jordan20.miz



    begin

    theorem :: JORDAN20:1

    

     Th1: for P be Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & p in P holds ( Segment (P,p1,p2,p,p)) = {p}

    proof

      let P be Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: p in P;

      

       A3: ( Segment (P,p1,p2,p,p)) = { q where q be Point of ( TOP-REAL 2) : LE (p,q,P,p1,p2) & LE (q,p,P,p1,p2) } by JORDAN6: 26;

      

       A4: {p} c= ( Segment (P,p1,p2,p,p))

      proof

        let x be object;

        assume x in {p};

        then

         A5: x = p by TARSKI:def 1;

         LE (p,p,P,p1,p2) by A2, JORDAN5C: 9;

        hence thesis by A3, A5;

      end;

      ( Segment (P,p1,p2,p,p)) c= {p}

      proof

        let x be object;

        assume x in ( Segment (P,p1,p2,p,p));

        then

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

         A6: x = q and

         A7: LE (p,q,P,p1,p2) & LE (q,p,P,p1,p2) by A3;

        p = q by A1, A7, JORDAN5C: 12;

        hence thesis by A6, TARSKI:def 1;

      end;

      hence thesis by A4, XBOOLE_0:def 10;

    end;

    theorem :: JORDAN20:2

    

     Th2: for p1,p2,p be Point of ( TOP-REAL 2), a be Real st p in ( LSeg (p1,p2)) & (p1 `1 ) <= a & (p2 `1 ) <= a holds (p `1 ) <= a

    proof

      let p1,p2,p be Point of ( TOP-REAL 2), a be Real;

      assume that

       A1: p in ( LSeg (p1,p2)) and

       A2: (p1 `1 ) <= a and

       A3: (p2 `1 ) <= a;

      consider r be Real such that

       A4: p = (((1 - r) * p1) + (r * p2)) and

       A5: 0 <= r and

       A6: r <= 1 by A1;

      

       A7: (p `1 ) = ((((1 - r) * p1) `1 ) + ((r * p2) `1 )) by A4, TOPREAL3: 2

      .= ((((1 - r) * p1) `1 ) + (r * (p2 `1 ))) by TOPREAL3: 4

      .= (((1 - r) * (p1 `1 )) + (r * (p2 `1 ))) by TOPREAL3: 4;

      (1 - r) >= 0 by A6, XREAL_1: 48;

      then

       A8: ((1 - r) * (p1 `1 )) <= ((1 - r) * a) by A2, XREAL_1: 64;

      

       A9: (((1 - r) * a) + (r * a)) = a;

      (r * (p2 `1 )) <= (r * a) by A3, A5, XREAL_1: 64;

      hence thesis by A7, A8, A9, XREAL_1: 7;

    end;

    theorem :: JORDAN20:3

    

     Th3: for p1,p2,p be Point of ( TOP-REAL 2), a be Real st p in ( LSeg (p1,p2)) & (p1 `1 ) >= a & (p2 `1 ) >= a holds (p `1 ) >= a

    proof

      let p1,p2,p be Point of ( TOP-REAL 2), a be Real;

      assume that

       A1: p in ( LSeg (p1,p2)) and

       A2: (p1 `1 ) >= a and

       A3: (p2 `1 ) >= a;

      consider r be Real such that

       A4: p = (((1 - r) * p1) + (r * p2)) and

       A5: 0 <= r and

       A6: r <= 1 by A1;

      

       A7: (p `1 ) = ((((1 - r) * p1) `1 ) + ((r * p2) `1 )) by A4, TOPREAL3: 2

      .= ((((1 - r) * p1) `1 ) + (r * (p2 `1 ))) by TOPREAL3: 4

      .= (((1 - r) * (p1 `1 )) + (r * (p2 `1 ))) by TOPREAL3: 4;

      (1 - r) >= 0 by A6, XREAL_1: 48;

      then

       A8: ((1 - r) * (p1 `1 )) >= ((1 - r) * a) by A2, XREAL_1: 64;

      

       A9: (((1 - r) * a) + (r * a)) = a;

      (r * (p2 `1 )) >= (r * a) by A3, A5, XREAL_1: 64;

      hence thesis by A7, A8, A9, XREAL_1: 7;

    end;

    theorem :: JORDAN20:4

    for p1,p2,p be Point of ( TOP-REAL 2), a be Real st p in ( LSeg (p1,p2)) & (p1 `1 ) < a & (p2 `1 ) < a holds (p `1 ) < a

    proof

      let p1,p2,p be Point of ( TOP-REAL 2), a be Real;

      assume that

       A1: p in ( LSeg (p1,p2)) and

       A2: (p1 `1 ) < a and

       A3: (p2 `1 ) < a;

      consider r be Real such that

       A4: p = (((1 - r) * p1) + (r * p2)) and

       A5: 0 <= r and

       A6: r <= 1 by A1;

      

       A7: (p `1 ) = ((((1 - r) * p1) `1 ) + ((r * p2) `1 )) by A4, TOPREAL3: 2

      .= ((((1 - r) * p1) `1 ) + (r * (p2 `1 ))) by TOPREAL3: 4

      .= (((1 - r) * (p1 `1 )) + (r * (p2 `1 ))) by TOPREAL3: 4;

      per cases ;

        suppose 0 = r;

        

        then p = (p1 + ( 0 * p2)) by A4, RLVECT_1:def 8

        .= (p1 + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

        .= p1 by RLVECT_1: 4;

        hence thesis by A2;

      end;

        suppose

         A8: 0 <> r;

        

         A9: (((1 - r) * a) + (r * a)) = a;

        (1 - r) >= 0 by A6, XREAL_1: 48;

        then

         A10: ((1 - r) * (p1 `1 )) <= ((1 - r) * a) by A2, XREAL_1: 64;

        (r * (p2 `1 )) < (r * a) by A3, A5, A8, XREAL_1: 68;

        hence thesis by A7, A10, A9, XREAL_1: 8;

      end;

    end;

    theorem :: JORDAN20:5

    for p1,p2,p be Point of ( TOP-REAL 2), a be Real st p in ( LSeg (p1,p2)) & (p1 `1 ) > a & (p2 `1 ) > a holds (p `1 ) > a

    proof

      let p1,p2,p be Point of ( TOP-REAL 2), a be Real;

      assume that

       A1: p in ( LSeg (p1,p2)) and

       A2: (p1 `1 ) > a and

       A3: (p2 `1 ) > a;

      consider r be Real such that

       A4: p = (((1 - r) * p1) + (r * p2)) and

       A5: 0 <= r and

       A6: r <= 1 by A1;

      

       A7: (p `1 ) = ((((1 - r) * p1) `1 ) + ((r * p2) `1 )) by A4, TOPREAL3: 2

      .= ((((1 - r) * p1) `1 ) + (r * (p2 `1 ))) by TOPREAL3: 4

      .= (((1 - r) * (p1 `1 )) + (r * (p2 `1 ))) by TOPREAL3: 4;

      per cases ;

        suppose 0 = r;

        

        then p = (p1 + ( 0 * p2)) by A4, RLVECT_1:def 8

        .= (p1 + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

        .= p1 by RLVECT_1: 4;

        hence thesis by A2;

      end;

        suppose

         A8: 0 <> r;

        

         A9: (((1 - r) * a) + (r * a)) = a;

        (1 - r) >= 0 by A6, XREAL_1: 48;

        then

         A10: ((1 - r) * (p1 `1 )) >= ((1 - r) * a) by A2, XREAL_1: 64;

        (r * (p2 `1 )) > (r * a) by A3, A5, A8, XREAL_1: 68;

        hence thesis by A7, A10, A9, XREAL_1: 8;

      end;

    end;

    reserve j for Nat;

    theorem :: JORDAN20:6

    

     Th6: for f be S-Sequence_in_R2, p,q be Point of ( TOP-REAL 2) st 1 <= j & j < ( len f) & p in ( LSeg (f,j)) & q in ( LSeg (f,j)) & ((f /. j) `2 ) = ((f /. (j + 1)) `2 ) & ((f /. j) `1 ) > ((f /. (j + 1)) `1 ) & LE (p,q,( L~ f),(f /. 1),(f /. ( len f))) holds (p `1 ) >= (q `1 )

    proof

      let f be S-Sequence_in_R2, p,q be Point of ( TOP-REAL 2);

      assume that

       A1: 1 <= j and

       A2: j < ( len f) and

       A3: p in ( LSeg (f,j)) and

       A4: q in ( LSeg (f,j)) and

       A5: ((f /. j) `2 ) = ((f /. (j + 1)) `2 ) and

       A6: ((f /. j) `1 ) > ((f /. (j + 1)) `1 ) and

       A7: LE (p,q,( L~ f),(f /. 1),(f /. ( len f)));

      (j + 1) <= ( len f) by A2, NAT_1: 13;

      then

       A8: ( LSeg (f,j)) = ( LSeg ((f /. j),(f /. (j + 1)))) by A1, TOPREAL1:def 3;

      per cases ;

        suppose

         A9: (p `1 ) <> ((f /. j) `1 );

        ((f /. j) `1 ) >= (p `1 ) by A3, A6, A8, TOPREAL1: 3;

        then ((f /. j) `1 ) > (p `1 ) by A9, XXREAL_0: 1;

        then

         A10: (((f /. j) `1 ) - (p `1 )) > 0 by XREAL_1: 50;

        now

          reconsider a = ((((f /. j) `1 ) - (q `1 )) / (((f /. j) `1 ) - (p `1 ))) as Real;

          

           A11: (1 - a) = (((((f /. j) `1 ) - (p `1 )) / (((f /. j) `1 ) - (p `1 ))) - ((((f /. j) `1 ) - (q `1 )) / (((f /. j) `1 ) - (p `1 )))) by A10, XCMPLX_1: 60

          .= (((((f /. j) `1 ) - (p `1 )) - (((f /. j) `1 ) - (q `1 ))) / (((f /. j) `1 ) - (p `1 ))) by XCMPLX_1: 120

          .= (((q `1 ) - (p `1 )) / (((f /. j) `1 ) - (p `1 )));

          

           A12: ((((1 - a) * (f /. j)) + (a * p)) `1 ) = ((((1 - a) * (f /. j)) `1 ) + ((a * p) `1 )) by TOPREAL3: 2

          .= (((1 - a) * ((f /. j) `1 )) + ((a * p) `1 )) by TOPREAL3: 4

          .= (((1 * ((f /. j) `1 )) - (a * ((f /. j) `1 ))) + (a * (p `1 ))) by TOPREAL3: 4

          .= (((f /. j) `1 ) - (a * (((f /. j) `1 ) - (p `1 ))))

          .= (((f /. j) `1 ) - (((f /. j) `1 ) - (q `1 ))) by A10, XCMPLX_1: 87

          .= (q `1 );

          ((f /. j) `1 ) >= (q `1 ) by A4, A6, A8, TOPREAL1: 3;

          then

           A13: (((f /. j) `1 ) - (q `1 )) >= 0 by XREAL_1: 48;

          

           A14: (p `2 ) = ((f /. j) `2 ) by A3, A5, A8, GOBOARD7: 6;

          ((((1 - a) * (f /. j)) + (a * p)) `2 ) = ((((1 - a) * (f /. j)) `2 ) + ((a * p) `2 )) by TOPREAL3: 2

          .= (((1 - a) * ((f /. j) `2 )) + ((a * p) `2 )) by TOPREAL3: 4

          .= (((1 * ((f /. j) `2 )) - (a * ((f /. j) `2 ))) + (a * (p `2 ))) by TOPREAL3: 4

          .= (q `2 ) by A4, A5, A8, A14, GOBOARD7: 6;

          then

           A15: q = (((1 - a) * (f /. j)) + (a * p)) by A12, TOPREAL3: 6;

          assume

           A16: (p `1 ) < (q `1 );

          then ((q `1 ) - (p `1 )) > 0 by XREAL_1: 50;

          then ((1 - a) + a) >= ( 0 + a) by A10, A11, XREAL_1: 7;

          then q in ( LSeg ((f /. j),p)) by A10, A13, A15;

          then LE (q,p,( L~ f),(f /. 1),(f /. ( len f))) by A1, A2, A3, SPRECT_3: 23;

          hence contradiction by A7, A16, JORDAN5C: 12, TOPREAL1: 25;

        end;

        hence thesis;

      end;

        suppose (p `1 ) = ((f /. j) `1 );

        hence thesis by A4, A6, A8, TOPREAL1: 3;

      end;

    end;

    theorem :: JORDAN20:7

    

     Th7: for f be S-Sequence_in_R2, p,q be Point of ( TOP-REAL 2) st 1 <= j & j < ( len f) & p in ( LSeg (f,j)) & q in ( LSeg (f,j)) & ((f /. j) `2 ) = ((f /. (j + 1)) `2 ) & ((f /. j) `1 ) < ((f /. (j + 1)) `1 ) & LE (p,q,( L~ f),(f /. 1),(f /. ( len f))) holds (p `1 ) <= (q `1 )

    proof

      let f be S-Sequence_in_R2, p,q be Point of ( TOP-REAL 2);

      assume that

       A1: 1 <= j and

       A2: j < ( len f) and

       A3: p in ( LSeg (f,j)) and

       A4: q in ( LSeg (f,j)) and

       A5: ((f /. j) `2 ) = ((f /. (j + 1)) `2 ) and

       A6: ((f /. j) `1 ) < ((f /. (j + 1)) `1 ) and

       A7: LE (p,q,( L~ f),(f /. 1),(f /. ( len f)));

      (j + 1) <= ( len f) by A2, NAT_1: 13;

      then

       A8: ( LSeg (f,j)) = ( LSeg ((f /. j),(f /. (j + 1)))) by A1, TOPREAL1:def 3;

      per cases ;

        suppose

         A9: (p `1 ) <> ((f /. j) `1 );

        ((f /. j) `1 ) <= (p `1 ) by A3, A6, A8, TOPREAL1: 3;

        then ((f /. j) `1 ) < (p `1 ) by A9, XXREAL_0: 1;

        then

         A10: (((f /. j) `1 ) - (p `1 )) < 0 by XREAL_1: 49;

        now

          reconsider a = ((((f /. j) `1 ) - (q `1 )) / (((f /. j) `1 ) - (p `1 ))) as Real;

          

           A11: (1 - a) = (((((f /. j) `1 ) - (p `1 )) / (((f /. j) `1 ) - (p `1 ))) - ((((f /. j) `1 ) - (q `1 )) / (((f /. j) `1 ) - (p `1 )))) by A10, XCMPLX_1: 60

          .= (((((f /. j) `1 ) - (p `1 )) - (((f /. j) `1 ) - (q `1 ))) / (((f /. j) `1 ) - (p `1 ))) by XCMPLX_1: 120

