqrt (1 + (((r `1 ) / (r `2 )) ^2 )))) by A1, A7;

        hence (f . x) = (g3 . x) by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    theorem :: JGRAPH_3:36

    

     Th36: 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 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      reconsider g1 = ( proj1 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm7;

      reconsider g2 = ( proj2 | K1) as continuous Function of (( TOP-REAL 2) | K1), R^1 by Lm5;

      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 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^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 ;

      

       A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

      for q be Point of (( TOP-REAL 2) | K1) holds (g2 . 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;

        (g2 . q) = ( proj2 . q) by Lm4

        .= (q2 `2 ) by PSCOMP_1:def 6;

        hence thesis by A2;

      end;

      then

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

       A4: for q be Point of (( TOP-REAL 2) | K1), r1,r2 be Real st (g1 . q) = r1 & (g2 . q) = r2 holds (g3 . q) = (r1 * ( sqrt (1 + ((r1 / r2) ^2 )))) and

       A5: g3 is continuous by Th31;

       A6:

      now

        let x be object;

        assume

         A7: x in ( dom f);

        then

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

        x in the carrier of (( TOP-REAL 2) | K1) by A7;

        then x in K1 by PRE_TOPC: 8;

        then

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

        

         A8: ( proj2 . r) = (r `2 ) & ( proj1 . r) = (r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;

        

         A9: (g2 . s) = ( proj2 . s) & (g1 . s) = ( proj1 . s) by Lm4, Lm6;

        (f . r) = ((r `1 ) * ( sqrt (1 + (((r `1 ) / (r `2 )) ^2 )))) by A1, A7;

        hence (f . x) = (g3 . x) by A4, A9, A8;

      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 A5, A6, FUNCT_1: 2;

    end;

    

     Lm17: for K1 be non empty Subset of ( TOP-REAL 2) holds ( proj2 * (( Sq_Circ " ) | K1)) is Function of (( TOP-REAL 2) | K1), R^1

    proof

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

      

       A1: ( rng ( proj2 * (( Sq_Circ " ) | K1))) c= ( rng proj2 ) by RELAT_1: 26;

      

       A2: ( dom (( Sq_Circ " ) | K1)) c= ( dom ( proj2 * (( Sq_Circ " ) | K1)))

      proof

        let x be object;

        

         A3: ( rng ( Sq_Circ " )) c= the carrier of ( TOP-REAL 2) by Th29, RELAT_1:def 19;

        assume

         A4: x in ( dom (( Sq_Circ " ) | K1));

        then x in (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61;

        then x in ( dom ( Sq_Circ " )) by XBOOLE_0:def 4;

        then

         A5: (( Sq_Circ " ) . x) in ( rng ( Sq_Circ " )) by FUNCT_1: 3;

        ((( Sq_Circ " ) | K1) . x) = (( Sq_Circ " ) . x) by A4, FUNCT_1: 47;

        hence thesis by A4, A5, A3, Lm3, FUNCT_1: 11;

      end;

      ( dom ( proj2 * (( Sq_Circ " ) | K1))) c= ( dom (( Sq_Circ " ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj2 * (( Sq_Circ " ) | K1))) = ( dom (( Sq_Circ " ) | K1)) by A2

      .= (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

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

      hence thesis by A1, FUNCT_2: 2, TOPMETR: 17, XBOOLE_1: 1;

    end;

    

     Lm18: for K1 be non empty Subset of ( TOP-REAL 2) holds ( proj1 * (( Sq_Circ " ) | K1)) is Function of (( TOP-REAL 2) | K1), R^1

    proof

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

      

       A1: ( rng ( proj1 * (( Sq_Circ " ) | K1))) c= ( rng proj1 ) by RELAT_1: 26;

      

       A2: ( dom (( Sq_Circ " ) | K1)) c= ( dom ( proj1 * (( Sq_Circ " ) | K1)))

      proof

        let x be object;

        

         A3: ( rng ( Sq_Circ " )) c= the carrier of ( TOP-REAL 2) by Th29, RELAT_1:def 19;

        assume

         A4: x in ( dom (( Sq_Circ " ) | K1));

        then x in (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61;

        then x in ( dom ( Sq_Circ " )) by XBOOLE_0:def 4;

        then

         A5: (( Sq_Circ " ) . x) in ( rng ( Sq_Circ " )) by FUNCT_1: 3;

        ((( Sq_Circ " ) | K1) . x) = (( Sq_Circ " ) . x) by A4, FUNCT_1: 47;

        hence thesis by A4, A5, A3, Lm2, FUNCT_1: 11;

      end;

      ( dom ( proj1 * (( Sq_Circ " ) | K1))) c= ( dom (( Sq_Circ " ) | K1)) by RELAT_1: 25;

      

      then ( dom ( proj1 * (( Sq_Circ " ) | K1))) = ( dom (( Sq_Circ " ) | K1)) by A2

      .= (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

      .= K1 by XBOOLE_1: 28

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

      hence thesis by A1, FUNCT_2: 2, TOPMETR: 17, XBOOLE_1: 1;

    end;

    theorem :: JGRAPH_3:37

    

     Th37: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = (( Sq_Circ " ) | 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);

      assume

       A1: f = (( Sq_Circ " ) | 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)) };

      then ( 1.REAL 2) in K0 by Lm9, Lm10;

      then

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

      reconsider g1 = ( proj1 * (( Sq_Circ " ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by Lm18;

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

      proof

        

         A2: ( dom (( Sq_Circ " ) | K1)) = (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

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

        

         A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A4: 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 A1, A3;

        then

         A5: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Th28;

        ((( Sq_Circ " ) | K1) . p) = (( Sq_Circ " ) . p) by A4, A3, FUNCT_1: 49;

        

        then (g1 . p) = ( proj1 . |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) by A4, A2, A3, A5, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A6: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))));

      reconsider g2 = ( proj2 * (( Sq_Circ " ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by Lm17;

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

      proof

        

         A7: ( dom (( Sq_Circ " ) | K1)) = (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

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

        

         A8: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A9: 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 A1, A8;

        then

         A10: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Th28;

        ((( Sq_Circ " ) | K1) . p) = (( Sq_Circ " ) . p) by A9, A8, FUNCT_1: 49;

        

        then (g2 . p) = ( proj2 . |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|) by A9, A7, A8, A10, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A11: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))));

      

       A12: 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);

        

         A13: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

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

        then

         A14: 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 A1, A13;

        now

          assume

           A15: (q `1 ) = 0 ;

          then (q `2 ) = 0 by A14;

          hence contradiction by A14, A15, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      then

       A16: f1 is continuous by A6, Th33;

       A17:

      now

        let x,y,r,s be Real;

        assume that

         A18: |[x, y]| in K1 and

         A19: r = (f1 . |[x, y]|) & s = (f2 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A20: 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 A1, A18;

        

         A21: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A22: (f1 . p99) = ((p99 `1 ) * ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 )))) by A6, A18;

        ((( Sq_Circ " ) | K0) . |[x, y]|) = (( Sq_Circ " ) . |[x, y]|) by A18, FUNCT_1: 49

        .= |[((p99 `1 ) * ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 )))), ((p99 `2 ) * ( sqrt (1 + (((p99 `2 ) / (p99 `1 )) ^2 ))))]| by A20, Th28

        .= |[r, s]| by A11, A18, A19, A21, A22;

        hence (f . |[x, y]|) = |[r, s]| by A1;

      end;

      f2 is continuous by A12, A11, Th34;

      hence thesis by A1, A16, A17, Lm13, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_3:38

    

     Th38: for K0,B0 be Subset of ( TOP-REAL 2), f be Function of (( TOP-REAL 2) | K0), (( TOP-REAL 2) | B0) st f = (( Sq_Circ " ) | 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);

      assume

       A1: f = (( Sq_Circ " ) | 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)) };

      then ( 1.REAL 2) in K0 by Lm14, Lm15;

      then

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

      reconsider g1 = ( proj2 * (( Sq_Circ " ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by Lm17;

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

      proof

        

         A2: ( dom (( Sq_Circ " ) | K1)) = (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

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

        

         A3: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A4: 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 A1, A3;

        then

         A5: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th30;

        ((( Sq_Circ " ) | K1) . p) = (( Sq_Circ " ) . p) by A4, A3, FUNCT_1: 49;

        

        then (g1 . p) = ( proj2 . |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A4, A2, A3, A5, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A6: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f1 . p) = ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))));

      reconsider g2 = ( proj1 * (( Sq_Circ " ) | K1)) as Function of (( TOP-REAL 2) | K1), R^1 by Lm18;

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

      proof

        

         A7: ( dom (( Sq_Circ " ) | K1)) = (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

        .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

        .= K1 by XBOOLE_1: 28;

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

        

         A8: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        assume

         A9: 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 A1, A8;

        then

         A10: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th30;

        ((( Sq_Circ " ) | K1) . p) = (( Sq_Circ " ) . p) by A9, A8, FUNCT_1: 49;

        

        then (g2 . p) = ( proj1 . |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|) by A9, A7, A8, A10, FUNCT_1: 13

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

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

        hence thesis;

      end;

      then

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

       A11: for p be Point of ( TOP-REAL 2) st p in the carrier of (( TOP-REAL 2) | K1) holds (f2 . p) = ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))));

      

       A12: 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);

        

         A13: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

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

        then

         A14: 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 A1, A13;

        now

          assume

           A15: (q `2 ) = 0 ;

          then (q `1 ) = 0 by A14;

          hence contradiction by A14, A15, EUCLID: 53, EUCLID: 54;

        end;

        hence thesis;

      end;

      then

       A16: f1 is continuous by A6, Th35;

      

       A17: 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

         A18: |[x, y]| in K1 and

         A19: s = (f2 . |[x, y]|) & r = (f1 . |[x, y]|);

        set p99 = |[x, y]|;

        

         A20: 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 A1, A18;

        

         A21: the carrier of (( TOP-REAL 2) | K1) = K1 by PRE_TOPC: 8;

        then

         A22: (f1 . p99) = ((p99 `2 ) * ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 )))) by A6, A18;

        ((( Sq_Circ " ) | K0) . |[x, y]|) = (( Sq_Circ " ) . |[x, y]|) by A18, FUNCT_1: 49

        .= |[((p99 `1 ) * ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 )))), ((p99 `2 ) * ( sqrt (1 + (((p99 `1 ) / (p99 `2 )) ^2 ))))]| by A20, Th30