          .= (((q `1 ) - (p `1 )) / (((f /. j) `1 ) - (p `1 )));

          

           A12: ((((1 - a) * (f /. j)) + (a * p)) `1 ) = ((((1 - a) * (f /. j)) `1 ) + ((a * p) `1 )) by TOPREAL3: 2

          .= (((1 - a) * ((f /. j) `1 )) + ((a * p) `1 )) by TOPREAL3: 4

          .= (((1 * ((f /. j) `1 )) - (a * ((f /. j) `1 ))) + (a * (p `1 ))) by TOPREAL3: 4

          .= (((f /. j) `1 ) - (a * (((f /. j) `1 ) - (p `1 ))))

          .= (((f /. j) `1 ) - (((f /. j) `1 ) - (q `1 ))) by A10, XCMPLX_1: 87

          .= (q `1 );

          ((f /. j) `1 ) <= (q `1 ) by A4, A6, A8, TOPREAL1: 3;

          then

           A13: (((f /. j) `1 ) - (q `1 )) <= 0 by XREAL_1: 47;

          

           A14: (p `2 ) = ((f /. j) `2 ) by A3, A5, A8, GOBOARD7: 6;

          ((((1 - a) * (f /. j)) + (a * p)) `2 ) = ((((1 - a) * (f /. j)) `2 ) + ((a * p) `2 )) by TOPREAL3: 2

          .= (((1 - a) * ((f /. j) `2 )) + ((a * p) `2 )) by TOPREAL3: 4

          .= (((1 * ((f /. j) `2 )) - (a * ((f /. j) `2 ))) + (a * (p `2 ))) by TOPREAL3: 4

          .= (q `2 ) by A4, A5, A8, A14, GOBOARD7: 6;

          then

           A15: q = (((1 - a) * (f /. j)) + (a * p)) by A12, TOPREAL3: 6;

          assume

           A16: (p `1 ) > (q `1 );

          then ((q `1 ) - (p `1 )) < 0 by XREAL_1: 49;

          then ((1 - a) + a) >= ( 0 + a) by A10, A11, XREAL_1: 7;

          then q in ( LSeg ((f /. j),p)) by A10, A13, A15;

          then LE (q,p,( L~ f),(f /. 1),(f /. ( len f))) by A1, A2, A3, SPRECT_3: 23;

          hence contradiction by A7, A16, JORDAN5C: 12, TOPREAL1: 25;

        end;

        hence thesis;

      end;

        suppose (p `1 ) = ((f /. j) `1 );

        hence thesis by A4, A6, A8, TOPREAL1: 3;

      end;

    end;

    definition

      let P be Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      :: JORDAN20:def1

      pred p is_Lin P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p4 be Point of ( TOP-REAL 2) st (p4 `1 ) < e & LE (p4,p,P,p1,p2) & for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) <= e;

      :: JORDAN20:def2

      pred p is_Rin P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p4 be Point of ( TOP-REAL 2) st (p4 `1 ) > e & LE (p4,p,P,p1,p2) & for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) >= e;

      :: JORDAN20:def3

      pred p is_Lout P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p4 be Point of ( TOP-REAL 2) st (p4 `1 ) < e & LE (p,p4,P,p1,p2) & for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) <= e;

      :: JORDAN20:def4

      pred p is_Rout P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p4 be Point of ( TOP-REAL 2) st (p4 `1 ) > e & LE (p,p4,P,p1,p2) & for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) >= e;

      :: JORDAN20:def5

      pred p is_OSin P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p7 be Point of ( TOP-REAL 2) st LE (p7,p,P,p1,p2) & (for p8 be Point of ( TOP-REAL 2) st LE (p7,p8,P,p1,p2) & LE (p8,p,P,p1,p2) holds (p8 `1 ) = e) & for p4 be Point of ( TOP-REAL 2) st LE (p4,p7,P,p1,p2) & p4 <> p7 holds (ex p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,p7,P,p1,p2) & (p5 `1 ) > e) & ex p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p7,P,p1,p2) & (p6 `1 ) < e;

      :: JORDAN20:def6

      pred p is_OSout P,p1,p2,e means P is_an_arc_of (p1,p2) & p in P & (p `1 ) = e & ex p7 be Point of ( TOP-REAL 2) st LE (p,p7,P,p1,p2) & (for p8 be Point of ( TOP-REAL 2) st LE (p8,p7,P,p1,p2) & LE (p,p8,P,p1,p2) holds (p8 `1 ) = e) & for p4 be Point of ( TOP-REAL 2) st LE (p7,p4,P,p1,p2) & p4 <> p7 holds (ex p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (p7,p5,P,p1,p2) & (p5 `1 ) > e) & ex p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p7,p6,P,p1,p2) & (p6 `1 ) < e;

      correctness ;

    end

    theorem :: JORDAN20:8

    for P be Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real st P is_an_arc_of (p1,p2) & (p1 `1 ) <= e & (p2 `1 ) >= e holds ex p3 be Point of ( TOP-REAL 2) st p3 in P & (p3 `1 ) = e

    proof

      let P be Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      set x = the Element of (P /\ ( Vertical_Line e));

      assume P is_an_arc_of (p1,p2) & (p1 `1 ) <= e & (p2 `1 ) >= e;

      then P meets ( Vertical_Line e) by JORDAN6: 49;

      then

       A1: (P /\ ( Vertical_Line e)) <> {} by XBOOLE_0:def 7;

      then x in ( Vertical_Line e) by XBOOLE_0:def 4;

      then x in { p3 where p3 be Point of ( TOP-REAL 2) : (p3 `1 ) = e } by JORDAN6:def 6;

      then

       A2: ex p4 be Point of ( TOP-REAL 2) st p4 = x & (p4 `1 ) = e;

      x in P by A1, XBOOLE_0:def 4;

      hence thesis by A2;

    end;

    theorem :: JORDAN20:9

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real st P is_an_arc_of (p1,p2) & (p1 `1 ) < e & p in P & (p `1 ) = e holds p is_Lin (P,p1,p2,e) or p is_Rin (P,p1,p2,e) or p is_OSin (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: (p1 `1 ) < e and

       A3: p in P and

       A4: (p `1 ) = e;

      now

        reconsider pr1a = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider pro1 = (pr1a | P) as Function of (( TOP-REAL 2) | P), R^1 by PRE_TOPC: 9;

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

         A5: f is being_homeomorphism and

         A6: (f . 0 ) = p1 and

         A7: (f . 1) = p2 by A1, TOPREAL1:def 1;

        

         A8: f is continuous by A5, TOPS_2:def 5;

        

         A9: ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A5, TOPS_2:def 5;

        then p in ( rng f) by A3, PRE_TOPC:def 5;

        then

        consider xs be object such that

         A10: xs in ( dom f) and

         A11: p = (f . xs) by FUNCT_1:def 3;

        

         A12: ( dom f) = ( [#] I[01] ) by A5, TOPS_2:def 5;

        then

        reconsider s2 = xs as Element of REAL by A10, BORSUK_1: 40;

        

         A13: 0 <= s2 by A10, BORSUK_1: 40, XXREAL_1: 1;

        

         A14: 0 in REAL by XREAL_0:def 1;

        for q be Point of ( TOP-REAL 2) st q = (f . 0 ) holds (q `1 ) <> e by A2, A6;

        then

         A15: 0 in { s where s be Element of REAL : 0 <= s & s <= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) } by A13, A14;

        { s where s be Element of REAL : 0 <= s & s <= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e } c= REAL

        proof

          let x be object;

          assume x in { s where s be Element of REAL : 0 <= s & s <= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) };

          then ex s be Element of REAL st s = x & 0 <= s & s <= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e;

          hence thesis;

        end;

        then

        reconsider R = { s where s be Element of REAL : 0 <= s & s <= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) } as non empty Subset of REAL by A15;

        

         A16: s2 <= 1 by A10, BORSUK_1: 40, XXREAL_1: 1;

        R c= [. 0 , 1.]

        proof

          let x be object;

          assume x in R;

          then

          consider s be Element of REAL such that

           A17: s = x & 0 <= s and

           A18: s <= s2 and for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e;

          s <= 1 by A16, A18, XXREAL_0: 2;

          hence thesis by A17, XXREAL_1: 1;

        end;

        then

        reconsider R99 = R as Subset of I[01] by BORSUK_1: 40;

        reconsider s0 = ( upper_bound R) as Element of REAL by XREAL_0:def 1;

        

         A19: for s be Real st s in R holds s < s2

        proof

          let s be Real;

          assume s in R;

          then

           A20: ex s3 be Element of REAL st s3 = s & 0 <= s3 & s3 <= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s3) holds (q `1 ) <> e;

          then s <> s2 by A4, A11;

          hence thesis by A20, XXREAL_0: 1;

        end;

        then for s be Real st s in R holds s <= s2;

        then

         A21: s0 <= s2 by SEQ_4: 45;

        then

         A22: s0 <= 1 by A16, XXREAL_0: 2;

        R99 = R;

        then

         A23: 0 <= s0 by A15, BORSUK_4: 26;

        then s0 in ( dom f) by A12, A22, BORSUK_1: 40, XXREAL_1: 1;

        then (f . s0) in ( rng f) by FUNCT_1:def 3;

        then (f . s0) in P by A9, PRE_TOPC:def 5;

        then

        reconsider p9 = (f . s0) as Point of ( TOP-REAL 2);

        

         A24: LE (p9,p,P,p1,p2) by A1, A5, A6, A7, A11, A16, A23, A21, A22, JORDAN5C: 8;

        for p7 be Point of (( TOP-REAL 2) | P) holds (pro1 . p7) = ( proj1 . p7)

        proof

          let p7 be Point of (( TOP-REAL 2) | P);

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

          hence thesis by FUNCT_1: 49;

        end;

        then

         A25: pro1 is continuous by JGRAPH_2: 29;

        reconsider h = (pro1 * f) as Function of I[01] , R^1 ;

        

         A26: ( dom h) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

        for s be ExtReal st s in R holds s <= s2 by A19;

        then s2 is UpperBound of R by XXREAL_2:def 1;

        then

         A27: R is bounded_above by XXREAL_2:def 10;

        

         A28: ( rng f) = P by A9, PRE_TOPC:def 5;

        

         A29: for p8 be Point of ( TOP-REAL 2) st LE (p9,p8,P,p1,p2) & LE (p8,p,P,p1,p2) holds (p8 `1 ) = e

        proof

          let p8 be Point of ( TOP-REAL 2);

          assume that

           A30: LE (p9,p8,P,p1,p2) and

           A31: LE (p8,p,P,p1,p2);

          

           A32: p8 in P by A30, JORDAN5C:def 3;

          then

          consider x8 be object such that

           A33: x8 in ( dom f) and

           A34: p8 = (f . x8) by A28, FUNCT_1:def 3;

          reconsider s8 = x8 as Element of REAL by A12, A33, BORSUK_1: 40;

          

           A35: s8 <= 1 by A33, BORSUK_1: 40, XXREAL_1: 1;

          then

           A36: s8 <= s2 by A5, A6, A7, A11, A13, A16, A31, A34, JORDAN5C:def 3;

          

           A37: 0 <= s8 by A33, BORSUK_1: 40, XXREAL_1: 1;

          then

           A38: s0 <= s8 by A5, A6, A7, A22, A30, A34, A35, JORDAN5C:def 3;

          now

            reconsider s8n = s8 as Point of RealSpace by METRIC_1:def 13;

            reconsider s8m = s8 as Point of ( Closed-Interval-MSpace ( 0 ,1)) by A33, BORSUK_1: 40, TOPMETR: 10;

            reconsider ee = ( |.((p8 `1 ) - e).| / 2) as Real;

            reconsider w = (p8 `1 ) as Element of RealSpace by METRIC_1:def 13, XREAL_0:def 1;

            reconsider B = ( Ball (w,ee)) as Subset of R^1 by METRIC_1:def 13, TOPMETR: 17;

            

             A39: B = { s7 where s7 be Real : ((p8 `1 ) - ee) < s7 & s7 < ((p8 `1 ) + ee) } by JORDAN2B: 17

            .= ].((p8 `1 ) - ee), ((p8 `1 ) + ee).[ by RCOMP_1:def 2;

            assume

             A40: (p8 `1 ) <> e;

            then ((p8 `1 ) - e) <> 0 ;

            then |.((p8 `1 ) - e).| > 0 by COMPLEX1: 47;

            then

             A41: w in ( Ball (w,ee)) by GOBOARD6: 1, XREAL_1: 139;

            

             A42: (h " B) is open & I[01] = ( TopSpaceMetr ( Closed-Interval-MSpace ( 0 ,1))) by A8, A25, TOPMETR: 20, TOPMETR:def 6, TOPMETR:def 7, UNIFORM1: 2;

            (h . s8) = (pro1 . (f . s8)) by A26, A33, BORSUK_1: 40, FUNCT_1: 12

            .= ( proj1 . p8) by A32, A34, FUNCT_1: 49

            .= (p8 `1 ) by PSCOMP_1:def 5;

            then s8 in (h " B) by A26, A33, A41, BORSUK_1: 40, FUNCT_1:def 7;

            then

            consider r0 be Real such that

             A43: r0 > 0 and

             A44: ( Ball (s8m,r0)) c= (h " B) by A42, TOPMETR: 15;

            reconsider r0 as Real;

            reconsider r01 = ( min ((s2 - s8),r0)) as Real;

            s8 < s2 by A4, A11, A34, A36, A40, XXREAL_0: 1;

            then (s2 - s8) > 0 by XREAL_1: 50;

            then

             A45: r01 > 0 by A43, XXREAL_0: 21;

            then

             A46: ((r01 - (r01 / 2)) + (r01 / 2)) > ( 0 + (r01 / 2)) by XREAL_1: 6;

            then

             A47: (s8 + (r01 / 2)) < (s8 + r01) by XREAL_1: 6;

            reconsider s70 = (s8 + (r01 / 2)) as Real;

            the carrier of ( Closed-Interval-MSpace ( 0 ,1)) = [. 0 , 1.] & ( Ball (s8n,r01)) = ].(s8 - r01), (s8 + r01).[ by FRECHET: 7, TOPMETR: 10;

            then

             A48: ( Ball (s8m,r01)) = ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) by TOPMETR: 9;

            (s2 - s8) >= r01 by XXREAL_0: 17;

            then

             A49: ((s2 - s8) + s8) >= (r01 + s8) by XREAL_1: 7;

            then

             A50: s70 <= s2 by A47, XXREAL_0: 2;

            (s8 + r01) <= 1 by A16, A49, XXREAL_0: 2;

            then (s8 + (r01 / 2)) < 1 by A47, XXREAL_0: 2;

            then

             A51: (s8 + (r01 / 2)) in [. 0 , 1.] by A37, A45, XXREAL_1: 1;

            ( Ball (s8m,r01)) c= ( Ball (s8m,r0)) by PCOMPS_1: 1, XXREAL_0: 17;

            then

             A52: ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) c= (h " B) by A44, A48;

            (s8 + 0 ) < (s8 + ((r01 / 2) + r01)) by A45, XREAL_1: 6;

            then ((s8 - r01) + r01) < ((s8 + (r01 / 2)) + r01);

            then

             A53: (s8 - r01) < (s8 + (r01 / 2)) by XREAL_1: 6;

            (s8 + (r01 / 2)) < (s8 + r01) by A46, XREAL_1: 6;

            then (s8 + (r01 / 2)) in ].(s8 - r01), (s8 + r01).[ by A53, XXREAL_1: 4;

            then

             A54: (s8 + (r01 / 2)) in ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) by A51, XBOOLE_0:def 4;

            then

             A55: (h . (s8 + (r01 / 2))) in B by A52, FUNCT_1:def 7;

            

             A56: (s8 + (r01 / 2)) in ( dom h) by A52, A54, FUNCT_1:def 7;

            

             A57: for p7 be Point of ( TOP-REAL 2) st p7 = (f . s70) holds (p7 `1 ) <> e

            proof

              let p7 be Point of ( TOP-REAL 2);

              assume

               A58: p7 = (f . s70);

              s70 <= 1 by A16, A50, XXREAL_0: 2;

              then s70 in [. 0 , 1.] by A37, A45, XXREAL_1: 1;

              then

               A59: p7 in ( rng f) by A12, A58, BORSUK_1: 40, FUNCT_1:def 3;

              

               A60: ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A5, TOPS_2:def 5

              .= P by PRE_TOPC:def 5;

              

               A61: (h . s70) = (pro1 . (f . s70)) by A56, FUNCT_1: 12

              .= (pr1a . p7) by A58, A59, A60, FUNCT_1: 49

              .= (p7 `1 ) by PSCOMP_1:def 5;

              then

               A62: (p7 `1 ) < ((p8 `1 ) + ee) by A39, A55, XXREAL_1: 4;

              

               A63: ((p8 `1 ) - ee) < (p7 `1 ) by A39, A55, A61, XXREAL_1: 4;

              now

                assume

                 A64: (p7 `1 ) = e;

                now

                  per cases ;

                    case

                     A65: ((p8 `1 ) - e) >= 0 ;

                    then ((p8 `1 ) - (((p8 `1 ) - e) / 2)) < e by A63, A64, ABSVALUE:def 1;

                    then (((p8 `1 ) / 2) + (e / 2)) < ((e / 2) + (e / 2));

                    then ((p8 `1 ) / 2) < (e / 2) by XREAL_1: 7;

                    then

                     A66: (((p8 `1 ) / 2) - (e / 2)) < ((e / 2) - (e / 2)) by XREAL_1: 14;

                    (((p8 `1 ) - e) / 2) >= ( 0 / 2) by A65;

                    hence contradiction by A66;

                  end;

                    case

                     A67: ((p8 `1 ) - e) < 0 ;

                    then e < ((p8 `1 ) + (( - ((p8 `1 ) - e)) / 2)) by A62, A64, ABSVALUE:def 1;

                    then (((p8 `1 ) / 2) + (e / 2)) > ((e / 2) + (e / 2));

                    then ((p8 `1 ) / 2) > (e / 2) by XREAL_1: 7;

                    then

                     A68: (((p8 `1 ) / 2) - (e / 2)) > ((e / 2) - (e / 2)) by XREAL_1: 14;

                    (((p8 `1 ) - e) / 2) <= ( 0 / 2) by A67;

                    hence contradiction by A68;

                  end;

                end;

                hence contradiction;

              end;

              hence thesis;

            end;

            s8 < s70 by A45, XREAL_1: 29, XREAL_1: 139;

            then

            consider s7 be Real such that

             A69: s8 < s7 and

             A70: 0 <= s7 & s7 <= s2 & for p7 be Point of ( TOP-REAL 2) st p7 = (f . s7) holds (p7 `1 ) <> e by A37, A50, A57;

            reconsider s7 as Element of REAL by XREAL_0:def 1;

            s7 in R by A70;

            then s7 <= s0 by A27, SEQ_4:def 1;

            hence contradiction by A38, A69, XXREAL_0: 2;

          end;

          hence thesis;

        end;

        assume not p is_OSin (P,p1,p2,e);

        then

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

         A71: LE (p4,p9,P,p1,p2) and

         A72: p4 <> p9 and

         A73: (for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,p9,P,p1,p2) holds (p5 `1 ) <= e) or for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p9,P,p1,p2) holds (p6 `1 ) >= e by A1, A3, A4, A24, A29;

        

         A74: p9 in P by A71, JORDAN5C:def 3;

        now

          per cases by A73;

            case

             A75: for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,p9,P,p1,p2) holds (p5 `1 ) <= e;

             A76:

            now

              p4 in P by A71, JORDAN5C:def 3;

              then p4 in ( rng f) by A9, PRE_TOPC:def 5;

              then

              consider xs4 be object such that

               A77: xs4 in ( dom f) and

               A78: p4 = (f . xs4) by FUNCT_1:def 3;

              reconsider s4 = xs4 as Real by A77;

              

               A79: 0 <= s4 by A77, BORSUK_1: 40, XXREAL_1: 1;

              

               A80: s4 <= 1 by A77, BORSUK_1: 40, XXREAL_1: 1;

              assume

               A81: not ex p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) & (p51 `1 ) < e;

              

               A82: for p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) holds (p51 `1 ) = e

              proof

                let p51 be Point of ( TOP-REAL 2);

                assume LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2);

                then (p51 `1 ) >= e & (p51 `1 ) <= e by A75, A81;

                hence thesis by XXREAL_0: 1;

              end;

               A83:

              now

                assume s4 < s0;

                then

                 A84: (s0 - s4) > 0 by XREAL_1: 50;

                then

                 A85: s4 < (s4 + ((s0 - s4) / 2)) by XREAL_1: 29, XREAL_1: 139;

                ((s0 - s4) / 2) > 0 by A84, XREAL_1: 139;

                then

                consider r be Real such that

                 A86: r in R and

                 A87: (s0 - ((s0 - s4) / 2)) < r by A27, SEQ_4:def 1;

                reconsider rss = r as Real;

                

                 A88: ex s7 be Element of REAL st s7 = r & 0 <= s7 & s7 <= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s7) holds (q `1 ) <> e by A86;

                then

                 A89: r <= 1 by A16, XXREAL_0: 2;

                then r in [. 0 , 1.] by A79, A85, A87, XXREAL_1: 1;

                then (f . rss) in ( rng f) by A12, BORSUK_1: 40, FUNCT_1:def 3;

                then (f . rss) in P by A9, PRE_TOPC:def 5;

                then

                reconsider pss = (f . rss) as Point of ( TOP-REAL 2);

                s4 < r by A85, A87, XXREAL_0: 2;

                then

                 A90: LE (p4,pss,P,p1,p2) by A1, A5, A6, A7, A78, A79, A80, A89, JORDAN5C: 8;

                r <= s0 by A27, A86, SEQ_4:def 1;

                then LE (pss,p9,P,p1,p2) by A1, A5, A6, A7, A22, A79, A85, A87, A89, JORDAN5C: 8;

                then (pss `1 ) = e by A82, A90;

                hence contradiction by A88;

              end;

              s4 <= s0 by A5, A6, A7, A23, A22, A71, A78, A80, JORDAN5C:def 3;

              hence contradiction by A72, A78, A83, XXREAL_0: 1;

            end;

            now

              assume ex p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) & (p51 `1 ) < e;

              then

              consider p51 be Point of ( TOP-REAL 2) such that

               A91: LE (p4,p51,P,p1,p2) and

               A92: LE (p51,p9,P,p1,p2) and

               A93: (p51 `1 ) < e;

              

               A94: for p5 be Point of ( TOP-REAL 2) st LE (p51,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A95: LE (p51,p5,P,p1,p2) and

                 A96: LE (p5,p,P,p1,p2);

                

                 A97: LE (p4,p5,P,p1,p2) by A91, A95, JORDAN5C: 13;

                

                 A98: p5 in P by A95, JORDAN5C:def 3;

                then

                 A99: p5 = p9 implies LE (p9,p5,P,p1,p2) by JORDAN5C: 9;

                now

                  per cases by A1, A74, A98, A99, JORDAN5C: 14;

                    case LE (p5,p9,P,p1,p2);

                    hence thesis by A75, A97;

                  end;

                    case LE (p9,p5,P,p1,p2);

                    hence thesis by A29, A96;

                  end;

                end;

                hence thesis;

              end;

               LE (p51,p,P,p1,p2) by A24, A92, JORDAN5C: 13;

              hence p is_Lin (P,p1,p2,e) by A1, A3, A4, A93, A94;

            end;

            hence p is_Lin (P,p1,p2,e) or p is_Rin (P,p1,p2,e) by A76;

          end;

            case

             A100: for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p9,P,p1,p2) holds (p6 `1 ) >= e;

             A101:

            now

              p4 in P by A71, JORDAN5C:def 3;

              then p4 in ( rng f) by A9, PRE_TOPC:def 5;

              then

              consider xs4 be object such that

               A102: xs4 in ( dom f) and

               A103: p4 = (f . xs4) by FUNCT_1:def 3;

              reconsider s4 = xs4 as Real by A102;

              

               A104: 0 <= s4 by A102, BORSUK_1: 40, XXREAL_1: 1;

              

               A105: s4 <= 1 by A102, BORSUK_1: 40, XXREAL_1: 1;

              assume

               A106: not ex p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) & (p51 `1 ) > e;

              

               A107: for p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) holds (p51 `1 ) = e

              proof

                let p51 be Point of ( TOP-REAL 2);

                assume LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2);

                then (p51 `1 ) <= e & (p51 `1 ) >= e by A100, A106;

                hence thesis by XXREAL_0: 1;

              end;

               A108:

              now

                assume s4 < s0;

                then

                 A109: (s0 - s4) > 0 by XREAL_1: 50;

                then

                 A110: s4 < (s4 + ((s0 - s4) / 2)) by XREAL_1: 29, XREAL_1: 139;

                ((s0 - s4) / 2) > 0 by A109, XREAL_1: 139;

                then

                consider r be Real such that

                 A111: r in R and

                 A112: (s0 - ((s0 - s4) / 2)) < r by A27, SEQ_4:def 1;

                reconsider rss = r as Real;

                

                 A113: ex s7 be Element of REAL st s7 = r & 0 <= s7 & s7 <= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s7) holds (q `1 ) <> e by A111;

                then

                 A114: r <= 1 by A16, XXREAL_0: 2;

                then r in [. 0 , 1.] by A104, A110, A112, XXREAL_1: 1;

                then (f . rss) in ( rng f) by A12, BORSUK_1: 40, FUNCT_1:def 3;

                then (f . rss) in P by A9, PRE_TOPC:def 5;

                then

                reconsider pss = (f . rss) as Point of ( TOP-REAL 2);

                s4 < r by A110, A112, XXREAL_0: 2;

                then

                 A115: LE (p4,pss,P,p1,p2) by A1, A5, A6, A7, A103, A104, A105, A114, JORDAN5C: 8;

                r <= s0 by A27, A111, SEQ_4:def 1;

                then LE (pss,p9,P,p1,p2) by A1, A5, A6, A7, A22, A104, A110, A112, A114, JORDAN5C: 8;

                then (pss `1 ) = e by A107, A115;

                hence contradiction by A113;

              end;

              s4 <= s0 by A5, A6, A7, A23, A22, A71, A103, A105, JORDAN5C:def 3;

              hence contradiction by A72, A103, A108, XXREAL_0: 1;

            end;

            now

              assume ex p51 be Point of ( TOP-REAL 2) st LE (p4,p51,P,p1,p2) & LE (p51,p9,P,p1,p2) & (p51 `1 ) > e;

              then

              consider p51 be Point of ( TOP-REAL 2) such that

               A116: LE (p4,p51,P,p1,p2) and

               A117: LE (p51,p9,P,p1,p2) and

               A118: (p51 `1 ) > e;

              

               A119: for p5 be Point of ( TOP-REAL 2) st LE (p51,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A120: LE (p51,p5,P,p1,p2) and

                 A121: LE (p5,p,P,p1,p2);

                

                 A122: LE (p4,p5,P,p1,p2) by A116, A120, JORDAN5C: 13;

                

                 A123: p5 in P by A120, JORDAN5C:def 3;

                then

                 A124: p5 = p9 implies LE (p9,p5,P,p1,p2) by JORDAN5C: 9;

                now

                  per cases by A1, A74, A123, A124, JORDAN5C: 14;

                    case LE (p5,p9,P,p1,p2);

                    hence thesis by A100, A122;

                  end;

                    case LE (p9,p5,P,p1,p2);

                    hence thesis by A29, A121;

                  end;

                end;

                hence thesis;

              end;

               LE (p51,p,P,p1,p2) by A24, A117, JORDAN5C: 13;

              hence p is_Rin (P,p1,p2,e) by A1, A3, A4, A118, A119;

            end;

            hence p is_Lin (P,p1,p2,e) or p is_Rin (P,p1,p2,e) by A101;

          end;

        end;

        hence p is_Lin (P,p1,p2,e) or p is_Rin (P,p1,p2,e);

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:10

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real st P is_an_arc_of (p1,p2) & (p2 `1 ) > e & p in P & (p `1 ) = e holds p is_Lout (P,p1,p2,e) or p is_Rout (P,p1,p2,e) or p is_OSout (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: (p2 `1 ) > e and

       A3: p in P and

       A4: (p `1 ) = e;

      now

        reconsider pr1a = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

        reconsider pro1 = (pr1a | P) as Function of (( TOP-REAL 2) | P), R^1 by PRE_TOPC: 9;

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

         A5: f is being_homeomorphism and

         A6: (f . 0 ) = p1 and

         A7: (f . 1) = p2 by A1, TOPREAL1:def 1;

        

         A8: f is continuous by A5, TOPS_2:def 5;

        

         A9: ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A5, TOPS_2:def 5;

        then p in ( rng f) by A3, PRE_TOPC:def 5;

        then

        consider xs be object such that

         A10: xs in ( dom f) and

         A11: p = (f . xs) by FUNCT_1:def 3;

        

         A12: ( dom f) = ( [#] I[01] ) by A5, TOPS_2:def 5;

        reconsider s2 = xs as Real by A10;