        .= |[s, r]| by A11, A18, A19, A21, A22;

        hence thesis by A1;

      end;

      f2 is continuous by A12, A11, Th36;

      hence thesis by A1, A16, A17, Lm13, JGRAPH_2: 35;

    end;

    theorem :: JGRAPH_3:39

    

     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 = (( Sq_Circ " ) | 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 JGRAPH_2: 47;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `1 )) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 47;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      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 )));

      set b0 = ( NonZero ( TOP-REAL 2));

      defpred P0[ Point of ( TOP-REAL 2)] means (($1 `2 ) <= ($1 `1 ) & ( - ($1 `1 )) <= ($1 `2 ) or ($1 `2 ) >= ($1 `1 ) & ($1 `2 ) <= ( - ($1 `1 )));

      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);

      set k0 = { p : P0[p] & p <> ( 0. ( TOP-REAL 2)) };

      assume that

       A1: f = (( Sq_Circ " ) | K0) and

       A2: B0 = b0 & K0 = k0;

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      k0 c= ( NonZero ( TOP-REAL 2)) from TopIncl;

      then

       A3: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by A2, PRE_TOPC: 7;

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

      

       A4: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A5: x in ((K2 /\ K3) \/ (K4 /\ K5));

        per cases by A5, XBOOLE_0:def 3;

          suppose

           A6: x in (K2 /\ K3);

          then x in K3 by XBOOLE_0:def 4;

          then

           A7: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `1 )) <= (p8 `2 );

          x in K2 by A6, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) <= (p7 `1 );

          hence thesis by A7;

        end;

          suppose

           A8: x in (K4 /\ K5);

          then x in K5 by XBOOLE_0:def 4;

          then

           A9: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `2 ) <= ( - (p8 `1 ));

          x in K4 by A8, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `2 ) >= (p7 `1 );

          hence thesis by A9;

        end;

      end;

      

       A10: (K2 /\ K3) is closed & (K4 /\ K5) is closed by TOPS_1: 8;

      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 A4;

      then

       A11: K1 is closed by A10, TOPS_1: 9;

      k0 = ({ p7 where p7 be Point of ( TOP-REAL 2) : P0[p7] } /\ b0) from TopInter;

      then K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A2, PRE_TOPC:def 5;

      hence thesis by A1, A2, A3, A11, Th37, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_3:40

    

     Th40: 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 = (( Sq_Circ " ) | 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 JGRAPH_2: 48;

      reconsider K4 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `2 ) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      reconsider K3 = { p7 where p7 be Point of ( TOP-REAL 2) : ( - (p7 `2 )) <= (p7 `1 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 48;

      reconsider K2 = { p7 where p7 be Point of ( TOP-REAL 2) : (p7 `1 ) <= (p7 `2 ) } as closed Subset of ( TOP-REAL 2) by JGRAPH_2: 46;

      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 )));

      defpred P0[ Point of ( TOP-REAL 2)] means (($1 `1 ) <= ($1 `2 ) & ( - ($1 `2 )) <= ($1 `1 ) or ($1 `1 ) >= ($1 `2 ) & ($1 `1 ) <= ( - ($1 `2 )));

      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);

      set k0 = { p : P0[p] & p <> ( 0. ( TOP-REAL 2)) }, b0 = ( NonZero ( TOP-REAL 2));

      assume that

       A1: f = (( Sq_Circ " ) | K0) and

       A2: B0 = b0 & K0 = k0;

      the carrier of (( TOP-REAL 2) | B0) = B0 by PRE_TOPC: 8;

      then

      reconsider K1 = K0 as Subset of ( TOP-REAL 2) by XBOOLE_1: 1;

      { p : P[p] & p <> ( 0. ( TOP-REAL 2)) } c= ( NonZero ( TOP-REAL 2)) from TopIncl;

      then

       A3: ((( TOP-REAL 2) | B0) | K0) = (( TOP-REAL 2) | K1) by A2, PRE_TOPC: 7;

      set k1 = { p7 where p7 be Point of ( TOP-REAL 2) : P0[p7] };

      

       A4: (K2 /\ K3) is closed & (K4 /\ K5) is closed by TOPS_1: 8;

      reconsider K1 = k1 as Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

      

       A5: ((K2 /\ K3) \/ (K4 /\ K5)) c= K1

      proof

        let x be object;

        assume

         A6: x in ((K2 /\ K3) \/ (K4 /\ K5));

        per cases by A6, XBOOLE_0:def 3;

          suppose

           A7: x in (K2 /\ K3);

          then x in K3 by XBOOLE_0:def 4;

          then

           A8: ex p8 be Point of ( TOP-REAL 2) st p8 = x & ( - (p8 `2 )) <= (p8 `1 );

          x in K2 by A7, XBOOLE_0:def 4;

          then ex p7 be Point of ( TOP-REAL 2) st p7 = x & (p7 `1 ) <= (p7 `2 );

          hence thesis by A8;

        end;

          suppose

           A9: x in (K4 /\ K5);

          then x in K5 by XBOOLE_0:def 4;

          then

           A10: ex p8 be Point of ( TOP-REAL 2) st p8 = x & (p8 `1 ) <= ( - (p8 `2 ));

          x in K4 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;

      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 A5;

      then

       A11: K1 is closed by A4, TOPS_1: 9;

      k0 = (k1 /\ b0) from TopInter;

      then K0 = (K1 /\ ( [#] (( TOP-REAL 2) | B0))) by A2, PRE_TOPC:def 5;

      hence thesis by A1, A2, A3, A11, Th38, PRE_TOPC: 13;

    end;

    theorem :: JGRAPH_3:41

    

     Th41: 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 = (( Sq_Circ " ) | D) & h is continuous

    proof

      set Y1 = |[( - 1), 1]|;

      set B0 = {( 0. ( TOP-REAL 2))};

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

      

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

      ( dom ( Sq_Circ " )) = the carrier of ( TOP-REAL 2) by Th29, FUNCT_2:def 1;

      

      then

       A2: ( dom (( Sq_Circ " ) | D)) = (the carrier of ( TOP-REAL 2) /\ D) by RELAT_1: 61

      .= the carrier of (( TOP-REAL 2) | D) by A1, XBOOLE_1: 28;

      assume

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

      

      then

       A4: D = (B0 ` )

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

      

       A5: { 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

         A6: 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 A6, XBOOLE_0:def 5;

          then x in (D ` ) by SUBSET_1:def 4;

          hence contradiction by A3, A6, TARSKI:def 1;

        end;

        hence thesis by PRE_TOPC: 8;

      end;

      ( 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 Lm9, Lm10;

      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 A5;

      

       A7: K0 = the carrier of ((( TOP-REAL 2) | D) | K0) by PRE_TOPC: 8;

      

       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 A3, A9, TARSKI:def 1;

        end;

        hence thesis by PRE_TOPC: 8;

      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 JGRAPH_2: 3;

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

      

       A11: D c= (K0 \/ K1)

      proof

        let x be object;

        assume

         A12: x in D;

        then

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

         not x in {( 0. ( TOP-REAL 2))} by A4, A12, 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;

      

       A13: the carrier of (( TOP-REAL 2) | D) = ( [#] (( TOP-REAL 2) | D))

      .= ( NonZero ( TOP-REAL 2)) by A4, PRE_TOPC:def 5;

      

       A14: K0 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K0;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `2 ) <= (p8 `1 ) & ( - (p8 `1 )) <= (p8 `2 ) or (p8 `2 ) >= (p8 `1 ) & (p8 `2 ) <= ( - (p8 `1 ))) & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      

       A15: ( rng (( Sq_Circ " ) | K0)) c= the carrier of ((( TOP-REAL 2) | D) | K0)

      proof

        reconsider K00 = K0 as Subset of ( TOP-REAL 2) by A14;

        let y be object;

        

         A16: 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);

          

           A17: the carrier of (( TOP-REAL 2) | K00) = K0 by PRE_TOPC: 8;

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

          then

           A18: 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 A17;

          now

            assume

             A19: (q `1 ) = 0 ;

            then (q `2 ) = 0 by A18;

            hence contradiction by A18, A19, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

        assume y in ( rng (( Sq_Circ " ) | K0));

        then

        consider x be object such that

         A20: x in ( dom (( Sq_Circ " ) | K0)) and

         A21: y = ((( Sq_Circ " ) | K0) . x) by FUNCT_1:def 3;

        

         A22: x in (( dom ( Sq_Circ " )) /\ K0) by A20, RELAT_1: 61;

        then

         A23: x in K0 by XBOOLE_0:def 4;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A14;

        K00 = the carrier of (( TOP-REAL 2) | K00) by PRE_TOPC: 8;

        then p in the carrier of (( TOP-REAL 2) | K00) by A22, XBOOLE_0:def 4;

        then

         A24: (p `1 ) <> 0 by A16;

        set p9 = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|;

        

         A25: (p9 `1 ) = ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & (p9 `2 ) = ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by EUCLID: 52;

        

         A26: 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 A23;

        then

         A27: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by Th28;

        

         A28: ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        then ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & (( - (p `1 )) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) or ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) >= ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= (( - (p `1 )) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A26, XREAL_1: 64;

        then

         A29: ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ( - ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))) <= ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) or ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) >= ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) & ((p `2 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) <= ( - ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))));

        

         A30: (p9 `1 ) = ((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by EUCLID: 52;

         A31:

        now

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

          then ( 0 / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) = (((p `1 ) * ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A30, EUCLID: 52, EUCLID: 54;

          hence contradiction by A24, A28, XCMPLX_1: 89;

        end;

        (( Sq_Circ " ) . p) = y by A21, A23, FUNCT_1: 49;

        then y in K0 by A31, A27, A29, A25;

        hence thesis by PRE_TOPC: 8;

      end;

      ( dom (( Sq_Circ " ) | K0)) = (( dom ( Sq_Circ " )) /\ K0) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K0) by Th29, FUNCT_2:def 1

      .= K0 by A14, XBOOLE_1: 28;

      then

      reconsider f = (( Sq_Circ " ) | K0) as Function of ((( TOP-REAL 2) | D) | K0), (( TOP-REAL 2) | D) by A7, A15, FUNCT_2: 2, XBOOLE_1: 1;

      

       A32: K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      

       A33: K1 c= the carrier of ( TOP-REAL 2)

      proof

        let z be object;

        assume z in K1;

        then ex p8 be Point of ( TOP-REAL 2) st p8 = z & ((p8 `1 ) <= (p8 `2 ) & ( - (p8 `2 )) <= (p8 `1 ) or (p8 `1 ) >= (p8 `2 ) & (p8 `1 ) <= ( - (p8 `2 ))) & p8 <> ( 0. ( TOP-REAL 2));

        hence thesis;

      end;

      

       A34: ( rng (( Sq_Circ " ) | K1)) c= the carrier of ((( TOP-REAL 2) | D) | K1)

      proof

        reconsider K10 = K1 as Subset of ( TOP-REAL 2) by A33;

        let y be object;

        

         A35: 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);

          

           A36: the carrier of (( TOP-REAL 2) | K10) = K1 by PRE_TOPC: 8;

          assume q in the carrier of (( TOP-REAL 2) | K10);

          then

           A37: 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 A36;

          now

            assume

             A38: (q `2 ) = 0 ;

            then (q `1 ) = 0 by A37;

            hence contradiction by A37, A38, EUCLID: 53, EUCLID: 54;

          end;

          hence thesis;

        end;

        assume y in ( rng (( Sq_Circ " ) | K1));

        then

        consider x be object such that

         A39: x in ( dom (( Sq_Circ " ) | K1)) and

         A40: y = ((( Sq_Circ " ) | K1) . x) by FUNCT_1:def 3;

        

         A41: x in (( dom ( Sq_Circ " )) /\ K1) by A39, RELAT_1: 61;

        then

         A42: x in K1 by XBOOLE_0:def 4;

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A33;

        K10 = the carrier of (( TOP-REAL 2) | K10) by PRE_TOPC: 8;

        then p in the carrier of (( TOP-REAL 2) | K10) by A41, XBOOLE_0:def 4;

        then

         A43: (p `2 ) <> 0 by A35;

        set p9 = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|;

        

         A44: (p9 `2 ) = ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & (p9 `1 ) = ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by EUCLID: 52;

        

         A45: 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 A42;

        then

         A46: (( Sq_Circ " ) . p) = |[((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by Th30;

        

         A47: ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        then ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & (( - (p `2 )) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) or ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) >= ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= (( - (p `2 )) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A45, XREAL_1: 64;

        then

         A48: ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ( - ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))) <= ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) or ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) >= ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) & ((p `1 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) <= ( - ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))));

        

         A49: (p9 `2 ) = ((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by EUCLID: 52;

         A50:

        now

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

          then ( 0 / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) = (((p `2 ) * ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A49, EUCLID: 52, EUCLID: 54;

          hence contradiction by A43, A47, XCMPLX_1: 89;

        end;

        (( Sq_Circ " ) . p) = y by A40, A42, FUNCT_1: 49;

        then y in K1 by A50, A46, A48, A44;

        hence thesis by PRE_TOPC: 8;

      end;

      ( dom (( Sq_Circ " ) | K1)) = (( dom ( Sq_Circ " )) /\ K1) by RELAT_1: 61

      .= (the carrier of ( TOP-REAL 2) /\ K1) by Th29, FUNCT_2:def 1

      .= K1 by A33, XBOOLE_1: 28;

      then

      reconsider g = (( Sq_Circ " ) | K1) as Function of ((( TOP-REAL 2) | D) | K1), (( TOP-REAL 2) | D) by A10, A34, FUNCT_2: 2, XBOOLE_1: 1;

      

       A51: ( dom g) = K1 by A10, FUNCT_2:def 1;

      g = (( Sq_Circ " ) | K1);

      then

       A52: K1 is closed by A4, Th40;

      

       A53: K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) by PRE_TOPC:def 5;

       A54:

      now

        let x be object;

        assume

         A55: x in (( [#] ((( TOP-REAL 2) | D) | K0)) /\ ( [#] ((( TOP-REAL 2) | D) | K1)));

        then x in K0 by A53, XBOOLE_0:def 4;

        then (f . x) = (( Sq_Circ " ) . x) by FUNCT_1: 49;

        hence (f . x) = (g . x) by A32, A55, FUNCT_1: 49;

      end;

      f = (( Sq_Circ " ) | K0);

      then

       A56: K0 is closed by A4, Th39;

      

       A57: ( dom f) = K0 by A7, FUNCT_2:def 1;

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

      then

       A58: (( [#] ((( TOP-REAL 2) | D) | K0)) \/ ( [#] ((( TOP-REAL 2) | D) | K1))) = ( [#] (( TOP-REAL 2) | D)) by A53, A32, A11;

      

       A59: f is continuous & g is continuous by A4, Th39, Th40;

      then

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

       A60: h = (f +* g) and h is continuous by A53, A32, A58, A56, A52, A54, JGRAPH_2: 1;

      K0 = ( [#] ((( TOP-REAL 2) | D) | K0)) & K1 = ( [#] ((( TOP-REAL 2) | D) | K1)) by PRE_TOPC:def 5;

      then

       A61: f tolerates g by A54, A57, A51, PARTFUN1:def 4;

      

       A62: for x be object st x in ( dom h) holds (h . x) = ((( Sq_Circ " ) | D) . x)

      proof

        let x be object;

        assume

         A63: x in ( dom h);

        then

        reconsider p = x as Point of ( TOP-REAL 2) by A13, XBOOLE_0:def 5;

         not x in {( 0. ( TOP-REAL 2))} by A13, A63, XBOOLE_0:def 5;

        then

         A64: x <> ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        x in (the carrier of ( TOP-REAL 2) \ (D ` )) by A3, A13, A63;

        then

         A65: x in ((D ` ) ` ) by SUBSET_1:def 4;

        per cases ;

          suppose

           A66: x in K0;

          

           A67: ((( Sq_Circ " ) | D) . p) = (( Sq_Circ " ) . p) by A65, FUNCT_1: 49

          .= (f . p) by A66, FUNCT_1: 49;

          (h . p) = ((g +* f) . p) by A60, A61, FUNCT_4: 34

          .= (f . p) by A57, A66, FUNCT_4: 13;

          hence thesis by A67;

        end;

          suppose not x in K0;

          then not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ))) by A64;

          then (p `1 ) <= (p `2 ) & ( - (p `2 )) <= (p `1 ) or (p `1 ) >= (p `2 ) & (p `1 ) <= ( - (p `2 )) by XREAL_1: 26;

          then

           A68: x in K1 by A64;

          ((( Sq_Circ " ) | D) . p) = (( Sq_Circ " ) . p) by A65, FUNCT_1: 49

          .= (g . p) by A68, FUNCT_1: 49;

          hence thesis by A60, A51, A68, FUNCT_4: 13;

        end;

      end;

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

      then (f +* g) = (( Sq_Circ " ) | D) by A60, A2, A62;

      hence thesis by A53, A32, A58, A56, A59, A52, A54, JGRAPH_2: 1;

    end;

    theorem :: JGRAPH_3:42

    

     Th42: ex h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st h = ( Sq_Circ " ) & h is continuous

    proof

      reconsider f = ( Sq_Circ " ) as Function of ( TOP-REAL 2), ( TOP-REAL 2) by Th29;

      reconsider D = ( NonZero ( TOP-REAL 2)) as non empty Subset of ( TOP-REAL 2) by JGRAPH_2: 9;

      

       A1: (f . ( 0. ( TOP-REAL 2))) = ( 0. ( TOP-REAL 2)) by Th28;

      

       A2: for p be Point of (( TOP-REAL 2) | D) holds (f . p) <> (f . ( 0. ( TOP-REAL 2)))

      proof

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

        

         A3: ( [#] (( TOP-REAL 2) | D)) = D by PRE_TOPC:def 5;

        then

        reconsider q = p as Point of ( TOP-REAL 2) by XBOOLE_0:def 5;

         not p in {( 0. ( TOP-REAL 2))} by A3, XBOOLE_0:def 5;

        then

         A4: not p = ( 0. ( TOP-REAL 2)) by TARSKI:def 1;

        per cases ;

          suppose

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

          then

           A6: (q `2 ) <> 0 ;

          set q9 = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

          

           A7: (q9 `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

          

           A8: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

          now

            assume q9 = ( 0. ( TOP-REAL 2));

            then ( 0 * (q `2 )) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A7, EUCLID: 52, EUCLID: 54;

            then ( 0 * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = (((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))));

            hence contradiction by A6, A8, XCMPLX_1: 89;

          end;

          hence thesis by A1, A5, Th28;

        end;

          suppose

           A9: (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

           A10:

          now

            assume

             A11: (q `1 ) = 0 ;

            then (q `2 ) = 0 by A9;

            hence contradiction by A4, A11, EUCLID: 53, EUCLID: 54;

          end;

          set q9 = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

          

           A12: (q9 `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

          

           A13: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

          now

            assume q9 = ( 0. ( TOP-REAL 2));

            then ( 0 / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A12, EUCLID: 52, EUCLID: 54;

            hence contradiction by A10, A13, XCMPLX_1: 89;

          end;

          hence thesis by A1, A4, A9, Th28;

        end;

      end;

      

       A14: for V be Subset of ( TOP-REAL 2) st (f . ( 0. ( TOP-REAL 2))) in V & V is open holds ex W be Subset of ( TOP-REAL 2) st ( 0. ( TOP-REAL 2)) in W & W is open & (f .: W) c= V

      proof

        reconsider u0 = ( 0. ( TOP-REAL 2)) as Point of ( Euclid 2) by EUCLID: 67;

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

        reconsider VV = V as Subset of ( TopSpaceMetr ( Euclid 2)) by Lm16;

        assume that

         A15: (f . ( 0. ( TOP-REAL 2))) in V and

         A16: V is open;

        VV is open by A16, Lm16, PRE_TOPC: 30;

        then

        consider r be Real such that

         A17: r > 0 and

         A18: ( Ball (u0,r)) c= V by A1, A15, TOPMETR: 15;

        reconsider r as Real;

        reconsider W1 = ( Ball (u0,r)), V1 = ( Ball (u0,(r / ( sqrt 2)))) as Subset of ( TOP-REAL 2) by EUCLID: 67;

        

         A19: (f .: V1) c= W1

        proof

          let z be object;

          