        

         A13: s2 <= 1 by A10, BORSUK_1: 40, XXREAL_1: 1;

        for q be Point of ( TOP-REAL 2) st q = (f . 1) holds (q `1 ) <> e by A2, A7;

        then

         A14: 1 in { s where s be Real : 1 >= s & s >= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) } by A13;

        { s where s be Real : 1 >= s & s >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e } c= REAL

        proof

          let x be object;

          assume x in { s where s be Real : 1 >= s & s >= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) };

          then

          consider s be Real such that

           A15: s = x & 1 >= s & s >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e;

          s in REAL by XREAL_0:def 1;

          hence thesis by A15;

        end;

        then

        reconsider R = { s where s be Real : 1 >= s & s >= s2 & (for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e) } as non empty Subset of REAL by A14;

        reconsider s0 = ( lower_bound R) as Real;

        

         A16: for s be Real st s in R holds s > s2

        proof

          let s be Real;

          assume s in R;

          then

           A17: ex s3 be Real st s3 = s & 1 >= s3 & s3 >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s3) holds (q `1 ) <> e;

          then s <> s2 by A4, A11;

          hence thesis by A17, XXREAL_0: 1;

        end;

        then for s be Real st s in R holds s >= s2;

        then

         A18: s0 >= s2 by SEQ_4: 43;

        for p7 be Point of (( TOP-REAL 2) | P) holds (pro1 . p7) = ( proj1 . p7)

        proof

          let p7 be Point of (( TOP-REAL 2) | P);

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

          hence thesis by FUNCT_1: 49;

        end;

        then

         A19: pro1 is continuous by JGRAPH_2: 29;

        reconsider h = (pro1 * f) as Function of I[01] , R^1 ;

        

         A20: ( dom h) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

        for s be ExtReal st s in R holds s >= s2 by A16;

        then s2 is LowerBound of R by XXREAL_2:def 2;

        then

         A21: R is bounded_below by XXREAL_2:def 9;

        

         A22: 0 <= s2 by A10, BORSUK_1: 40, XXREAL_1: 1;

        R c= [. 0 , 1.]

        proof

          let x be object;

          assume x in R;

          then ex s be Real st s = x & 1 >= s & s >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s) holds (q `1 ) <> e;

          hence thesis by A22, XXREAL_1: 1;

        end;

        then

         A23: 1 >= s0 by A14, BORSUK_1: 40, BORSUK_4: 26;

        then s0 in ( dom f) by A12, A22, A18, BORSUK_1: 40, XXREAL_1: 1;

        then (f . s0) in ( rng f) by FUNCT_1:def 3;

        then (f . s0) in P by A9, PRE_TOPC:def 5;

        then

        reconsider p9 = (f . s0) as Point of ( TOP-REAL 2);

        

         A24: LE (p,p9,P,p1,p2) by A1, A5, A6, A7, A11, A22, A13, A23, A18, JORDAN5C: 8;

        

         A25: ( rng f) = P by A9, PRE_TOPC:def 5;

        

         A26: for p8 be Point of ( TOP-REAL 2) st LE (p8,p9,P,p1,p2) & LE (p,p8,P,p1,p2) holds (p8 `1 ) = e

        proof

          let p8 be Point of ( TOP-REAL 2);

          assume that

           A27: LE (p8,p9,P,p1,p2) and

           A28: LE (p,p8,P,p1,p2);

          

           A29: p8 in P by A27, JORDAN5C:def 3;

          then

          consider x8 be object such that

           A30: x8 in ( dom f) and

           A31: p8 = (f . x8) by A25, FUNCT_1:def 3;

          reconsider s8 = x8 as Element of REAL by A12, A30, BORSUK_1: 40;

          

           A32: s8 <= 1 by A30, BORSUK_1: 40, XXREAL_1: 1;

           0 <= s8 by A30, BORSUK_1: 40, XXREAL_1: 1;

          then

           A33: s8 >= s2 by A5, A6, A7, A11, A13, A28, A31, A32, JORDAN5C:def 3;

          

           A34: s0 >= s8 by A5, A6, A7, A22, A23, A18, A27, A31, A32, JORDAN5C:def 3;

          now

            reconsider s8n = s8 as Point of RealSpace by METRIC_1:def 13;

            reconsider s8m = s8 as Point of ( Closed-Interval-MSpace ( 0 ,1)) by A30, BORSUK_1: 40, TOPMETR: 10;

            reconsider ee = ( |.((p8 `1 ) - e).| / 2) as Real;

            reconsider w = (p8 `1 ) as Element of RealSpace by METRIC_1:def 13, XREAL_0:def 1;

            reconsider B = ( Ball (w,ee)) as Subset of R^1 by METRIC_1:def 13, TOPMETR: 17;

            

             A35: B = { s7 where s7 be Real : ((p8 `1 ) - ee) < s7 & s7 < ((p8 `1 ) + ee) } by JORDAN2B: 17

            .= ].((p8 `1 ) - ee), ((p8 `1 ) + ee).[ by RCOMP_1:def 2;

            assume

             A36: (p8 `1 ) <> e;

            then ((p8 `1 ) - e) <> 0 ;

            then |.((p8 `1 ) - e).| > 0 by COMPLEX1: 47;

            then

             A37: w in ( Ball (w,ee)) by GOBOARD6: 1, XREAL_1: 139;

            

             A38: (h " B) is open & I[01] = ( TopSpaceMetr ( Closed-Interval-MSpace ( 0 ,1))) by A8, A19, TOPMETR: 20, TOPMETR:def 6, TOPMETR:def 7, UNIFORM1: 2;

            (h . s8) = (pro1 . (f . s8)) by A20, A30, BORSUK_1: 40, FUNCT_1: 12

            .= ( proj1 . p8) by A29, A31, FUNCT_1: 49

            .= (p8 `1 ) by PSCOMP_1:def 5;

            then s8 in (h " B) by A20, A30, A37, BORSUK_1: 40, FUNCT_1:def 7;

            then

            consider r0 be Real such that

             A39: r0 > 0 and

             A40: ( Ball (s8m,r0)) c= (h " B) by A38, TOPMETR: 15;

            reconsider r0 as Real;

            reconsider r01 = ( min ((s8 - s2),r0)) as Real;

            the carrier of ( Closed-Interval-MSpace ( 0 ,1)) = [. 0 , 1.] & ( Ball (s8n,r01)) = ].(s8 - r01), (s8 + r01).[ by FRECHET: 7, TOPMETR: 10;

            then

             A41: ( Ball (s8m,r01)) = ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) by TOPMETR: 9;

            s8 > s2 by A4, A11, A31, A33, A36, XXREAL_0: 1;

            then (s8 - s2) > 0 by XREAL_1: 50;

            then

             A42: r01 > 0 by A39, XXREAL_0: 21;

            then

             A43: ((r01 - (r01 / 2)) + (r01 / 2)) > ( 0 + (r01 / 2)) by XREAL_1: 6;

            then

             A44: (s8 - r01) < (s8 - (r01 / 2)) by XREAL_1: 10;

            

             A45: (r01 / 2) > 0 by A42, XREAL_1: 139;

            then

             A46: (s8 + ( - (r01 / 2))) < (s8 + (r01 / 2)) by XREAL_1: 8;

            (s8 + (r01 / 2)) < (s8 + r01) by A43, XREAL_1: 8;

            then (s8 - (r01 / 2)) < (s8 + r01) by A46, XXREAL_0: 2;

            then

             A47: (s8 - (r01 / 2)) in ].(s8 - r01), (s8 + r01).[ by A44, XXREAL_1: 4;

            

             A48: (s8 - (r01 / 2)) > (s8 - r01) by A43, XREAL_1: 10;

            ( Ball (s8m,r01)) c= ( Ball (s8m,r0)) by PCOMPS_1: 1, XXREAL_0: 17;

            then

             A49: ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) c= (h " B) by A40, A41;

            reconsider s70 = (s8 - (r01 / 2)) as Real;

            (s8 - s2) >= r01 by XXREAL_0: 17;

            then ( - (s8 - s2)) <= ( - r01) by XREAL_1: 24;

            then

             A50: ((s2 - s8) + s8) <= (( - r01) + s8) by XREAL_1: 7;

            ( - ( - (r01 / 2))) > 0 by A42, XREAL_1: 139;

            then ( - (r01 / 2)) < 0 ;

            then

             A51: (s8 + 0 ) > (s8 + ( - (r01 / 2))) by XREAL_1: 8;

            then

             A52: 1 >= s70 by A32, XXREAL_0: 2;

            (1 - 0 ) > (s8 - (r01 / 2)) by A32, A45, XREAL_1: 15;

            then (s8 - (r01 / 2)) in [. 0 , 1.] by A22, A50, A48, XXREAL_1: 1;

            then

             A53: (s8 - (r01 / 2)) in ( ].(s8 - r01), (s8 + r01).[ /\ [. 0 , 1.]) by A47, XBOOLE_0:def 4;

            then

             A54: (h . (s8 - (r01 / 2))) in B by A49, FUNCT_1:def 7;

            

             A55: (s8 - (r01 / 2)) in ( dom h) by A49, A53, FUNCT_1:def 7;

            

             A56: for p7 be Point of ( TOP-REAL 2) st p7 = (f . s70) holds (p7 `1 ) <> e

            proof

              let p7 be Point of ( TOP-REAL 2);

              assume

               A57: p7 = (f . s70);

              s70 in [. 0 , 1.] by A22, A50, A44, A52, XXREAL_1: 1;

              then

               A58: p7 in ( rng f) by A12, A57, BORSUK_1: 40, FUNCT_1:def 3;

              

               A59: ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A5, TOPS_2:def 5

              .= P by PRE_TOPC:def 5;

              

               A60: (h . s70) = (pro1 . (f . s70)) by A55, FUNCT_1: 12

              .= (pr1a . p7) by A57, A58, A59, FUNCT_1: 49

              .= (p7 `1 ) by PSCOMP_1:def 5;

              then

               A61: (p7 `1 ) < ((p8 `1 ) + ee) by A35, A54, XXREAL_1: 4;

              

               A62: ((p8 `1 ) - ee) < (p7 `1 ) by A35, A54, A60, XXREAL_1: 4;

              now

                assume

                 A63: (p7 `1 ) = e;

                now

                  per cases ;

                    case

                     A64: ((p8 `1 ) - e) >= 0 ;

                    then ((p8 `1 ) - (((p8 `1 ) - e) / 2)) < e by A62, A63, ABSVALUE:def 1;

                    then (((p8 `1 ) / 2) + (e / 2)) < ((e / 2) + (e / 2));

                    then ((p8 `1 ) / 2) < (e / 2) by XREAL_1: 7;

                    then

                     A65: (((p8 `1 ) / 2) - (e / 2)) < ((e / 2) - (e / 2)) by XREAL_1: 14;

                    (((p8 `1 ) - e) / 2) >= ( 0 / 2) by A64;

                    hence contradiction by A65;

                  end;

                    case

                     A66: ((p8 `1 ) - e) < 0 ;

                    then e < ((p8 `1 ) + (( - ((p8 `1 ) - e)) / 2)) by A61, A63, ABSVALUE:def 1;

                    then (((p8 `1 ) / 2) + (e / 2)) > ((e / 2) + (e / 2));

                    then ((p8 `1 ) / 2) > (e / 2) by XREAL_1: 7;

                    then

                     A67: (((p8 `1 ) / 2) - (e / 2)) > ((e / 2) - (e / 2)) by XREAL_1: 14;

                    (((p8 `1 ) - e) / 2) <= ( 0 / 2) by A66;

                    hence contradiction by A67;

                  end;

                end;

                hence contradiction;

              end;

              hence thesis;

            end;

            s70 >= s2 by A50, A44, XXREAL_0: 2;

            then

            consider s7 be Real such that

             A68: s8 > s7 and

             A69: 1 >= s7 & s7 >= s2 & for p7 be Point of ( TOP-REAL 2) st p7 = (f . s7) holds (p7 `1 ) <> e by A51, A52, A56;

            s7 in R by A69;

            then s7 >= s0 by A21, SEQ_4:def 2;

            hence contradiction by A34, A68, XXREAL_0: 2;

          end;

          hence thesis;

        end;

        assume not p is_OSout (P,p1,p2,e);

        then

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

         A70: LE (p9,p4,P,p1,p2) and

         A71: p4 <> p9 and

         A72: (for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (p9,p5,P,p1,p2) holds (p5 `1 ) <= e) or for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p9,p6,P,p1,p2) holds (p6 `1 ) >= e by A1, A3, A4, A24, A26;

        

         A73: p9 in P by A70, JORDAN5C:def 3;

        now

          per cases by A72;

            case

             A74: for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (p9,p5,P,p1,p2) holds (p5 `1 ) <= e;

             A75:

            now

              p4 in P by A70, JORDAN5C:def 3;

              then p4 in ( rng f) by A9, PRE_TOPC:def 5;

              then

              consider xs4 be object such that

               A76: xs4 in ( dom f) and

               A77: p4 = (f . xs4) by FUNCT_1:def 3;

              reconsider s4 = xs4 as Real by A76;

              

               A78: s4 <= 1 by A76, BORSUK_1: 40, XXREAL_1: 1;

              assume

               A79: not ex p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) & (p51 `1 ) < e;

              

               A80: for p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) holds (p51 `1 ) = e

              proof

                let p51 be Point of ( TOP-REAL 2);

                assume LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2);

                then (p51 `1 ) >= e & (p51 `1 ) <= e by A74, A79;

                hence thesis by XXREAL_0: 1;

              end;

               A81:

              now

                assume s4 > s0;

                then ( - ( - (s4 - s0))) > 0 by XREAL_1: 50;

                then ( - (s4 - s0)) < 0 ;

                then

                 A82: ((s0 - s4) / 2) < 0 by XREAL_1: 141;

                then ( - ((s0 - s4) / 2)) > 0 by XREAL_1: 58;

                then

                consider r be Real such that

                 A83: r in R and

                 A84: r < (s0 + ( - ((s0 - s4) / 2))) by A21, SEQ_4:def 2;

                reconsider rss = r as Real;

                

                 A85: ex s7 be Real st s7 = r & 1 >= s7 & s7 >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s7) holds (q `1 ) <> e by A83;

                then r in [. 0 , 1.] by A22, XXREAL_1: 1;

                then (f . rss) in ( rng f) by A12, BORSUK_1: 40, FUNCT_1:def 3;

                then (f . rss) in P by A9, PRE_TOPC:def 5;

                then

                reconsider pss = (f . rss) as Point of ( TOP-REAL 2);

                (s4 + 0 ) > (s4 + ((s0 - s4) / 2)) by A82, XREAL_1: 8;

                then

                 A86: s4 > r by A84, XXREAL_0: 2;

                then

                 A87: 1 > r by A78, XXREAL_0: 2;

                

                 A88: r >= s0 by A21, A83, SEQ_4:def 2;

                then

                 A89: LE (p9,pss,P,p1,p2) by A1, A5, A6, A7, A22, A23, A18, A87, JORDAN5C: 8;

                 LE (pss,p4,P,p1,p2) by A1, A5, A6, A7, A22, A18, A77, A78, A88, A86, A87, JORDAN5C: 8;

                then (pss `1 ) = e by A80, A89;

                hence contradiction by A85;

              end;

               0 <= s4 by A76, BORSUK_1: 40, XXREAL_1: 1;

              then s4 >= s0 by A5, A6, A7, A23, A70, A77, A78, JORDAN5C:def 3;

              hence contradiction by A71, A77, A81, XXREAL_0: 1;

            end;

            now

              assume ex p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) & (p51 `1 ) < e;

              then

              consider p51 be Point of ( TOP-REAL 2) such that

               A90: LE (p51,p4,P,p1,p2) and

               A91: LE (p9,p51,P,p1,p2) and

               A92: (p51 `1 ) < e;

              

               A93: for p5 be Point of ( TOP-REAL 2) st LE (p5,p51,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A94: LE (p5,p51,P,p1,p2) and

                 A95: LE (p,p5,P,p1,p2);

                

                 A96: LE (p5,p4,P,p1,p2) by A90, A94, JORDAN5C: 13;

                

                 A97: p5 in P by A94, JORDAN5C:def 3;

                then

                 A98: p5 = p9 implies LE (p9,p5,P,p1,p2) by JORDAN5C: 9;

                now

                  per cases by A1, A73, A97, A98, JORDAN5C: 14;

                    case LE (p5,p9,P,p1,p2);

                    hence thesis by A26, A95;

                  end;

                    case LE (p9,p5,P,p1,p2);

                    hence thesis by A74, A96;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p51,P,p1,p2) by A24, A91, JORDAN5C: 13;

              hence p is_Lout (P,p1,p2,e) by A1, A3, A4, A92, A93;

            end;

            hence p is_Lout (P,p1,p2,e) or p is_Rout (P,p1,p2,e) by A75;

          end;

            case

             A99: for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p9,p6,P,p1,p2) holds (p6 `1 ) >= e;

             A100:

            now

              p4 in P by A70, JORDAN5C:def 3;

              then p4 in ( rng f) by A9, PRE_TOPC:def 5;

              then

              consider xs4 be object such that

               A101: xs4 in ( dom f) and

               A102: p4 = (f . xs4) by FUNCT_1:def 3;

              reconsider s4 = xs4 as Real by A101;

              

               A103: s4 <= 1 by A101, BORSUK_1: 40, XXREAL_1: 1;

              assume

               A104: not ex p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) & (p51 `1 ) > e;

              

               A105: for p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) holds (p51 `1 ) = e

              proof

                let p51 be Point of ( TOP-REAL 2);

                assume LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2);

                then (p51 `1 ) <= e & (p51 `1 ) >= e by A99, A104;

                hence thesis by XXREAL_0: 1;

              end;

               A106:

              now

                assume s4 > s0;

                then ( - ( - (s4 - s0))) > 0 by XREAL_1: 50;

                then ( - (s4 - s0)) < 0 ;

                then

                 A107: ((s0 - s4) / 2) < 0 by XREAL_1: 141;

                then ( - ((s0 - s4) / 2)) > 0 by XREAL_1: 58;

                then

                consider r be Real such that

                 A108: r in R and

                 A109: r < (s0 + ( - ((s0 - s4) / 2))) by A21, SEQ_4:def 2;

                reconsider rss = r as Real;

                

                 A110: ex s7 be Real st s7 = r & 1 >= s7 & s7 >= s2 & for q be Point of ( TOP-REAL 2) st q = (f . s7) holds (q `1 ) <> e by A108;

                then r in [. 0 , 1.] by A22, XXREAL_1: 1;

                then (f . rss) in ( rng f) by A12, BORSUK_1: 40, FUNCT_1:def 3;

                then (f . rss) in P by A9, PRE_TOPC:def 5;

                then

                reconsider pss = (f . rss) as Point of ( TOP-REAL 2);

                (s4 + 0 ) > (s4 + ((s0 - s4) / 2)) by A107, XREAL_1: 8;

                then

                 A111: s4 > r by A109, XXREAL_0: 2;

                then

                 A112: 1 > r by A103, XXREAL_0: 2;

                

                 A113: r >= s0 by A21, A108, SEQ_4:def 2;

                then

                 A114: LE (p9,pss,P,p1,p2) by A1, A5, A6, A7, A22, A23, A18, A112, JORDAN5C: 8;

                 LE (pss,p4,P,p1,p2) by A1, A5, A6, A7, A22, A18, A102, A103, A113, A111, A112, JORDAN5C: 8;

                then (pss `1 ) = e by A105, A114;

                hence contradiction by A110;

              end;

               0 <= s4 by A101, BORSUK_1: 40, XXREAL_1: 1;

              then s4 >= s0 by A5, A6, A7, A23, A70, A102, A103, JORDAN5C:def 3;

              hence contradiction by A71, A102, A106, XXREAL_0: 1;

            end;

            now

              assume ex p51 be Point of ( TOP-REAL 2) st LE (p51,p4,P,p1,p2) & LE (p9,p51,P,p1,p2) & (p51 `1 ) > e;

              then

              consider p51 be Point of ( TOP-REAL 2) such that

               A115: LE (p51,p4,P,p1,p2) and

               A116: LE (p9,p51,P,p1,p2) and

               A117: (p51 `1 ) > e;

              

               A118: for p5 be Point of ( TOP-REAL 2) st LE (p5,p51,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A119: LE (p5,p51,P,p1,p2) and

                 A120: LE (p,p5,P,p1,p2);

                

                 A121: LE (p5,p4,P,p1,p2) by A115, A119, JORDAN5C: 13;

                

                 A122: p5 in P by A119, JORDAN5C:def 3;

                then

                 A123: p5 = p9 implies LE (p9,p5,P,p1,p2) by JORDAN5C: 9;

                now

                  per cases by A1, A73, A122, A123, JORDAN5C: 14;

                    case LE (p9,p5,P,p1,p2);

                    hence thesis by A99, A121;

                  end;

                    case LE (p5,p9,P,p1,p2);

                    hence thesis by A26, A120;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p51,P,p1,p2) by A24, A116, JORDAN5C: 13;

              hence p is_Rout (P,p1,p2,e) by A1, A3, A4, A117, A118;

            end;

            hence p is_Lout (P,p1,p2,e) or p is_Rout (P,p1,p2,e) by A100;

          end;

        end;

        hence p is_Lout (P,p1,p2,e) or p is_Rout (P,p1,p2,e);

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:11

    

     Th11: for P be Subset of I[01] , s be Real st P = [. 0 , s.[ holds P is open

    proof

      

       A1: ( [#] I[01] ) = [. 0 , 1.] by TOPMETR: 18, TOPMETR: 20;

      let P be Subset of I[01] , s be Real;

      assume

       A2: P = [. 0 , s.[;

      per cases ;

        suppose

         A3: s in [. 0 , 1.];

        reconsider T = [. 0 , 1.] as Subset of R^1 by TOPMETR: 17;

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

        then [. 0 , s.[ c= [. 0 , s.] & [. 0 , s.] c= [. 0 , 1.] by A3, XXREAL_1: 24, XXREAL_2:def 12;

        then [. 0 , s.[ c= [. 0 , 1.];

        then P c= ( [#] ( R^1 | T)) by A2, PRE_TOPC:def 5;

        then

        reconsider P2 = P as Subset of ( R^1 | T);

        reconsider Q = ].( - 1), s.[ as Subset of R^1 by TOPMETR: 17;

        

         A4: s <= 1 by A3, XXREAL_1: 1;

        

         A5: [. 0 , s.[ c= ( ].( - 1), s.[ /\ [. 0 , 1.])

        proof

          let x be object;

          assume

           A6: x in [. 0 , s.[;

          then

          reconsider sx = x as Real;

          

           A7: 0 <= sx by A6, XXREAL_1: 3;

          

           A8: sx < s by A6, XXREAL_1: 3;

          then sx <= 1 by A4, XXREAL_0: 2;

          then

           A9: x in [. 0 , 1.] by A7, XXREAL_1: 1;

          ( - 1) < sx by A7, XXREAL_0: 2;

          then x in ].( - 1), s.[ by A8, XXREAL_1: 4;

          hence thesis by A9, XBOOLE_0:def 4;

        end;

        ( ].( - 1), s.[ /\ [. 0 , 1.]) c= [. 0 , s.[

        proof

          let x be object;

          assume

           A10: x in ( ].( - 1), s.[ /\ [. 0 , 1.]);

          then

          reconsider sx = x as Real;

          x in [. 0 , 1.] by A10, XBOOLE_0:def 4;

          then

           A11: 0 <= sx by XXREAL_1: 1;

          x in ].( - 1), s.[ by A10, XBOOLE_0:def 4;

          then sx < s by XXREAL_1: 4;

          hence thesis by A11, XXREAL_1: 3;

        end;

        then [. 0 , s.[ = ( ].( - 1), s.[ /\ [. 0 , 1.]) by A5, XBOOLE_0:def 10;

        then

         A12: P2 = (Q /\ ( [#] ( R^1 | T))) by A2, PRE_TOPC:def 5;

        Q is open & ( Closed-Interval-TSpace ( 0 ,1)) = ( R^1 | T) by JORDAN6: 35, TOPMETR: 19;

        hence thesis by A12, TOPMETR: 20, TOPS_2: 24;

      end;

        suppose

         A13: not s in [. 0 , 1.];

        now

          per cases by A13, XXREAL_1: 1;

            case s < 0 ;

            then [. 0 , s.[ = {} by XXREAL_1: 27;

            then P in the topology of I[01] by A2, PRE_TOPC: 1;

            hence thesis by PRE_TOPC:def 2;

          end;

            case

             A14: s > 1;

            

             A15: for r be Real st 0 <= r & r < s holds r <= 1

            proof

              let r be Real;

              assume 0 <= r & r < s;

              then r in [. 0 , s.[ by XXREAL_1: 3;

              hence thesis by A2, A1, XXREAL_1: 1;

            end;

            consider t be Real such that

             A16: 1 < t and

             A17: t < s by A14, XREAL_1: 5;

            reconsider t as Real;

            thus contradiction by A16, A17, A15;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN20:12

    

     Th12: for P be Subset of I[01] , s be Real st P = ].s, 1.] holds P is open

    proof

      

       A1: ( [#] I[01] ) = [. 0 , 1.] by TOPMETR: 18, TOPMETR: 20;

      let P be Subset of I[01] , s be Real;

      assume

       A2: P = ].s, 1.];

      per cases ;

        suppose

         A3: s in [. 0 , 1.];

        reconsider T = [. 0 , 1.] as Subset of R^1 by TOPMETR: 17;

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

        then ].s, 1.] c= [.s, 1.] & [.s, 1.] c= [. 0 , 1.] by A3, XXREAL_1: 23, XXREAL_2:def 12;

        then ].s, 1.] c= [. 0 , 1.];

        then P c= ( [#] ( R^1 | T)) by A2, PRE_TOPC:def 5;

        then

        reconsider P2 = P as Subset of ( R^1 | T);

        reconsider Q = ].s, 2.[ as Subset of R^1 by TOPMETR: 17;

        

         A4: 0 <= s by A3, XXREAL_1: 1;

        

         A5: ].s, 1.] c= ( ].s, 2.[ /\ [. 0 , 1.])

        proof

          let x be object;

          assume

           A6: x in ].s, 1.];

          then

          reconsider sx = x as Real;

          

           A7: s < sx by A6, XXREAL_1: 2;

          

           A8: sx <= 1 by A6, XXREAL_1: 2;

          then 2 > sx by XXREAL_0: 2;

          then

           A9: x in ].s, 2.[ by A7, XXREAL_1: 4;

          x in [. 0 , 1.] by A4, A7, A8, XXREAL_1: 1;

          hence thesis by A9, XBOOLE_0:def 4;

        end;

        ( ].s, 2.[ /\ [. 0 , 1.]) c= ].s, 1.]

        proof

          let x be object;

          assume

           A10: x in ( ].s, 2.[ /\ [. 0 , 1.]);

          then

          reconsider sx = x as Real;

          x in [. 0 , 1.] by A10, XBOOLE_0:def 4;

          then

           A11: sx <= 1 by XXREAL_1: 1;

          x in ].s, 2.[ by A10, XBOOLE_0:def 4;

          then s < sx by XXREAL_1: 4;

          hence thesis by A11, XXREAL_1: 2;

        end;

        then ].s, 1.] = ( ].s, 2.[ /\ [. 0 , 1.]) by A5, XBOOLE_0:def 10;

        then

         A12: P2 = (Q /\ ( [#] ( R^1 | T))) by A2, PRE_TOPC:def 5;

        Q is open & ( Closed-Interval-TSpace ( 0 ,1)) = ( R^1 | T) by JORDAN6: 35, TOPMETR: 19;

        hence thesis by A12, TOPMETR: 20, TOPS_2: 24;

      end;

        suppose

         A13: not s in [. 0 , 1.];

        now

          per cases by A13, XXREAL_1: 1;

            case s > 1;

            then ].s, 1.] = {} by XXREAL_1: 26;

            then P in the topology of I[01] by A2, PRE_TOPC: 1;

            hence thesis by PRE_TOPC:def 2;

          end;

            case

             A14: s < 0 ;

            

             A15: for r be Real st s < r & r <= 1 holds r >= 0

            proof

              let r be Real;

              assume s < r & r <= 1;

              then r in ].s, 1.] by XXREAL_1: 2;

              hence thesis by A2, A1, XXREAL_1: 1;

            end;

            consider t be Real such that

             A16: s < t and

             A17: t < 0 by A14, XREAL_1: 5;

            reconsider t as Real;

            thus contradiction by A16, A17, A15;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN20:13

    

     Th13: for P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), Q be Subset of I[01] , f be Function of I[01] , (( TOP-REAL 2) | P), s be Real st s <= 1 & P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } & Q = [. 0 , s.[ holds (f .: Q) = P1

    proof

      let P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), Q be Subset of I[01] , f be Function of I[01] , (( TOP-REAL 2) | P), s be Real;

      assume that

       A1: s <= 1 and

       A2: P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } and

       A3: Q = [. 0 , s.[;