           A20: ( sqrt 2) > 0 by SQUARE_1: 25;

          assume z in (f .: V1);

          then

          consider y be object such that

           A21: y in ( dom f) and

           A22: y in V1 and

           A23: z = (f . y) by FUNCT_1:def 6;

          z in ( rng f) by A21, A23, FUNCT_1:def 3;

          then

          reconsider qz = z as Point of ( TOP-REAL 2);

          reconsider pz = qz as Point of ( Euclid 2) by EUCLID: 67;

          reconsider q = y as Point of ( TOP-REAL 2) by A21;

          reconsider qy = q as Point of ( Euclid 2) by EUCLID: 67;

          

           A24: ((q `1 ) ^2 ) >= 0 by XREAL_1: 63;

          

           A25: ((q `2 ) ^2 ) >= 0 by XREAL_1: 63;

          ( dist (u0,qy)) < (r / ( sqrt 2)) by A22, METRIC_1: 11;

          then |.(( 0. ( TOP-REAL 2)) - q).| < (r / ( sqrt 2)) by JGRAPH_1: 28;

          then ( sqrt ((((( 0. ( TOP-REAL 2)) - q) `1 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - q) `2 ) ^2 ))) < (r / ( sqrt 2)) by JGRAPH_1: 30;

          then ( sqrt ((((( 0. ( TOP-REAL 2)) `1 ) - (q `1 )) ^2 ) + (((( 0. ( TOP-REAL 2)) - q) `2 ) ^2 ))) < (r / ( sqrt 2)) by TOPREAL3: 3;

          then ( sqrt ((((( 0. ( TOP-REAL 2)) `1 ) - (q `1 )) ^2 ) + (((( 0. ( TOP-REAL 2)) `2 ) - (q `2 )) ^2 ))) < (r / ( sqrt 2)) by TOPREAL3: 3;

          then (( sqrt (((q `1 ) ^2 ) + ((q `2 ) ^2 ))) * ( sqrt 2)) < ((r / ( sqrt 2)) * ( sqrt 2)) by A20, JGRAPH_2: 3, XREAL_1: 68;

          then ( sqrt ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) * 2)) < ((r / ( sqrt 2)) * ( sqrt 2)) by A24, A25, SQUARE_1: 29;

          then

           A26: ( sqrt ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) * 2)) < r by A20, XCMPLX_1: 87;

          per cases ;

            suppose q = ( 0. ( TOP-REAL 2));

            then z = ( 0. ( TOP-REAL 2)) by A23, Th28;

            hence thesis by A17, GOBOARD6: 1;

          end;

            suppose

             A27: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

             A28:

            now

              assume ((q `1 ) ^2 ) <= 0 ;

              then ((q `1 ) ^2 ) = 0 by XREAL_1: 63;

              then

               A29: (q `1 ) = 0 by XCMPLX_1: 6;

              then (q `2 ) = 0 by A27;

              hence contradiction by A27, A29, EUCLID: 53, EUCLID: 54;

            end;

            

             A30: (( Sq_Circ " ) . q) = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A27, Th28;

            then (qz `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A23, EUCLID: 52;

            then

             A31: ((qz `1 ) ^2 ) = (((q `1 ) ^2 ) * (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ));

            (qz `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A23, A30, EUCLID: 52;

            then

             A32: ((qz `2 ) ^2 ) = (((q `2 ) ^2 ) * (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ));

            

             A33: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

            now

              per cases by A27;

                case

                 A34: (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 );

                now

                  per cases ;

                    case 0 <= (q `2 );

                    hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 ) by A34, SQUARE_1: 15;

                  end;

                    case

                     A35: 0 > (q `2 );

                    ( - ( - (q `1 ))) >= ( - (q `2 )) by A34, XREAL_1: 24;

                    then (( - (q `2 )) ^2 ) <= ((q `1 ) ^2 ) by A35, SQUARE_1: 15;

                    hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 );

                  end;

                end;

                hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 );

              end;

                case

                 A36: (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

                now

                  per cases ;

                    case

                     A37: 0 >= (q `2 );

                    ( - (q `2 )) <= ( - (q `1 )) by A36, XREAL_1: 24;

                    then (( - (q `2 )) ^2 ) <= (( - (q `1 )) ^2 ) by A37, SQUARE_1: 15;

                    hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 );

                  end;

                    case 0 < (q `2 );

                    then ((q `2 ) ^2 ) <= (( - (q `1 )) ^2 ) by A36, SQUARE_1: 15;

                    hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 );

                  end;

                end;

                hence ((q `2 ) ^2 ) <= ((q `1 ) ^2 );

              end;

            end;

            then (((q `2 ) ^2 ) / ((q `1 ) ^2 )) <= (((q `1 ) ^2 ) / ((q `1 ) ^2 )) by A28, XREAL_1: 72;

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

            then (((q `2 ) / (q `1 )) ^2 ) <= 1 by A28, XCMPLX_1: 60;

            then

             A38: (1 + (((q `2 ) / (q `1 )) ^2 )) <= (1 + 1) by XREAL_1: 7;

            then (((q `2 ) ^2 ) * (1 + (((q `2 ) / (q `1 )) ^2 ))) <= (((q `2 ) ^2 ) * 2) by A25, XREAL_1: 64;

            then

             A39: ((qz `2 ) ^2 ) <= (((q `2 ) ^2 ) * 2) by A33, A32, SQUARE_1:def 2;

            (((q `1 ) ^2 ) * (1 + (((q `2 ) / (q `1 )) ^2 ))) <= (((q `1 ) ^2 ) * 2) by A24, A38, XREAL_1: 64;

            then ((qz `1 ) ^2 ) <= (((q `1 ) ^2 ) * 2) by A33, A31, SQUARE_1:def 2;

            then

             A40: (((qz `1 ) ^2 ) + ((qz `2 ) ^2 )) <= ((((q `1 ) ^2 ) * 2) + (((q `2 ) ^2 ) * 2)) by A39, XREAL_1: 7;

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

            then

             A41: ( sqrt (((qz `1 ) ^2 ) + ((qz `2 ) ^2 ))) <= ( sqrt ((((q `1 ) ^2 ) * 2) + (((q `2 ) ^2 ) * 2))) by A40, SQUARE_1: 26;

            

             A42: ((( 0. ( TOP-REAL 2)) - qz) `2 ) = ((( 0. ( TOP-REAL 2)) `2 ) - (qz `2 )) by TOPREAL3: 3

            .= ( - (qz `2 )) by JGRAPH_2: 3;

            ((( 0. ( TOP-REAL 2)) - qz) `1 ) = ((( 0. ( TOP-REAL 2)) `1 ) - (qz `1 )) by TOPREAL3: 3

            .= ( - (qz `1 )) by JGRAPH_2: 3;

            then ( sqrt ((((( 0. ( TOP-REAL 2)) - qz) `1 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - qz) `2 ) ^2 ))) < r by A26, A42, A41, XXREAL_0: 2;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by JGRAPH_1: 30;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

            suppose

             A43: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

             A44:

            now

              assume ((q `2 ) ^2 ) <= 0 ;

              then ((q `2 ) ^2 ) = 0 by XREAL_1: 63;

              then (q `2 ) = 0 by XCMPLX_1: 6;

              hence contradiction by A43;

            end;

            now

              per cases by A43, JGRAPH_2: 13;

                case

                 A45: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 );

                now

                  per cases ;

                    case 0 <= (q `1 );

                    hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 ) by A45, SQUARE_1: 15;

                  end;

                    case

                     A46: 0 > (q `1 );

                    ( - ( - (q `2 ))) >= ( - (q `1 )) by A45, XREAL_1: 24;

                    then (( - (q `1 )) ^2 ) <= ((q `2 ) ^2 ) by A46, SQUARE_1: 15;

                    hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 );

                  end;

                end;

                hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 );

              end;

                case

                 A47: (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 ));

                now

                  per cases ;

                    case

                     A48: 0 >= (q `1 );

                    ( - (q `1 )) <= ( - (q `2 )) by A47, XREAL_1: 24;

                    then (( - (q `1 )) ^2 ) <= (( - (q `2 )) ^2 ) by A48, SQUARE_1: 15;

                    hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 );

                  end;

                    case 0 < (q `1 );

                    then ((q `1 ) ^2 ) <= (( - (q `2 )) ^2 ) by A47, SQUARE_1: 15;

                    hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 );

                  end;

                end;

                hence ((q `1 ) ^2 ) <= ((q `2 ) ^2 );

              end;

            end;

            then (((q `1 ) ^2 ) / ((q `2 ) ^2 )) <= (((q `2 ) ^2 ) / ((q `2 ) ^2 )) by A44, XREAL_1: 72;

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

            then (((q `1 ) / (q `2 )) ^2 ) <= 1 by A44, XCMPLX_1: 60;

            then

             A49: (1 + (((q `1 ) / (q `2 )) ^2 )) <= (1 + 1) by XREAL_1: 7;

            then

             A50: (((q `2 ) ^2 ) * (1 + (((q `1 ) / (q `2 )) ^2 ))) <= (((q `2 ) ^2 ) * 2) by A25, XREAL_1: 64;

            (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by Lm1;

            then

             A51: (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ) = (1 + (((q `1 ) / (q `2 )) ^2 )) by SQUARE_1:def 2;

            

             A52: (((q `1 ) ^2 ) * (1 + (((q `1 ) / (q `2 )) ^2 ))) <= (((q `1 ) ^2 ) * 2) by A24, A49, XREAL_1: 64;

            

             A53: (( Sq_Circ " ) . q) = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A43, Th28;

            then (qz `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A23, EUCLID: 52;

            then

             A54: ((qz `1 ) ^2 ) <= (((q `1 ) ^2 ) * 2) by A52, A51, SQUARE_1: 9;

            (qz `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A23, A53, EUCLID: 52;

            then ((qz `2 ) ^2 ) <= (((q `2 ) ^2 ) * 2) by A50, A51, SQUARE_1: 9;

            then

             A55: (((qz `2 ) ^2 ) + ((qz `1 ) ^2 )) <= ((((q `2 ) ^2 ) * 2) + (((q `1 ) ^2 ) * 2)) by A54, XREAL_1: 7;

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

            then

             A56: ( sqrt (((qz `2 ) ^2 ) + ((qz `1 ) ^2 ))) <= ( sqrt ((((q `2 ) ^2 ) * 2) + (((q `1 ) ^2 ) * 2))) by A55, SQUARE_1: 26;

            

             A57: ((( 0. ( TOP-REAL 2)) - qz) `2 ) = ((( 0. ( TOP-REAL 2)) `2 ) - (qz `2 )) by TOPREAL3: 3

            .= ( - (qz `2 )) by JGRAPH_2: 3;

            ((( 0. ( TOP-REAL 2)) - qz) `1 ) = ((( 0. ( TOP-REAL 2)) `1 ) - (qz `1 )) by TOPREAL3: 3

            .= ( - (qz `1 )) by JGRAPH_2: 3;

            then ( sqrt ((((( 0. ( TOP-REAL 2)) - qz) `2 ) ^2 ) + (((( 0. ( TOP-REAL 2)) - qz) `1 ) ^2 ))) < r by A26, A57, A56, XXREAL_0: 2;

            then |.(( 0. ( TOP-REAL 2)) - qz).| < r by JGRAPH_1: 30;

            then ( dist (u0,pz)) < r by JGRAPH_1: 28;

            hence thesis by METRIC_1: 11;

          end;

        end;

        

         A58: V1 is open by GOBOARD6: 3;

        ( sqrt 2) > 0 by SQUARE_1: 25;

        then u0 in V1 by A17, GOBOARD6: 1, XREAL_1: 139;

        hence thesis by A18, A58, A19, XBOOLE_1: 1;

      end;

      

       A59: (D ` ) = {( 0. ( TOP-REAL 2))} by Th20;

      then ex h be Function of (( TOP-REAL 2) | D), (( TOP-REAL 2) | D) st h = (( Sq_Circ " ) | D) & h is continuous by Th41;

      hence thesis by A1, A59, A2, A14, Th3;

    end;

    theorem :: JGRAPH_3:43

    

     Th43: Sq_Circ is Function of ( TOP-REAL 2), ( TOP-REAL 2) & ( rng Sq_Circ ) = the carrier of ( TOP-REAL 2) & for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ holds f is being_homeomorphism

    proof

      thus Sq_Circ is Function of ( TOP-REAL 2), ( TOP-REAL 2);

      

       A1: for f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ holds ( rng Sq_Circ ) = the carrier of ( TOP-REAL 2) & f is being_homeomorphism

      proof

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

        assume

         A2: f = Sq_Circ ;

        reconsider g = (f /" ) as Function of ( TOP-REAL 2), ( TOP-REAL 2);

        

         A3: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

        the carrier of ( TOP-REAL 2) c= ( rng f)

        proof

          let y be object;

          assume y in the carrier of ( TOP-REAL 2);

          then

          reconsider p2 = y as Point of ( TOP-REAL 2);

          set q = p2;

          now

            per cases ;

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

              then y = ( Sq_Circ . q) by Def1;

              hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A2, A3;

            end;

              case

               A4: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

              set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;

              

               A5: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

               A6:

              now

                assume that

                 A7: (px `1 ) = 0 and

                 A8: (px `2 ) = 0 ;

                ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A8, EUCLID: 52;

                then

                 A9: (q `2 ) = 0 by A5, XCMPLX_1: 6;

                ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A7, EUCLID: 52;

                then (q `1 ) = 0 by A5, XCMPLX_1: 6;

                hence contradiction by A4, A9, EUCLID: 53, EUCLID: 54;

              end;

              

               A10: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              

               A11: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

              

               A12: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

              then

               A13: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A11, A5, XCMPLX_1: 91;

              then

               A14: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A12, A5, XCMPLX_1: 89;

              (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A4, A5, XREAL_1: 64;

              then (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A11, A12, A5, XREAL_1: 64;

              then ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A11, A5, EUCLID: 52, XREAL_1: 64;

              then

               A15: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A11, A12, A6, Def1, JGRAPH_2: 3;

              ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A11, A5, A13, XCMPLX_1: 89;

              hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A15, A14, A10, EUCLID: 53;

            end;

              case

               A16: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

              set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;

              

               A17: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

               A18:

              now

                assume that

                 A19: (px `2 ) = 0 and (px `1 ) = 0 ;

                ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A19, EUCLID: 52;

                then (q `2 ) = 0 by A17, XCMPLX_1: 6;

                hence contradiction by A16;

              end;

              

               A20: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

              

               A21: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

              then

               A22: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A20, A17, XCMPLX_1: 91;

              then

               A23: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A21, A17, XCMPLX_1: 89;

              (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A16, JGRAPH_2: 13;

              then (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A17, XREAL_1: 64;

              then (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A20, A21, A17, XREAL_1: 64;

              then ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A20, A17, EUCLID: 52, XREAL_1: 64;

              then

               A24: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A20, A21, A18, Th4, JGRAPH_2: 3;

              

               A25: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

              ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A20, A17, A22, XCMPLX_1: 89;

              hence ex x be set st x in ( dom Sq_Circ ) & y = ( Sq_Circ . x) by A24, A23, A25, EUCLID: 53;

            end;

          end;

          hence thesis by A2, FUNCT_1:def 3;

        end;

        then ( rng f) = the carrier of ( TOP-REAL 2);

        then

         A26: f is onto by FUNCT_2:def 3;

        

         A27: ( rng f) = ( dom (f qua Function " )) by A2, FUNCT_1: 33

        .= ( dom (f /" )) by A2, A26, TOPS_2:def 4

        .= ( [#] ( TOP-REAL 2)) by FUNCT_2:def 1;

        g = ( Sq_Circ " ) by A26, A2, TOPS_2:def 4;

        hence thesis by A2, A3, A27, Th21, Th42, TOPS_2:def 5;

      end;

      hence ( rng Sq_Circ ) = the carrier of ( TOP-REAL 2);

      thus thesis by A1;

    end;

     Lm19:

    now

      let pz2,pz1 be Real;

      assume ((((pz2 ^2 ) + (pz1 ^2 )) - 1) * (pz2 ^2 )) <= (pz1 ^2 );

      then ((((pz2 ^2 ) * (pz2 ^2 )) + ((pz2 ^2 ) * ((pz1 ^2 ) - 1))) - (pz1 ^2 )) <= ((pz1 ^2 ) - (pz1 ^2 )) by XREAL_1: 9;

      hence (((pz2 ^2 ) - 1) * ((pz2 ^2 ) + (pz1 ^2 ))) <= 0 ;

    end;

     Lm20:

    now

      let px1 be Real;

      assume ((px1 ^2 ) - 1) = 0 ;

      then ((px1 - 1) * (px1 + 1)) = 0 ;

      then (px1 - 1) = 0 or (px1 + 1) = 0 by XCMPLX_1: 6;

      hence px1 = 1 or px1 = ( - 1);

    end;

    theorem :: JGRAPH_3:44

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

    proof

      

       A1: ( dom ( Sq_Circ " )) = the carrier of ( TOP-REAL 2) by Th29, FUNCT_2:def 1;

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

      assume

       A2: O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXN & (f . I) in KXP & (g . O) in KYN & (g . I) in KYP & ( rng f) c= C0 & ( rng g) c= C0;

      then

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

       A3: (f . O) = p1 and

       A4: |.p1.| = 1 and

       A5: (p1 `2 ) >= (p1 `1 ) and

       A6: (p1 `2 ) <= ( - (p1 `1 ));

      reconsider gg = (( Sq_Circ " ) * g) as Function of I[01] , ( TOP-REAL 2) by Th29, FUNCT_2: 13;

      

       A7: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      reconsider ff = (( Sq_Circ " ) * f) as Function of I[01] , ( TOP-REAL 2) by Th29, FUNCT_2: 13;

      

       A8: ( dom gg) = the carrier of I[01] by FUNCT_2:def 1;

      

       A9: ( dom ff) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A10: (ff . O) = (( Sq_Circ " ) . (f . O)) by FUNCT_1: 12;

      

       A11: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      

       A12: for r be Point of I[01] holds ( - 1) <= ((ff . r) `1 ) & ((ff . r) `1 ) <= 1 & ( - 1) <= ((gg . r) `1 ) & ((gg . r) `1 ) <= 1 & ( - 1) <= ((ff . r) `2 ) & ((ff . r) `2 ) <= 1 & ( - 1) <= ((gg . r) `2 ) & ((gg . r) `2 ) <= 1

      proof

        let r be Point of I[01] ;

        (f . r) in ( rng f) by A11, FUNCT_1: 3;

        then (f . r) in C0 by A2;

        then

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

         A13: (f . r) = p1 and

         A14: |.p1.| <= 1 by A2;

        (g . r) in ( rng g) by A7, FUNCT_1: 3;

        then (g . r) in C0 by A2;

        then

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

         A15: (g . r) = p2 and

         A16: |.p2.| <= 1 by A2;

        

         A17: (gg . r) = (( Sq_Circ " ) . (g . r)) by A8, FUNCT_1: 12;

         A18:

        now

          per cases ;

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

            hence ( - 1) <= ((gg . r) `1 ) & ((gg . r) `1 ) <= 1 & ( - 1) <= ((gg . r) `2 ) & ((gg . r) `2 ) <= 1 by A17, A15, Th28, JGRAPH_2: 3;

          end;

            case

             A19: p2 <> ( 0. ( TOP-REAL 2)) & ((p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 )));

            set px = (gg . r);

            

             A20: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A19, Th28;

            then

             A21: (px `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A17, A15, EUCLID: 52;

            ( |.p2.| ^2 ) <= |.p2.| by A16, SQUARE_1: 42;

            then

             A22: ( |.p2.| ^2 ) <= 1 by A16, XXREAL_0: 2;

            

             A23: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A24: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A25: (px `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A17, A15, A20, EUCLID: 52;

            

             A26: ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            then (p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A19, XREAL_1: 64;

            then

             A27: (p2 `2 ) <= (p2 `1 ) & (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A21, A25, A26, XREAL_1: 64;

            then

             A28: ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A17, A15, A20, A21, A26, EUCLID: 52, XREAL_1: 64;

             A29:

            now

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

              then (p2 `1 ) = 0 & (p2 `2 ) = 0 by A21, A25, A26, XCMPLX_1: 6;

              hence contradiction by A19, EUCLID: 53, EUCLID: 54;

            end;

            then

             A30: (px `1 ) <> 0 by A21, A25, A26, A27, XREAL_1: 64;

            set q = px;

            

             A31: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            

             A32: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

            