      

       A4: the carrier of (( TOP-REAL 2) | P) = P by PRE_TOPC: 8;

      

       A5: (f .: Q) c= P1

      proof

        let y be object;

        assume y in (f .: Q);

        then

        consider z be object such that

         A6: z in ( dom f) and

         A7: z in Q and

         A8: (f . z) = y by FUNCT_1:def 6;

        reconsider ss = z as Real by A6;

        y in ( rng f) by A6, A8, FUNCT_1:def 3;

        then y in P by A4;

        then

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

         0 <= ss & ss < s by A3, A7, XXREAL_1: 3;

        then ex ss be Real st 0 <= ss & ss < s & q = (f . ss) by A8;

        hence thesis by A2;

      end;

      P1 c= (f .: Q)

      proof

        let x be object;

        

         A9: ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

        assume x in P1;

        then

        consider q0 be Point of ( TOP-REAL 2) such that

         A10: q0 = x and

         A11: ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) by A2;

        consider ss be Real such that

         A12: 0 <= ss and

         A13: ss < s and

         A14: q0 = (f . ss) by A11;

        ss < 1 by A1, A13, XXREAL_0: 2;

        then

         A15: ss in ( dom f) by A12, A9, XXREAL_1: 1;

        ss in Q by A3, A12, A13, XXREAL_1: 3;

        hence thesis by A10, A14, A15, FUNCT_1:def 6;

      end;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    theorem :: JORDAN20:14

    

     Th14: for P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), Q be Subset of I[01] , f be Function of I[01] , (( TOP-REAL 2) | P), s be Real st s >= 0 & P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } & Q = ].s, 1.] holds (f .: Q) = P1

    proof

      let P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), Q be Subset of I[01] , f be Function of I[01] , (( TOP-REAL 2) | P), s be Real;

      assume that

       A1: s >= 0 and

       A2: P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } and

       A3: Q = ].s, 1.];

      

       A4: the carrier of (( TOP-REAL 2) | P) = P by PRE_TOPC: 8;

      

       A5: (f .: Q) c= P1

      proof

        let y be object;

        assume y in (f .: Q);

        then

        consider z be object such that

         A6: z in ( dom f) and

         A7: z in Q and

         A8: (f . z) = y by FUNCT_1:def 6;

        reconsider ss = z as Real by A6;

        y in ( rng f) by A6, A8, FUNCT_1:def 3;

        then y in P by A4;

        then

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

        s < ss & ss <= 1 by A3, A7, XXREAL_1: 2;

        then ex ss be Real st s < ss & ss <= 1 & q = (f . ss) by A8;

        hence thesis by A2;

      end;

      P1 c= (f .: Q)

      proof

        let x be object;

        assume x in P1;

        then

        consider q0 be Point of ( TOP-REAL 2) such that

         A9: q0 = x and

         A10: ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) by A2;

        consider ss be Real such that

         A11: s < ss & ss <= 1 and

         A12: q0 = (f . ss) by A10;

        

         A13: ss in Q by A3, A11, XXREAL_1: 2;

        ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

        then ss in ( dom f) by A1, A11, XXREAL_1: 1;

        hence thesis by A9, A12, A13, FUNCT_1:def 6;

      end;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    

     Lm1: ( [#] I[01] ) <> {} ;

    theorem :: JORDAN20:15

    

     Th15: for P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), f be Function of I[01] , (( TOP-REAL 2) | P), s be Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } holds P1 is open

    proof

      let P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), f be Function of I[01] , (( TOP-REAL 2) | P), s be Real;

      assume that

       A1: s <= 1 and

       A2: f is being_homeomorphism and

       A3: P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) };

      f is one-to-one & ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A2, TOPS_2:def 5;

      then

       A4: ((f " ) " ) = f by TOPS_2: 51;

       [. 0 , s.[ c= [. 0 , 1.]

      proof

        let x be object;

        assume

         A5: x in [. 0 , s.[;

        then

        reconsider sx = x as Real;

        sx < s by A5, XXREAL_1: 3;

        then

         A6: sx < 1 by A1, XXREAL_0: 2;

         0 <= sx by A5, XXREAL_1: 3;

        hence thesis by A6, XXREAL_1: 1;

      end;

      then

      reconsider Q = [. 0 , s.[ as Subset of I[01] by TOPMETR: 18, TOPMETR: 20;

      

       A7: Q is open by Th11;

      

       A8: (f " ) is being_homeomorphism by A2, TOPS_2: 56;

      then

       A9: (f " ) is one-to-one by TOPS_2:def 5;

      ( rng (f " )) = ( [#] I[01] ) by A8, TOPS_2:def 5;

      then (f " ) is onto by FUNCT_2:def 3;

      then ((f " ) " ) = ((f " ) qua Function " ) by A9, TOPS_2:def 4;

      then

       A10: (((f " ) " ) .: Q) = ((f " ) " Q) by A9, FUNCT_1: 85;

      

       A11: P1 = (f .: Q) by A1, A3, Th13;

      (f " ) is continuous by A2, TOPS_2:def 5;

      hence thesis by A7, A4, A10, Lm1, A11, TOPS_2: 43;

    end;

    theorem :: JORDAN20:16

    

     Th16: for P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), f be Function of I[01] , (( TOP-REAL 2) | P), s be Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } holds P1 is open

    proof

      let P be non empty Subset of ( TOP-REAL 2), P1 be Subset of (( TOP-REAL 2) | P), f be Function of I[01] , (( TOP-REAL 2) | P), s be Real;

      assume that

       A1: s >= 0 and

       A2: f is being_homeomorphism and

       A3: P1 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) };

      f is one-to-one & ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A2, TOPS_2:def 5;

      then

       A4: ((f " ) " ) = f by TOPS_2: 51;

       ].s, 1.] c= [. 0 , 1.]

      proof

        let x be object;

        assume

         A5: x in ].s, 1.];

        then

        reconsider sx = x as Real;

         0 < sx & sx <= 1 by A1, A5, XXREAL_1: 2;

        hence thesis by XXREAL_1: 1;

      end;

      then

      reconsider Q = ].s, 1.] as Subset of I[01] by TOPMETR: 18, TOPMETR: 20;

      

       A6: ( [#] I[01] ) <> {} & Q is open by Th12;

      

       A7: (f " ) is being_homeomorphism by A2, TOPS_2: 56;

      then

       A8: (f " ) is one-to-one by TOPS_2:def 5;

      ( rng (f " )) = ( [#] I[01] ) by A7, TOPS_2:def 5;

      then (f " ) is onto by FUNCT_2:def 3;

      then ((f " ) " ) = ((f " ) qua Function " ) by A8, TOPS_2:def 4;

      then

       A9: (((f " ) " ) .: Q) = ((f " ) " Q) by A8, FUNCT_1: 85;

      P1 = (f .: Q) & (f " ) is continuous by A1, A2, A3, Th14, TOPS_2:def 5;

      hence thesis by A6, A4, A9, TOPS_2: 43;

    end;

    theorem :: JORDAN20:17

    

     Th17: for T be non empty TopStruct, Q1,Q2 be Subset of T, p1,p2 be Point of T st (Q1 /\ Q2) = {} & (Q1 \/ Q2) = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds not ex P be Function of I[01] , T st P is continuous & (P . 0 ) = p1 & (P . 1) = p2

    proof

      let T be non empty TopStruct, Q1,Q2 be Subset of T, p1,p2 be Point of T;

      assume that

       A1: (Q1 /\ Q2) = {} and

       A2: (Q1 \/ Q2) = the carrier of T and

       A3: p1 in Q1 and

       A4: p2 in Q2 and

       A5: Q1 is open & Q2 is open;

      assume ex P be Function of I[01] , T st P is continuous & (P . 0 ) = p1 & (P . 1) = p2;

      then

      consider P be Function of I[01] , T such that

       A6: P is continuous and

       A7: (P . 0 ) = p1 and

       A8: (P . 1) = p2;

      ( [#] T) <> {} ;

      then

       A9: (P " Q1) is open & (P " Q2) is open by A5, A6, TOPS_2: 43;

      

       A10: ( [#] I[01] ) = [. 0 , 1.] by TOPMETR: 18, TOPMETR: 20;

      then 0 in the carrier of I[01] by XXREAL_1: 1;

      then 0 in ( dom P) by FUNCT_2:def 1;

      then

       A11: ( [#] I[01] ) = the carrier of I[01] & (P " Q1) <> ( {} I[01] ) by A3, A7, FUNCT_1:def 7;

      ((P " Q1) /\ (P " Q2)) = (P " (Q1 /\ Q2)) by FUNCT_1: 68

      .= {} by A1;

      then

       A12: not (P " Q1) meets (P " Q2) by XBOOLE_0:def 7;

      1 in the carrier of I[01] by A10, XXREAL_1: 1;

      then 1 in ( dom P) by FUNCT_2:def 1;

      then

       A13: (P " Q2) <> ( {} I[01] ) by A4, A8, FUNCT_1:def 7;

      ((P " Q1) \/ (P " Q2)) = (P " (Q1 \/ Q2)) by RELAT_1: 140

      .= the carrier of I[01] by A2, FUNCT_2: 40;

      hence contradiction by A9, A11, A13, A12, CONNSP_1: 11, TREAL_1: 19;

    end;

    theorem :: JORDAN20:18

    

     Th18: for P be non empty Subset of ( TOP-REAL 2), Q be Subset of (( TOP-REAL 2) | P), p1,p2,q be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & q in P & q <> p1 & q <> p2 & Q = (P \ {q}) holds not Q is connected & not ex R be Function of I[01] , ((( TOP-REAL 2) | P) | Q) st R is continuous & (R . 0 ) = p1 & (R . 1) = p2

    proof

      let P be non empty Subset of ( TOP-REAL 2), Q be Subset of (( TOP-REAL 2) | P), p1,p2,q be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: q in P and

       A3: q <> p1 and

       A4: q <> p2 and

       A5: Q = (P \ {q});

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

       A6: f is being_homeomorphism and

       A7: (f . 0 ) = p1 and

       A8: (f . 1) = p2 by A1, TOPREAL1:def 1;

      

       A9: ( rng f) = ( [#] (( TOP-REAL 2) | P)) by A6, TOPS_2:def 5;

      

       A10: ( [#] I[01] ) = [. 0 , 1.] by TOPMETR: 18, TOPMETR: 20;

      

       A11: ( [#] (( TOP-REAL 2) | P)) = P by PRE_TOPC:def 5;

      then

      consider xs be object such that

       A12: xs in ( dom f) and

       A13: (f . xs) = q by A2, A9, FUNCT_1:def 3;

      

       A14: ( dom f) = ( [#] I[01] ) by A6, TOPS_2:def 5;

      reconsider s = xs as Real by A12;

      { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } c= the carrier of (( TOP-REAL 2) | P)

      proof

        let z be object;

        assume z in { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) };

        then

        consider q0 be Point of ( TOP-REAL 2) such that

         A15: q0 = z and

         A16: ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss);

        consider ss be Real such that

         A17: s < ss and

         A18: ss <= 1 and

         A19: q0 = (f . ss) by A16;

        ss > 0 by A12, A10, A17, XXREAL_1: 1;

        then ss in ( dom f) by A14, A10, A18, XXREAL_1: 1;

        then q0 in ( rng f) by A19, FUNCT_1:def 3;

        hence thesis by A15;

      end;

      then

      reconsider P29 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } as Subset of (( TOP-REAL 2) | P);

      

       A20: 0 <= s by A12, A10, XXREAL_1: 1;

      then

       A21: P29 is open by A6, Th16;

      

       A22: P29 c= Q

      proof

        let x be object;

        assume x in P29;

        then

        consider q00 be Point of ( TOP-REAL 2) such that

         A23: q00 = x and

         A24: ex ss be Real st s < ss & ss <= 1 & q00 = (f . ss);

        consider ss be Real such that

         A25: s < ss and

         A26: ss <= 1 and

         A27: q00 = (f . ss) by A24;

        ss > 0 by A12, A10, A25, XXREAL_1: 1;

        then

         A28: ss in ( dom f) by A14, A10, A26, XXREAL_1: 1;

        now

          assume

           A29: q00 = q;

          f is one-to-one by A6, TOPS_2:def 5;

          hence contradiction by A12, A13, A25, A27, A28, A29, FUNCT_1:def 4;

        end;

        then

         A30: not q00 in {q} by TARSKI:def 1;

        q00 in P by A9, A11, A27, A28, FUNCT_1:def 3;

        hence thesis by A5, A23, A30, XBOOLE_0:def 5;

      end;

      { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } c= the carrier of (( TOP-REAL 2) | P)

      proof

        let z be object;

        assume z in { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) };

        then

        consider q0 be Point of ( TOP-REAL 2) such that

         A31: q0 = z and

         A32: ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss);

        consider ss be Real such that

         A33: 0 <= ss and

         A34: ss < s and

         A35: q0 = (f . ss) by A32;

        s <= 1 by A12, A10, XXREAL_1: 1;

        then ss < 1 by A34, XXREAL_0: 2;

        then ss in ( dom f) by A14, A10, A33, XXREAL_1: 1;

        then q0 in ( rng f) by A35, FUNCT_1:def 3;

        hence thesis by A31;

      end;

      then

      reconsider P19 = { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } as Subset of (( TOP-REAL 2) | P);

      

       A36: s <= 1 by A12, A10, XXREAL_1: 1;

      then

       A37: P19 is open by A6, Th15;

      

       A38: Q c= (P19 \/ P29)

      proof

        let x be object;

        assume

         A39: x in Q;

        then

        consider xt be object such that

         A40: xt in ( dom f) and

         A41: (f . xt) = x by A9, FUNCT_1:def 3;

        reconsider t = xt as Real by A40;

        

         A42: t <= 1 by A10, A40, XXREAL_1: 1;

        reconsider qq = x as Point of ( TOP-REAL 2) by A5, A39;

         not x in {q} by A5, A39, XBOOLE_0:def 5;

        then

         A43: not x = q by TARSKI:def 1;

        

         A44: 0 <= t by A10, A40, XXREAL_1: 1;

        now

          per cases ;

            case t < s;

            then ex ss be Real st 0 <= ss & ss < s & qq = (f . ss) by A41, A44;

            then x in P19;

            hence thesis by XBOOLE_0:def 3;

          end;

            case t >= s;

            then t > s by A13, A41, A43, XXREAL_0: 1;

            then ex ss be Real st s < ss & ss <= 1 & qq = (f . ss) by A41, A42;

            then x in P29;

            hence thesis by XBOOLE_0:def 3;

          end;

        end;

        hence thesis;

      end;