             A33: p2 = ( Sq_Circ . px) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A17, A15, Th43, EUCLID: 52, FUNCT_1: 32;

            ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A21, A25, A29, A28, Def1, JGRAPH_2: 3;

            

            then ( |.p2.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A33, A31, JGRAPH_1: 29

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

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

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A32, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A32, SQUARE_1:def 2

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

            then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) <= (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A32, A22, XREAL_1: 64;

            then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) <= (1 + (((q `2 ) / (q `1 )) ^2 )) by A32, XCMPLX_1: 87;

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

            then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) <= (((px `2 ) ^2 ) / ((px `1 ) ^2 )) by XREAL_1: 20;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A24, XREAL_1: 64;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((px `2 ) ^2 ) by A30, XCMPLX_1: 6, XCMPLX_1: 87;

            then

             A34: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) <= 0 by Lm19;

            (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A29, COMPLEX1: 1;

            then

             A35: (((px `1 ) ^2 ) - 1) <= 0 by A24, A34, A23, XREAL_1: 129;

            then

             A36: (px `1 ) >= ( - 1) by SQUARE_1: 43;

            

             A37: (px `1 ) <= 1 by A35, SQUARE_1: 43;

            then (q `2 ) <= 1 & ( - ( - (q `1 ))) >= ( - (q `2 )) or (q `2 ) >= ( - 1) & (q `2 ) <= ( - (q `1 )) by A21, A25, A28, A36, XREAL_1: 24, XXREAL_0: 2;

            then (q `2 ) <= 1 & (q `1 ) >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - ( - (q `1 ))) by XREAL_1: 24;

            then (q `2 ) <= 1 & 1 >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= (q `1 ) by A37, XXREAL_0: 2;

            then (q `2 ) <= 1 & ( - 1) <= ( - ( - (q `2 ))) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - 1) by A36, XREAL_1: 24, XXREAL_0: 2;

            hence ( - 1) <= ((gg . r) `1 ) & ((gg . r) `1 ) <= 1 & ( - 1) <= ((gg . r) `2 ) & ((gg . r) `2 ) <= 1 by A35, SQUARE_1: 43, XREAL_1: 24;

          end;

            case

             A38: p2 <> ( 0. ( TOP-REAL 2)) & not ((p2 `2 ) <= (p2 `1 ) & ( - (p2 `1 )) <= (p2 `2 ) or (p2 `2 ) >= (p2 `1 ) & (p2 `2 ) <= ( - (p2 `1 )));

            set pz = (gg . r);

            

             A39: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A38, Th28;

            then

             A40: (pz `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A17, A15, EUCLID: 52;

            

             A41: (pz `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A17, A15, A39, EUCLID: 52;

            

             A42: ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 ) or (p2 `1 ) >= (p2 `2 ) & (p2 `1 ) <= ( - (p2 `2 )) by A38, JGRAPH_2: 13;

            then (p2 `1 ) <= (p2 `2 ) & ( - (p2 `2 )) <= (p2 `1 ) or (p2 `1 ) >= (p2 `2 ) & ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= (( - (p2 `2 )) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) by A42, XREAL_1: 64;

            then

             A43: (p2 `1 ) <= (p2 `2 ) & (( - (p2 `2 )) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A40, A41, A42, XREAL_1: 64;

            then

             A44: ((p2 `1 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A17, A15, A39, A40, A42, EUCLID: 52, XREAL_1: 64;

             A45:

            now

              assume that

               A46: (pz `2 ) = 0 and (pz `1 ) = 0 ;

              (p2 `2 ) = 0 by A40, A42, A46, XCMPLX_1: 6;

              hence contradiction by A38;

            end;

            then

             A47: (pz `2 ) <> 0 by A40, A41, A42, A43, XREAL_1: 64;

            

             A48: p2 = ( Sq_Circ . pz) & ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A17, A15, Th43, EUCLID: 52, FUNCT_1: 32;

            

             A49: ((pz `2 ) ^2 ) >= 0 by XREAL_1: 63;

            ( |.p2.| ^2 ) <= |.p2.| by A16, SQUARE_1: 42;

            then

             A50: ( |.p2.| ^2 ) <= 1 by A16, XXREAL_0: 2;

            

             A51: ((pz `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A52: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

            

             A53: (1 + (((pz `1 ) / (pz `2 )) ^2 )) > 0 by Lm1;

            ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A40, A41, A45, A44, Th4, JGRAPH_2: 3;

            

            then ( |.p2.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A48, A52, JGRAPH_1: 29

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

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

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A53, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A53, SQUARE_1:def 2

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

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) <= (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A53, A50, XREAL_1: 64;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <= (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A53, XCMPLX_1: 87;

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

            then ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) <= (((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) by XREAL_1: 20;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) <= ((((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) * ((pz `2 ) ^2 )) by A49, XREAL_1: 64;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) <= ((pz `1 ) ^2 ) by A47, XCMPLX_1: 6, XCMPLX_1: 87;

            then

             A54: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) <= 0 by Lm19;

            (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A45, COMPLEX1: 1;

            then

             A55: (((pz `2 ) ^2 ) - 1) <= 0 by A49, A54, A51, XREAL_1: 129;

            then

             A56: (pz `2 ) >= ( - 1) by SQUARE_1: 43;

            

             A57: (pz `2 ) <= 1 by A55, SQUARE_1: 43;

            then (pz `1 ) <= 1 & ( - ( - (pz `2 ))) >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & (pz `1 ) <= ( - (pz `2 )) by A40, A41, A44, A56, XREAL_1: 24, XXREAL_0: 2;

            then (pz `1 ) <= 1 & 1 >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & ( - (pz `1 )) >= ( - ( - (pz `2 ))) by A57, XREAL_1: 24, XXREAL_0: 2;

            then (pz `1 ) <= 1 & 1 >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & ( - (pz `1 )) >= ( - 1) by A56, XXREAL_0: 2;

            then (pz `1 ) <= 1 & ( - 1) <= ( - ( - (pz `1 ))) or (pz `1 ) >= ( - 1) & (pz `1 ) <= 1 by XREAL_1: 24;

            hence ( - 1) <= ((gg . r) `1 ) & ((gg . r) `1 ) <= 1 & ( - 1) <= ((gg . r) `2 ) & ((gg . r) `2 ) <= 1 by A55, SQUARE_1: 43;

          end;

        end;

        

         A58: (ff . r) = (( Sq_Circ " ) . (f . r)) by A9, FUNCT_1: 12;

        now

          per cases ;

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

            hence ( - 1) <= ((ff . r) `1 ) & ((ff . r) `1 ) <= 1 & ( - 1) <= ((ff . r) `2 ) & ((ff . r) `2 ) <= 1 by A58, A13, Th28, JGRAPH_2: 3;

          end;

            case

             A59: p1 <> ( 0. ( TOP-REAL 2)) & ((p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 )));

            set px = (ff . r);

            (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A59, Th28;

            then

             A60: (px `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) & (px `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A58, A13, EUCLID: 52;

            

             A61: ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            then (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A59, XREAL_1: 64;

            then

             A62: (p1 `2 ) <= (p1 `1 ) & (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A60, A61, XREAL_1: 64;

            then

             A63: (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A60, A61, XREAL_1: 64;

             A64:

            now

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

              then (p1 `1 ) = 0 & (p1 `2 ) = 0 by A60, A61, XCMPLX_1: 6;

              hence contradiction by A59, EUCLID: 53, EUCLID: 54;

            end;

            then

             A65: (px `1 ) <> 0 by A60, A61, A62, XREAL_1: 64;

            ( |.p1.| ^2 ) <= |.p1.| by A14, SQUARE_1: 42;

            then

             A66: ( |.p1.| ^2 ) <= 1 by A14, XXREAL_0: 2;

            

             A67: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A68: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

            set q = px;

            

             A69: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

            

             A70: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

            

             A71: p1 = ( Sq_Circ . px) & ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A58, A13, Th43, EUCLID: 52, FUNCT_1: 32;

            ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A64, A63, Def1, JGRAPH_2: 3;

            

            then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A71, A69, JGRAPH_1: 29

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

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

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A70, SQUARE_1:def 2

            .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A70, SQUARE_1:def 2

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

            then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) <= (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A70, A66, XREAL_1: 64;

            then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) <= (1 + (((q `2 ) / (q `1 )) ^2 )) by A70, XCMPLX_1: 87;

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

            then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) <= (((px `2 ) ^2 ) / ((px `1 ) ^2 )) by XREAL_1: 20;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A67, XREAL_1: 64;

            then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((px `2 ) ^2 ) by A65, XCMPLX_1: 6, XCMPLX_1: 87;

            then

             A72: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) <= 0 by Lm19;

            (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A64, COMPLEX1: 1;

            then

             A73: (((px `1 ) ^2 ) - 1) <= 0 by A67, A72, A68, XREAL_1: 129;

            then

             A74: (px `1 ) >= ( - 1) by SQUARE_1: 43;

            

             A75: (px `1 ) <= 1 by A73, SQUARE_1: 43;

            then (q `2 ) <= 1 & ( - ( - (q `1 ))) >= ( - (q `2 )) or (q `2 ) >= ( - 1) & (q `2 ) <= ( - (q `1 )) by A63, A74, XREAL_1: 24, XXREAL_0: 2;

            then (q `2 ) <= 1 & (q `1 ) >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - ( - (q `1 ))) by XREAL_1: 24;

            then (q `2 ) <= 1 & 1 >= ( - (q `2 )) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= (q `1 ) by A75, XXREAL_0: 2;

            then (q `2 ) <= 1 & ( - 1) <= ( - ( - (q `2 ))) or (q `2 ) >= ( - 1) & ( - (q `2 )) >= ( - 1) by A74, XREAL_1: 24, XXREAL_0: 2;

            hence ( - 1) <= ((ff . r) `1 ) & ((ff . r) `1 ) <= 1 & ( - 1) <= ((ff . r) `2 ) & ((ff . r) `2 ) <= 1 by A73, SQUARE_1: 43, XREAL_1: 24;

          end;

            case

             A76: p1 <> ( 0. ( TOP-REAL 2)) & not ((p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & (p1 `2 ) <= ( - (p1 `1 )));

            set pz = (ff . r);

            

             A77: (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 ))))]| by A76, Th28;

            then

             A78: (pz `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A58, A13, EUCLID: 52;

            

             A79: (pz `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A58, A13, A77, EUCLID: 52;

            

             A80: ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

            (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 ) or (p1 `1 ) >= (p1 `2 ) & (p1 `1 ) <= ( - (p1 `2 )) by A76, JGRAPH_2: 13;

            then (p1 `1 ) <= (p1 `2 ) & ( - (p1 `2 )) <= (p1 `1 ) or (p1 `1 ) >= (p1 `2 ) & ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= (( - (p1 `2 )) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) by A80, XREAL_1: 64;

            then

             A81: (p1 `1 ) <= (p1 `2 ) & (( - (p1 `2 )) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A78, A79, A80, XREAL_1: 64;

            then

             A82: ((p1 `1 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `1 ) / (p1 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A58, A13, A77, A78, A80, EUCLID: 52, XREAL_1: 64;

             A83:

            now

              assume that

               A84: (pz `2 ) = 0 and (pz `1 ) = 0 ;

              (p1 `2 ) = 0 by A78, A80, A84, XCMPLX_1: 6;

              hence contradiction by A76;

            end;

            then

             A85: (pz `2 ) <> 0 by A78, A79, A80, A81, XREAL_1: 64;

            

             A86: p1 = ( Sq_Circ . pz) & ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A58, A13, Th43, EUCLID: 52, FUNCT_1: 32;

            

             A87: ((pz `2 ) ^2 ) >= 0 by XREAL_1: 63;

            ( |.p1.| ^2 ) <= |.p1.| by A14, SQUARE_1: 42;

            then

             A88: ( |.p1.| ^2 ) <= 1 by A14, XXREAL_0: 2;

            

             A89: ((pz `1 ) ^2 ) >= 0 by XREAL_1: 63;

            

             A90: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `1 ) = ((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

            

             A91: (1 + (((pz `1 ) / (pz `2 )) ^2 )) > 0 by Lm1;

            ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A78, A79, A83, A82, Th4, JGRAPH_2: 3;

            

            then ( |.p1.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A86, A90, JGRAPH_1: 29

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

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

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A91, SQUARE_1:def 2

            .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A91, SQUARE_1:def 2

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

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) <= (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A91, A88, XREAL_1: 64;

            then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <= (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A91, XCMPLX_1: 87;

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

            then ((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) <= (((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) by XREAL_1: 20;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) <= ((((pz `1 ) ^2 ) / ((pz `2 ) ^2 )) * ((pz `2 ) ^2 )) by A87, XREAL_1: 64;

            then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) - 1) * ((pz `2 ) ^2 )) <= ((pz `1 ) ^2 ) by A85, XCMPLX_1: 6, XCMPLX_1: 87;

            then

             A92: ((((pz `2 ) ^2 ) - 1) * (((pz `2 ) ^2 ) + ((pz `1 ) ^2 ))) <= 0 by Lm19;

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

            then

             A93: (((pz `2 ) ^2 ) - 1) <= 0 by A87, A92, A89, XREAL_1: 129;

            then

             A94: (pz `2 ) >= ( - 1) by SQUARE_1: 43;

            

             A95: (pz `2 ) <= 1 by A93, SQUARE_1: 43;

            then (pz `1 ) <= 1 & ( - ( - (pz `2 ))) >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & (pz `1 ) <= ( - (pz `2 )) by A78, A79, A82, A94, XREAL_1: 24, XXREAL_0: 2;

            then (pz `1 ) <= 1 & 1 >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & ( - (pz `1 )) >= ( - ( - (pz `2 ))) by A95, XREAL_1: 24, XXREAL_0: 2;

            then (pz `1 ) <= 1 & 1 >= ( - (pz `1 )) or (pz `1 ) >= ( - 1) & ( - (pz `1 )) >= ( - 1) by A94, XXREAL_0: 2;

            then (pz `1 ) <= 1 & ( - 1) <= ( - ( - (pz `1 ))) or (pz `1 ) >= ( - 1) & (pz `1 ) <= 1 by XREAL_1: 24;

            hence ( - 1) <= ((ff . r) `1 ) & ((ff . r) `1 ) <= 1 & ( - 1) <= ((ff . r) `2 ) & ((ff . r) `2 ) <= 1 by A93, SQUARE_1: 43;

          end;

        end;

        hence thesis by A18;

      end;

      set y = the Element of (( rng ff) /\ ( rng gg));

      

       A96: p1 <> ( 0. ( TOP-REAL 2)) by A4, TOPRNS_1: 23;

      then

       A97: (( Sq_Circ " ) . p1) = |[((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A5, A6, Th28;

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

      proof

        set pz = (gg . O);

        set py = (ff . I);

        set px = (ff . O);

        set q = px;

        

         A98: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

        set pu = (gg . I);

        

         A99: ( |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| `1 ) = ((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) by EUCLID: 52;

        

         A100: ( |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| `2 ) = ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by EUCLID: 52;

        

         A101: (1 + (((pu `1 ) / (pu `2 )) ^2 )) > 0 by Lm1;

        (( Sq_Circ " ) . p1) = q by A9, A3, FUNCT_1: 12;

        then

         A102: p1 = ( Sq_Circ . px) by Th43, FUNCT_1: 32;

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

         A103: (g . I) = p4 and

         A104: |.p4.| = 1 and

         A105: (p4 `2 ) >= (p4 `1 ) and

         A106: (p4 `2 ) >= ( - (p4 `1 )) by A2;

        

         A107: ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A108: ( - (p4 `2 )) <= ( - ( - (p4 `1 ))) by A106, XREAL_1: 24;

        then

         A109: (p4 `1 ) <= (p4 `2 ) & (( - (p4 `2 )) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A105, A107, XREAL_1: 64;

        

         A110: (gg . I) = (( Sq_Circ " ) . (g . I)) by A8, FUNCT_1: 12;

        then

         A111: p4 = ( Sq_Circ . pu) by A103, Th43, FUNCT_1: 32;

        

         A112: p4 <> ( 0. ( TOP-REAL 2)) by A104, TOPRNS_1: 23;

        then

         A113: (( Sq_Circ " ) . p4) = |[((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))), ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 ))))]| by A105, A108, Th30;

        then

         A114: (pu `2 ) = ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A110, A103, EUCLID: 52;

        

         A115: (pu `1 ) = ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) by A110, A103, A113, EUCLID: 52;

         A116:

        now

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

          then (p4 `2 ) = 0 & (p4 `1 ) = 0 by A114, A115, A107, XCMPLX_1: 6;

          hence contradiction by A112, EUCLID: 53, EUCLID: 54;

        end;

        ((p4 `1 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) <= ((p4 `2 ) * ( sqrt (1 + (((p4 `1 ) / (p4 `2 )) ^2 )))) & ( - (pu `2 )) <= (pu `1 ) or (pu `1 ) >= (pu `2 ) & (pu `1 ) <= ( - (pu `2 )) by A110, A103, A113, A114, A107, A109, EUCLID: 52, XREAL_1: 64;

        then

         A117: ( Sq_Circ . pu) = |[((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))), ((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))))]| by A114, A115, A116, Th4, JGRAPH_2: 3;

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

        

        then ( |.p4.| ^2 ) = ((((pu `2 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 ) + (((pu `1 ) / ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 )))) ^2 )) by A111, A117, A100, JGRAPH_1: 29

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

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

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) ^2 ))) by A101, SQUARE_1:def 2

        .= ((((pu `2 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) + (((pu `1 ) ^2 ) / (1 + (((pu `1 ) / (pu `2 )) ^2 )))) by A101, SQUARE_1:def 2

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

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) / (1 + (((pu `1 ) / (pu `2 )) ^2 ))) * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) = (1 * (1 + (((pu `1 ) / (pu `2 )) ^2 ))) by A104;

        then (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) = (1 + (((pu `1 ) / (pu `2 )) ^2 )) by A101, XCMPLX_1: 87;

        then

         A118: ((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) = (((pu `1 ) ^2 ) / ((pu `2 ) ^2 )) by XCMPLX_1: 76;

        (pu `2 ) <> 0 by A114, A115, A107, A116, A109, XREAL_1: 64;

        then (((((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) - 1) * ((pu `2 ) ^2 )) = ((pu `1 ) ^2 ) by A118, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A119: ((((pu `2 ) ^2 ) - 1) * (((pu `2 ) ^2 ) + ((pu `1 ) ^2 ))) = 0 ;

        (((pu `2 ) ^2 ) + ((pu `1 ) ^2 )) <> 0 by A116, COMPLEX1: 1;

        then

         A120: (((pu `2 ) ^2 ) - 1) = 0 by A119, XCMPLX_1: 6;

        

         A121: ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A122: ( sqrt (1 + (((pu `1 ) / (pu `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

         A123:

        now

          assume

           A124: (pu `2 ) = ( - 1);

          then ( - (p4 `1 )) < 0 by A106, A111, A117, A100, A122, XREAL_1: 141;

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

          hence contradiction by A105, A111, A117, A122, A124, EUCLID: 52;

        end;

        

         A125: (1 + (((pz `1 ) / (pz `2 )) ^2 )) > 0 by Lm1;

        

         A126: (px `1 ) = ((p1 `1 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) & (px `2 ) = ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A10, A3, A97, EUCLID: 52;

         A127:

        now

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

          then (p1 `1 ) = 0 & (p1 `2 ) = 0 by A126, A121, XCMPLX_1: 6;

          hence contradiction by A96, EUCLID: 53, EUCLID: 54;

        end;

        (p1 `2 ) <= (p1 `1 ) & ( - (p1 `1 )) <= (p1 `2 ) or (p1 `2 ) >= (p1 `1 ) & ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) by A5, A6, A121, XREAL_1: 64;

        then

         A128: (p1 `2 ) <= (p1 `1 ) & (( - (p1 `1 )) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) <= ((p1 `2 ) * ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A126, A121, XREAL_1: 64;

        then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A126, A121, XREAL_1: 64;

        then

         A129: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A127, Def1, JGRAPH_2: 3;

        

         A130: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

         A131:

        now

          assume

           A132: (px `1 ) = 1;

          ( - (p1 `2 )) >= ( - ( - (p1 `1 ))) by A6, XREAL_1: 24;

          then ( - (p1 `2 )) > 0 by A102, A129, A98, A130, A132, XREAL_1: 139;

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

          hence contradiction by A5, A102, A129, A130, A132, EUCLID: 52;

        end;