       A45:

      now

        assume P19 meets P29;

        then

        consider p0 be object such that

         A46: p0 in P19 and

         A47: p0 in P29 by XBOOLE_0: 3;

        consider q00 be Point of ( TOP-REAL 2) such that

         A48: q00 = p0 and

         A49: ex ss be Real st 0 <= ss & ss < s & q00 = (f . ss) by A46;

        consider ss1 be Real such that

         A50: 0 <= ss1 and

         A51: ss1 < s and

         A52: q00 = (f . ss1) by A49;

        ss1 < 1 by A36, A51, XXREAL_0: 2;

        then

         A53: ss1 in ( dom f) by A14, A10, A50, XXREAL_1: 1;

        consider q01 be Point of ( TOP-REAL 2) such that

         A54: q01 = p0 and

         A55: ex ss be Real st s < ss & ss <= 1 & q01 = (f . ss) by A47;

        consider ss2 be Real such that

         A56: s < ss2 and

         A57: ss2 <= 1 and

         A58: q01 = (f . ss2) by A55;

        ss2 > 0 by A12, A10, A56, XXREAL_1: 1;

        then

         A59: ss2 in ( dom f) by A14, A10, A57, XXREAL_1: 1;

        f is one-to-one by A6, TOPS_2:def 5;

        hence contradiction by A48, A51, A52, A54, A56, A58, A53, A59, FUNCT_1:def 4;

      end;

      1 > s by A4, A8, A13, A36, XXREAL_0: 1;

      then

       A60: p2 in { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st s < ss & ss <= 1 & q0 = (f . ss) } by A8;

      then

      reconsider Q9 = Q as non empty Subset of (( TOP-REAL 2) | P) by A22;

      reconsider T = ((( TOP-REAL 2) | P) | Q9) as non empty TopSpace;

      

       A61: the carrier of T = ( [#] T);

      then

      reconsider P299 = P29 as Subset of T by A22, PRE_TOPC:def 5;

      (P29 /\ Q) <> {} by A60, A22, XBOOLE_1: 28;

      then

       A62: P29 meets Q by XBOOLE_0:def 7;

      

       A63: P19 c= Q

      proof

        let x be object;

        assume x in P19;

        then

        consider q00 be Point of ( TOP-REAL 2) such that

         A64: q00 = x and

         A65: ex ss be Real st 0 <= ss & ss < s & q00 = (f . ss);

        consider ss be Real such that

         A66: 0 <= ss and

         A67: ss < s and

         A68: q00 = (f . ss) by A65;

        ss < 1 by A36, A67, XXREAL_0: 2;

        then

         A69: ss in ( dom f) by A14, A10, A66, XXREAL_1: 1;

        now

          assume

           A70: q00 = q;

          f is one-to-one by A6, TOPS_2:def 5;

          hence contradiction by A12, A13, A67, A68, A69, A70, FUNCT_1:def 4;

        end;

        then

         A71: not q00 in {q} by TARSKI:def 1;

        q00 in P by A9, A11, A68, A69, FUNCT_1:def 3;

        hence thesis by A5, A64, A71, XBOOLE_0:def 5;

      end;

      then

      reconsider P199 = P19 as Subset of T by A61, PRE_TOPC:def 5;

      P199 = (P19 /\ the carrier of T) by XBOOLE_1: 28;

      then

       A72: P199 is open by A37, A61, TOPS_2: 24;

      s <> 0 by A3, A7, A13;

      then

       A73: p1 in { q0 where q0 be Point of ( TOP-REAL 2) : ex ss be Real st 0 <= ss & ss < s & q0 = (f . ss) } by A7, A20;

      then (P19 /\ Q) <> {} by A63, XBOOLE_1: 28;

      then P19 meets Q by XBOOLE_0:def 7;

      hence not Q is connected by A37, A21, A38, A62, A45, TOPREAL5: 1;

      the carrier of T = Q by A61, PRE_TOPC:def 5;

      then

       A74: (P199 \/ P299) = the carrier of ((( TOP-REAL 2) | P) | Q) by A38, XBOOLE_0:def 10;

      P299 = (P29 /\ the carrier of T) by XBOOLE_1: 28;

      then

       A75: P299 is open by A21, A61, TOPS_2: 24;

      (P199 /\ P299) = {} by A45, XBOOLE_0:def 7;

      hence thesis by A73, A60, A72, A75, A74, Th17;

    end;

    theorem :: JORDAN20:19

    

     Th19: for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & q1 in P & q2 in P holds LE (q1,q2,P,p1,p2) or LE (q2,q1,P,p1,p2)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: q1 in P and

       A3: q2 in P;

      per cases ;

        suppose q1 <> q2;

        hence thesis by A1, A2, A3, JORDAN5C: 14;

      end;

        suppose q1 = q2;

        hence thesis by A2, JORDAN5C: 9;

      end;

    end;

    theorem :: JORDAN20:20

    

     Th20: for n be Nat, p1,p2 be Point of ( TOP-REAL n), P,P1 be non empty Subset of ( TOP-REAL n) st P is_an_arc_of (p1,p2) & P1 is_an_arc_of (p1,p2) & P1 c= P holds P1 = P

    proof

      let n be Nat, p1,p2 be Point of ( TOP-REAL n), P,P1 be non empty Subset of ( TOP-REAL n);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: P1 is_an_arc_of (p1,p2) and

       A3: P1 c= P;

      P1 is_an_arc_of (p2,p1) by A2, JORDAN5B: 14;

      hence thesis by A1, A3, TOPMETR3: 14;

    end;

    theorem :: JORDAN20:21

    

     Th21: for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & q1 in P & p2 <> q1 holds ( Segment (P,p1,p2,q1,p2)) is_an_arc_of (q1,p2)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1 be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: q1 in P and

       A3: p2 <> q1;

       LE (q1,p2,P,p1,p2) by A1, A2, JORDAN5C: 10;

      hence thesis by A1, A3, JORDAN16: 21;

    end;

    theorem :: JORDAN20:22

    

     Th22: for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,q3 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & LE (q1,q2,P,p1,p2) & LE (q2,q3,P,p1,p2) holds (( Segment (P,p1,p2,q1,q2)) \/ ( Segment (P,p1,p2,q2,q3))) = ( Segment (P,p1,p2,q1,q3))

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,q3 be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: LE (q1,q2,P,p1,p2) and

       A3: LE (q2,q3,P,p1,p2);

      

       A4: q2 in P by A2, JORDAN5C:def 3;

      

       A5: ( Segment (P,p1,p2,q1,q3)) c= (( Segment (P,p1,p2,q1,q2)) \/ ( Segment (P,p1,p2,q2,q3)))

      proof

        let x be object;

        assume x in ( Segment (P,p1,p2,q1,q3));

        then x in { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q3,P,p1,p2) } by JORDAN6: 26;

        then

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

         A6: x = p3 and

         A7: LE (q1,p3,P,p1,p2) and

         A8: LE (p3,q3,P,p1,p2);

        

         A9: p3 in P by A7, JORDAN5C:def 3;

        now

          per cases ;

            suppose

             A10: p3 = q2;

            then LE (p3,q2,P,p1,p2) by A4, JORDAN5C: 9;

            then x in { p31 where p31 be Point of ( TOP-REAL 2) : LE (q1,p31,P,p1,p2) & LE (p31,q2,P,p1,p2) } by A2, A6, A10;

            hence x in ( Segment (P,p1,p2,q1,q2)) or x in ( Segment (P,p1,p2,q2,q3)) by JORDAN6: 26;

          end;

            suppose

             A11: p3 <> q2;

            now

              per cases by A1, A4, A9, A11, JORDAN5C: 14;

                case LE (p3,q2,P,p1,p2) & not LE (q2,p3,P,p1,p2);

                then x in { p31 where p31 be Point of ( TOP-REAL 2) : LE (q1,p31,P,p1,p2) & LE (p31,q2,P,p1,p2) } by A6, A7;

                hence x in ( Segment (P,p1,p2,q1,q2)) or x in ( Segment (P,p1,p2,q2,q3)) by JORDAN6: 26;

              end;

                case LE (q2,p3,P,p1,p2) & not LE (p3,q2,P,p1,p2);

                then x in { p31 where p31 be Point of ( TOP-REAL 2) : LE (q2,p31,P,p1,p2) & LE (p31,q3,P,p1,p2) } by A6, A8;

                hence x in ( Segment (P,p1,p2,q1,q2)) or x in ( Segment (P,p1,p2,q2,q3)) by JORDAN6: 26;

              end;

            end;

            hence x in ( Segment (P,p1,p2,q1,q2)) or x in ( Segment (P,p1,p2,q2,q3));

          end;

        end;

        hence thesis by XBOOLE_0:def 3;

      end;

      (( Segment (P,p1,p2,q1,q2)) \/ ( Segment (P,p1,p2,q2,q3))) c= ( Segment (P,p1,p2,q1,q3))

      proof

        let x be object;

        assume

         A12: x in (( Segment (P,p1,p2,q1,q2)) \/ ( Segment (P,p1,p2,q2,q3)));

        per cases by A12, XBOOLE_0:def 3;

          suppose x in ( Segment (P,p1,p2,q1,q2));

          then x in { p where p be Point of ( TOP-REAL 2) : LE (q1,p,P,p1,p2) & LE (p,q2,P,p1,p2) } by JORDAN6: 26;

          then

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

           A13: x = p & LE (q1,p,P,p1,p2) and

           A14: LE (p,q2,P,p1,p2);

           LE (p,q3,P,p1,p2) by A3, A14, JORDAN5C: 13;

          then x in { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q3,P,p1,p2) } by A13;

          hence thesis by JORDAN6: 26;

        end;

          suppose x in ( Segment (P,p1,p2,q2,q3));

          then x in { p where p be Point of ( TOP-REAL 2) : LE (q2,p,P,p1,p2) & LE (p,q3,P,p1,p2) } by JORDAN6: 26;

          then

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

           A15: x = p and

           A16: LE (q2,p,P,p1,p2) and

           A17: LE (p,q3,P,p1,p2);

           LE (q1,p,P,p1,p2) by A2, A16, JORDAN5C: 13;

          then x in { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q3,P,p1,p2) } by A15, A17;

          hence thesis by JORDAN6: 26;

        end;

      end;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    theorem :: JORDAN20:23

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,q3 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & LE (q1,q2,P,p1,p2) & LE (q2,q3,P,p1,p2) holds (( Segment (P,p1,p2,q1,q2)) /\ ( Segment (P,p1,p2,q2,q3))) = {q2}

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,q3 be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: LE (q1,q2,P,p1,p2) and

       A3: LE (q2,q3,P,p1,p2);

      

       A4: q2 in P by A2, JORDAN5C:def 3;

      

       A5: {q2} c= (( Segment (P,p1,p2,q1,q2)) /\ ( Segment (P,p1,p2,q2,q3)))

      proof

        set p3 = q2;

        let x be object;

        assume x in {q2};

        then

         A6: x = q2 by TARSKI:def 1;

         LE (q2,p3,P,p1,p2) by A4, JORDAN5C: 9;

        then x in { p31 where p31 be Point of ( TOP-REAL 2) : LE (q2,p31,P,p1,p2) & LE (p31,q3,P,p1,p2) } by A3, A6;

        then

         A7: x in ( Segment (P,p1,p2,q2,q3)) by JORDAN6: 26;

         LE (p3,q2,P,p1,p2) by A4, JORDAN5C: 9;

        then x in { p31 where p31 be Point of ( TOP-REAL 2) : LE (q1,p31,P,p1,p2) & LE (p31,q2,P,p1,p2) } by A2, A6;

        then x in ( Segment (P,p1,p2,q1,q2)) by JORDAN6: 26;

        hence thesis by A7, XBOOLE_0:def 4;

      end;

      (( Segment (P,p1,p2,q1,q2)) /\ ( Segment (P,p1,p2,q2,q3))) c= {q2}

      proof

        let x be object;

        assume

         A8: x in (( Segment (P,p1,p2,q1,q2)) /\ ( Segment (P,p1,p2,q2,q3)));

        then x in ( Segment (P,p1,p2,q2,q3)) by XBOOLE_0:def 4;

        then x in { p4 where p4 be Point of ( TOP-REAL 2) : LE (q2,p4,P,p1,p2) & LE (p4,q3,P,p1,p2) } by JORDAN6: 26;

        then

         A9: ex p4 be Point of ( TOP-REAL 2) st x = p4 & LE (q2,p4,P,p1,p2) & LE (p4,q3,P,p1,p2);

        x in ( Segment (P,p1,p2,q1,q2)) by A8, XBOOLE_0:def 4;

        then x in { p where p be Point of ( TOP-REAL 2) : LE (q1,p,P,p1,p2) & LE (p,q2,P,p1,p2) } by JORDAN6: 26;

        then ex p be Point of ( TOP-REAL 2) st x = p & LE (q1,p,P,p1,p2) & LE (p,q2,P,p1,p2);

        then x = q2 by A1, A9, JORDAN5C: 12;

        hence thesis by TARSKI:def 1;

      end;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    theorem :: JORDAN20:24

    

     Th24: for P be non empty Subset of ( TOP-REAL 2), p1,p2 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) holds ( Segment (P,p1,p2,p1,p2)) = P

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2 be Point of ( TOP-REAL 2);

      assume P is_an_arc_of (p1,p2);

      then

       A1: ( R_Segment (P,p1,p2,p1)) = P & ( L_Segment (P,p1,p2,p2)) = P by JORDAN6: 22, JORDAN6: 24;

      (( R_Segment (P,p1,p2,p1)) /\ ( L_Segment (P,p1,p2,p2))) = ( Segment (P,p1,p2,p1,p2)) by JORDAN6:def 5;

      hence thesis by A1;

    end;

    theorem :: JORDAN20:25

    

     Th25: for P,Q1 be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & Q1 is_an_arc_of (q1,q2) & LE (q1,q2,P,p1,p2) & Q1 c= P holds Q1 = ( Segment (P,p1,p2,q1,q2))

    proof

      let P,Q1 be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: P is_an_arc_of (p1,p2) and

       A2: Q1 is_an_arc_of (q1,q2) and

       A3: LE (q1,q2,P,p1,p2) and

       A4: Q1 c= P;

      reconsider Q0 = ( Segment (P,p1,p2,q1,q2)) as non empty Subset of ( TOP-REAL 2) by A3, JORDAN16: 18;

      

       A5: q1 <> q2 by A2, JORDAN6: 37;

      then

       A6: ( Segment (P,p1,p2,q1,q2)) is_an_arc_of (q1,q2) by A1, A3, JORDAN16: 21;

      

       A7: q2 in P by A3, JORDAN5C:def 3;