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

         A133: (f . I) = p2 and

         A134: |.p2.| = 1 and

         A135: (p2 `2 ) <= (p2 `1 ) and

         A136: (p2 `2 ) >= ( - (p2 `1 )) by A2;

        

         A137: (ff . I) = (( Sq_Circ " ) . (f . I)) by A9, FUNCT_1: 12;

        then

         A138: p2 = ( Sq_Circ . py) by A133, Th43, FUNCT_1: 32;

        

         A139: p2 <> ( 0. ( TOP-REAL 2)) by A134, TOPRNS_1: 23;

        then

         A140: (( Sq_Circ " ) . p2) = |[((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A135, A136, Th28;

        then

         A141: (py `1 ) = ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A137, A133, EUCLID: 52;

        

         A142: ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A143: (py `2 ) = ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) by A137, A133, A140, EUCLID: 52;

         A144:

        now

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

          then (p2 `1 ) = 0 & (p2 `2 ) = 0 by A141, A143, A142, XCMPLX_1: 6;

          hence contradiction by A139, EUCLID: 53, EUCLID: 54;

        end;

        

         A145: (p2 `2 ) <= (p2 `1 ) & (( - (p2 `1 )) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A135, A136, A142, XREAL_1: 64;

        then ((p2 `2 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) & ( - (py `1 )) <= (py `2 ) or (py `2 ) >= (py `1 ) & (py `2 ) <= ( - (py `1 )) by A137, A133, A140, A141, A142, EUCLID: 52, XREAL_1: 64;

        then

         A146: ( Sq_Circ . py) = |[((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))), ((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))))]| by A141, A143, A144, Def1, JGRAPH_2: 3;

        

         A147: ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

         A148:

        now

          assume

           A149: (py `1 ) = ( - 1);

          ( - (p2 `2 )) <= ( - ( - (p2 `1 ))) by A136, XREAL_1: 24;

          then ( - (p2 `2 )) < 0 by A138, A146, A99, A147, A149, XREAL_1: 141;

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

          hence contradiction by A135, A138, A146, A147, A149, EUCLID: 52;

        end;

        

         A150: (1 + (((py `2 ) / (py `1 )) ^2 )) > 0 by Lm1;

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

        

        then ( |.p2.| ^2 ) = ((((py `1 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 ) + (((py `2 ) / ( sqrt (1 + (((py `2 ) / (py `1 )) ^2 )))) ^2 )) by A138, A146, A99, JGRAPH_1: 29

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

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

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (( sqrt (1 + (((py `2 ) / (py `1 )) ^2 ))) ^2 ))) by A150, SQUARE_1:def 2

        .= ((((py `1 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 ))) + (((py `2 ) ^2 ) / (1 + (((py `2 ) / (py `1 )) ^2 )))) by A150, SQUARE_1:def 2

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

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) / (1 + (((py `2 ) / (py `1 )) ^2 ))) * (1 + (((py `2 ) / (py `1 )) ^2 ))) = (1 * (1 + (((py `2 ) / (py `1 )) ^2 ))) by A134;

        then (((py `1 ) ^2 ) + ((py `2 ) ^2 )) = (1 + (((py `2 ) / (py `1 )) ^2 )) by A150, XCMPLX_1: 87;

        then

         A151: ((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) = (((py `2 ) ^2 ) / ((py `1 ) ^2 )) by XCMPLX_1: 76;

        (py `1 ) <> 0 by A141, A143, A142, A144, A145, XREAL_1: 64;

        then (((((py `1 ) ^2 ) + ((py `2 ) ^2 )) - 1) * ((py `1 ) ^2 )) = ((py `2 ) ^2 ) by A151, XCMPLX_1: 6, XCMPLX_1: 87;

        then

         A152: ((((py `1 ) ^2 ) - 1) * (((py `1 ) ^2 ) + ((py `2 ) ^2 ))) = 0 ;

        (((py `1 ) ^2 ) + ((py `2 ) ^2 )) <> 0 by A144, COMPLEX1: 1;

        then

         A153: (((py `1 ) ^2 ) - 1) = 0 by A152, XCMPLX_1: 6;

        

         A154: ( |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| `2 ) = ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by EUCLID: 52;

        

         A155: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by Lm1;

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

        

        then ( |.p1.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A102, A129, A98, JGRAPH_1: 29

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

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

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A155, SQUARE_1:def 2

        .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A155, SQUARE_1:def 2

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

        then (((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) * (1 + (((q `2 ) / (q `1 )) ^2 ))) = (1 * (1 + (((q `2 ) / (q `1 )) ^2 ))) by A4;

        then (((q `1 ) ^2 ) + ((q `2 ) ^2 )) = (1 + (((q `2 ) / (q `1 )) ^2 )) by A155, XCMPLX_1: 87;

        then

         A156: ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) = (((px `2 ) ^2 ) / ((px `1 ) ^2 )) by XCMPLX_1: 76;

        (px `1 ) <> 0 by A126, A121, A127, A128, XREAL_1: 64;

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

        then

         A157: ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 ))) = 0 ;

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

         A158: (g . O) = p3 and

         A159: |.p3.| = 1 and

         A160: (p3 `2 ) <= (p3 `1 ) and

         A161: (p3 `2 ) <= ( - (p3 `1 )) by A2;

        

         A162: p3 <> ( 0. ( TOP-REAL 2)) by A159, TOPRNS_1: 23;

        

         A163: (gg . O) = (( Sq_Circ " ) . (g . O)) by A8, FUNCT_1: 12;

        then

         A164: p3 = ( Sq_Circ . pz) by A158, Th43, FUNCT_1: 32;

        

         A165: ( - (p3 `2 )) >= ( - ( - (p3 `1 ))) by A161, XREAL_1: 24;

        then

         A166: (( Sq_Circ " ) . p3) = |[((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A160, A162, Th30;

        then

         A167: (pz `2 ) = ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A163, A158, EUCLID: 52;

        

         A168: ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

        

         A169: (pz `1 ) = ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A163, A158, A166, EUCLID: 52;

         A170:

        now

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

          then (p3 `2 ) = 0 & (p3 `1 ) = 0 by A167, A169, A168, XCMPLX_1: 6;

          hence contradiction by A162, EUCLID: 53, EUCLID: 54;

        end;

        (p3 `1 ) <= (p3 `2 ) & ( - (p3 `2 )) <= (p3 `1 ) or (p3 `1 ) >= (p3 `2 ) & ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) by A160, A165, A168, XREAL_1: 64;

        then

         A171: (p3 `1 ) <= (p3 `2 ) & (( - (p3 `2 )) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A167, A169, A168, XREAL_1: 64;

        then ((p3 `1 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) <= ((p3 `2 ) * ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))) & ( - (pz `2 )) <= (pz `1 ) or (pz `1 ) >= (pz `2 ) & (pz `1 ) <= ( - (pz `2 )) by A163, A158, A166, A167, A168, EUCLID: 52, XREAL_1: 64;

        then

         A172: ( Sq_Circ . pz) = |[((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))), ((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))))]| by A167, A169, A170, Th4, JGRAPH_2: 3;

        

         A173: ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) > 0 by Lm1, SQUARE_1: 25;

         A174:

        now

          assume

           A175: (pz `2 ) = 1;

          then ( - (p3 `1 )) > 0 by A161, A164, A172, A154, A173, XREAL_1: 139;

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

          hence contradiction by A160, A164, A172, A173, A175, EUCLID: 52;

        end;

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

        

        then ( |.p3.| ^2 ) = ((((pz `2 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 ) + (((pz `1 ) / ( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 )))) ^2 )) by A164, A172, A154, JGRAPH_1: 29

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

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

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (( sqrt (1 + (((pz `1 ) / (pz `2 )) ^2 ))) ^2 ))) by A125, SQUARE_1:def 2

        .= ((((pz `2 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) + (((pz `1 ) ^2 ) / (1 + (((pz `1 ) / (pz `2 )) ^2 )))) by A125, SQUARE_1:def 2

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

        then (((((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) / (1 + (((pz `1 ) / (pz `2 )) ^2 ))) * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) = (1 * (1 + (((pz `1 ) / (pz `2 )) ^2 ))) by A159;

        then (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) = (1 + (((pz `1 ) / (pz `2 )) ^2 )) by A125, XCMPLX_1: 87;

        then

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

        (pz `2 ) <> 0 by A167, A169, A168, A170, A171, XREAL_1: 64;

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

        then

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

        (((pz `2 ) ^2 ) + ((pz `1 ) ^2 )) <> 0 by A170, COMPLEX1: 1;

        then

         A178: (((pz `2 ) ^2 ) - 1) = 0 by A177, XCMPLX_1: 6;

        (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <> 0 by A127, COMPLEX1: 1;

        then (((px `1 ) ^2 ) - 1) = 0 by A157, XCMPLX_1: 6;

        hence thesis by A131, A153, A148, A178, A174, A120, A123, Lm20;

      end;

      then ( rng ff) meets ( rng gg) by A2, A12, Th42, JGRAPH_1: 47;

      then

       A179: (( rng ff) /\ ( rng gg)) <> {} ;

      then y in ( rng ff) by XBOOLE_0:def 4;

      then

      consider x1 be object such that

       A180: x1 in ( dom ff) and

       A181: y = (ff . x1) by FUNCT_1:def 3;

      x1 in ( dom f) by A180, FUNCT_1: 11;

      then

       A182: (f . x1) in ( rng f) by FUNCT_1:def 3;

      y in ( rng gg) by A179, XBOOLE_0:def 4;

      then

      consider x2 be object such that

       A183: x2 in ( dom gg) and

       A184: y = (gg . x2) by FUNCT_1:def 3;

      

       A185: (gg . x2) = (( Sq_Circ " ) . (g . x2)) by A183, FUNCT_1: 12;

      x2 in ( dom g) by A183, FUNCT_1: 11;

      then

       A186: (g . x2) in ( rng g) by FUNCT_1:def 3;

      (ff . x1) = (( Sq_Circ " ) . (f . x1)) by A180, FUNCT_1: 12;

      then (f . x1) = (g . x2) by A181, A184, A1, A182, A186, A185, FUNCT_1:def 4;

      then (( rng f) /\ ( rng g)) <> {} by A182, A186, XBOOLE_0:def 4;

      hence thesis;

    end;