       A8:

      now

        assume

         A9: q1 = p2;

         LE (q2,p2,P,p1,p2) by A1, A7, JORDAN5C: 10;

        hence contradiction by A1, A2, A3, A9, JORDAN5C: 12, JORDAN6: 37;

      end;

      

       A10: q1 in P by A3, JORDAN5C:def 3;

       A11:

      now

        assume

         A12: q2 = p1;

         LE (p1,q1,P,p1,p2) by A1, A10, JORDAN5C: 10;

        hence contradiction by A1, A2, A3, A12, JORDAN5C: 12, JORDAN6: 37;

      end;

      

       A13: p1 in P & p2 in P by A1, TOPREAL1: 1;

      now

        

         A14: LE (p1,q1,P,p1,p2) by A1, A10, JORDAN5C: 10;

        then

         A15: (( Segment (P,p1,p2,p1,q1)) \/ ( Segment (P,p1,p2,q1,q2))) = ( Segment (P,p1,p2,p1,q2)) by A1, A3, Th22;

        

         A16: ( [#] (( TOP-REAL 2) | P)) = P by PRE_TOPC:def 5;

        

         A17: LE (q2,p2,P,p1,p2) by A1, A7, JORDAN5C: 10;

        

         A18: ( [#] I[01] ) = the carrier of I[01] ;

        Q0 is_an_arc_of (q1,q2) by A1, A3, A5, JORDAN16: 21;

        then

         A19: q2 in Q0 by TOPREAL1: 1;

        assume not Q1 c= Q0;

        then

        consider x8 be object such that

         A20: x8 in Q1 and

         A21: not x8 in Q0;

        reconsider q = x8 as Point of ( TOP-REAL 2) by A20;

        

         A22: q <> q1 by A3, A21, JORDAN16: 5;

         LE (p1,q2,P,p1,p2) by A3, A14, JORDAN5C: 13;

        

        then (( Segment (P,p1,p2,p1,q2)) \/ ( Segment (P,p1,p2,q2,p2))) = ( Segment (P,p1,p2,p1,p2)) by A1, A17, Th22

        .= P by A1, Th24;

        then

         A23: q in ( Segment (P,p1,p2,p1,q2)) or q in ( Segment (P,p1,p2,q2,p2)) by A4, A20, XBOOLE_0:def 3;

        now

          per cases by A21, A15, A23, XBOOLE_0:def 3;

            case

             A24: q in ( Segment (P,p1,p2,p1,q1));

            

             A25: not q in {q1} by A22, TARSKI:def 1;

             not q2 in {q1} by A5, TARSKI:def 1;

            then

            reconsider Qa = (P \ {q1}) as non empty Subset of (( TOP-REAL 2) | P) by A7, A16, XBOOLE_0:def 5, XBOOLE_1: 36;

            

             A26: the carrier of ((( TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC: 8;

            reconsider Qa9 = Qa as Subset of ( TOP-REAL 2);

            

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

            

             A28: ( Segment (Q1,q1,q2,q,q2)) is_an_arc_of (q,q2) by A2, A20, A21, A19, Th21;

            then

            consider f2 be Function of I[01] , (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q,q2))) such that

             A29: f2 is being_homeomorphism and

             A30: (f2 . 0 ) = q & (f2 . 1) = q2 by TOPREAL1:def 1;

            

             A31: ( rng f2) = ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q,q2)))) by A29, TOPS_2:def 5

            .= ( Segment (Q1,q1,q2,q,q2)) by PRE_TOPC:def 5;

            

             A32: ( not p2 in {q1}) & not q2 in {q1} by A5, A8, TARSKI:def 1;

            q in { p3 where p3 be Point of ( TOP-REAL 2) : LE (p1,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) } by A24, JORDAN6: 26;

            then

             A33: ex p3 be Point of ( TOP-REAL 2) st q = p3 & LE (p1,p3,P,p1,p2) & LE (p3,q1,P,p1,p2);

             A34:

            now

              assume

               A35: p1 = q1;

              then q = p1 by A1, A33, JORDAN5C: 12;

              hence contradiction by A6, A21, A35, TOPREAL1: 1;

            end;

            then not p1 in {q1} by TARSKI:def 1;

            then

            reconsider p19 = p1, q9 = q, q29 = q2, p29 = p2 as Point of ((( TOP-REAL 2) | P) | Qa) by A4, A7, A13, A20, A26, A32, A25, XBOOLE_0:def 5;

            now

              per cases ;

                case q <> p1;

                then

                 A36: ( Segment (P,p1,p2,p1,q)) is_an_arc_of (p1,q) by A1, A4, A20, JORDAN16: 24;

                then

                consider f1 be Function of I[01] , (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q))) such that

                 A37: f1 is being_homeomorphism and

                 A38: (f1 . 0 ) = p1 & (f1 . 1) = q by TOPREAL1:def 1;

                

                 A39: ( rng f1) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q)))) by A37, TOPS_2:def 5

                .= ( Segment (P,p1,p2,p1,q)) by PRE_TOPC:def 5;

                { p where p be Point of ( TOP-REAL 2) : LE (p1,p,P,p1,p2) & LE (p,q,P,p1,p2) } c= Qa

                proof

                  let x be object;

                  assume x in { p where p be Point of ( TOP-REAL 2) : LE (p1,p,P,p1,p2) & LE (p,q,P,p1,p2) };

                  then

                   A40: ex p be Point of ( TOP-REAL 2) st x = p & LE (p1,p,P,p1,p2) & LE (p,q,P,p1,p2);

                  then x <> q1 by A1, A22, A33, JORDAN5C: 12;

                  then

                   A41: not x in {q1} by TARSKI:def 1;

                  x in P by A40, JORDAN5C:def 3;

                  hence thesis by A41, XBOOLE_0:def 5;

                end;

                then

                 A42: ( Segment (P,p1,p2,p1,q)) c= Qa by JORDAN6: 26;

                ( dom f1) = the carrier of I[01] by A18, A37, TOPS_2:def 5;

                then

                reconsider g1 = f1 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A26, A39, A42, FUNCT_2: 2;

                

                 A43: f1 is continuous by A37, TOPS_2:def 5;

                

                 A44: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g1 " G) is open

                proof

                  let G be Subset of ((( TOP-REAL 2) | P) | Qa);

                  

                   A45: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

                  assume G is open;

                  then

                  consider G4 be Subset of ( TOP-REAL 2) such that

                   A46: G4 is open and

                   A47: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A45, TOPS_2: 24;

                  reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q))))) as Subset of (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q)));

                  

                   A48: G5 is open by A46, TOPS_2: 24;

                  

                   A49: ( rng g1) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q)))) by A37, TOPS_2:def 5

                  .= ( Segment (P,p1,p2,p1,q)) by PRE_TOPC:def 5;

                  

                   A50: p1 in ( Segment (P,p1,p2,p1,q)) by A36, TOPREAL1: 1;

                  

                   A51: (f1 " G5) = (g1 " (G4 /\ ( Segment (P,p1,p2,p1,q)))) by PRE_TOPC:def 5

                  .= ((g1 " G4) /\ (g1 " ( Segment (P,p1,p2,p1,q)))) by FUNCT_1: 68;

                  (g1 " G) = ((g1 " G4) /\ (g1 " ( [#] (( TOP-REAL 2) | Qa9)))) by A47, FUNCT_1: 68

                  .= ((g1 " G4) /\ (g1 " Qa9)) by PRE_TOPC:def 5

                  .= ((g1 " G4) /\ (g1 " (( rng g1) /\ Qa9))) by RELAT_1: 133

                  .= ((g1 " G4) /\ (g1 " ( Segment (P,p1,p2,p1,q)))) by A42, A49, XBOOLE_1: 28;

                  hence thesis by A43, A50, A48, A51, TOPS_2: 43;

                end;

                ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

                then

                 A52: g1 is continuous by A44, TOPS_2: 43;

                then (p19,q9) are_connected by A38, BORSUK_2:def 1;

                then g1 is Path of p19, q9 by A38, A52, BORSUK_2:def 2;

                hence ex G1 be Path of p19, q9 st G1 is continuous & (G1 . 0 ) = p19 & (G1 . 1) = q9 by A38, A52;

              end;

                case

                 A53: q = p1;

                consider g01 be Function of I[01] , ((( TOP-REAL 2) | P) | Qa) such that

                 A54: g01 is continuous & (g01 . 0 ) = p19 & (g01 . 1) = p19 by BORSUK_2: 3;

                (p19,p19) are_connected ;

                then g01 is Path of p19, p19 by A54, BORSUK_2:def 2;

                hence ex G1 be Path of p19, q9 st G1 is continuous & (G1 . 0 ) = p19 & (G1 . 1) = q9 by A53, A54;

              end;

            end;

            then

            consider G1 be Path of p19, q9 such that

             A55: G1 is continuous & (G1 . 0 ) = p19 & (G1 . 1) = q9;

            now

              per cases ;

                case q2 <> p2;

                then

                 A56: ( Segment (P,p1,p2,q2,p2)) is_an_arc_of (q2,p2) by A1, A7, Th21;

                then

                consider f3 be Function of I[01] , (( TOP-REAL 2) | ( Segment (P,p1,p2,q2,p2))) such that

                 A57: f3 is being_homeomorphism and

                 A58: (f3 . 0 ) = q2 & (f3 . 1) = p2 by TOPREAL1:def 1;

                

                 A59: ( rng f3) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q2,p2)))) by A57, TOPS_2:def 5

                .= ( Segment (P,p1,p2,q2,p2)) by PRE_TOPC:def 5;

                { p where p be Point of ( TOP-REAL 2) : LE (q2,p,P,p1,p2) & LE (p,p2,P,p1,p2) } c= Qa

                proof

                  let x be object;

                  assume x in { p where p be Point of ( TOP-REAL 2) : LE (q2,p,P,p1,p2) & LE (p,p2,P,p1,p2) };

                  then

                   A60: ex p be Point of ( TOP-REAL 2) st x = p & LE (q2,p,P,p1,p2) & LE (p,p2,P,p1,p2);

                  then x <> q1 by A1, A2, A3, JORDAN5C: 12, JORDAN6: 37;

                  then

                   A61: not x in {q1} by TARSKI:def 1;

                  x in P by A60, JORDAN5C:def 3;

                  hence thesis by A61, XBOOLE_0:def 5;

                end;

                then

                 A62: ( Segment (P,p1,p2,q2,p2)) c= Qa by JORDAN6: 26;

                

                 A63: the carrier of ((( TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC: 8;

                ( dom f3) = the carrier of I[01] by A18, A57, TOPS_2:def 5;

                then

                reconsider g3 = f3 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A59, A63, A62, FUNCT_2: 2;

                

                 A64: f3 is continuous by A57, TOPS_2:def 5;

                

                 A65: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g3 " G) is open

                proof

                  let G be Subset of ((( TOP-REAL 2) | P) | Qa);

                  

                   A66: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

                  assume G is open;

                  then

                  consider G4 be Subset of ( TOP-REAL 2) such that

                   A67: G4 is open and

                   A68: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A66, TOPS_2: 24;

                  reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q2,p2))))) as Subset of (( TOP-REAL 2) | ( Segment (P,p1,p2,q2,p2)));

                  

                   A69: G5 is open by A67, TOPS_2: 24;

                  

                   A70: ( rng g3) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q2,p2)))) by A57, TOPS_2:def 5

                  .= ( Segment (P,p1,p2,q2,p2)) by PRE_TOPC:def 5;

                  

                   A71: p2 in ( Segment (P,p1,p2,q2,p2)) by A56, TOPREAL1: 1;

                  

                   A72: (f3 " G5) = (g3 " (G4 /\ ( Segment (P,p1,p2,q2,p2)))) by PRE_TOPC:def 5

                  .= ((g3 " G4) /\ (g3 " ( Segment (P,p1,p2,q2,p2)))) by FUNCT_1: 68;

                  (g3 " G) = ((g3 " G4) /\ (g3 " ( [#] (( TOP-REAL 2) | Qa9)))) by A68, FUNCT_1: 68

                  .= ((g3 " G4) /\ (g3 " Qa9)) by PRE_TOPC:def 5

                  .= ((g3 " G4) /\ (g3 " (( rng g3) /\ Qa9))) by RELAT_1: 133

                  .= ((g3 " G4) /\ (g3 " ( Segment (P,p1,p2,q2,p2)))) by A62, A70, XBOOLE_1: 28;

                  hence thesis by A64, A71, A69, A72, TOPS_2: 43;

                end;

                ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

                then

                 A73: g3 is continuous by A65, TOPS_2: 43;

                then (q29,p29) are_connected by A58, BORSUK_2:def 1;

                then g3 is Path of q29, p29 by A58, A73, BORSUK_2:def 2;

                hence ex G3 be Path of q29, p29 st G3 is continuous & (G3 . 0 ) = q29 & (G3 . 1) = p29 by A58, A73;

              end;

                case

                 A74: q2 = p2;

                consider g01 be Function of I[01] , ((( TOP-REAL 2) | P) | Qa) such that

                 A75: g01 is continuous & (g01 . 0 ) = q29 & (g01 . 1) = q29 by BORSUK_2: 3;

                (q29,q29) are_connected ;

                then g01 is Path of q29, q29 by A75, BORSUK_2:def 2;

                hence ex G3 be Path of q29, p29 st G3 is continuous & (G3 . 0 ) = q29 & (G3 . 1) = p29 by A74, A75;

              end;

            end;

            then

            consider G3 be Path of q29, p29 such that

             A76: G3 is continuous & (G3 . 0 ) = q29 & (G3 . 1) = p29;

            { p where p be Point of ( TOP-REAL 2) : LE (q,p,Q1,q1,q2) & LE (p,q2,Q1,q1,q2) } c= Qa

            proof

              let x be object;

              assume x in { p where p be Point of ( TOP-REAL 2) : LE (q,p,Q1,q1,q2) & LE (p,q2,Q1,q1,q2) };

              then

               A77: ex p be Point of ( TOP-REAL 2) st x = p & LE (q,p,Q1,q1,q2) & LE (p,q2,Q1,q1,q2);

              now

                assume

                 A78: x = q1;

                 LE (q1,q,Q1,q1,q2) by A2, A20, JORDAN5C: 10;

                hence contradiction by A2, A22, A77, A78, JORDAN5C: 12;

              end;

              then

               A79: not x in {q1} by TARSKI:def 1;

              x in Q1 by A77, JORDAN5C:def 3;

              hence thesis by A4, A79, XBOOLE_0:def 5;

            end;

            then

             A80: ( Segment (Q1,q1,q2,q,q2)) c= Qa by JORDAN6: 26;

            ( dom f2) = the carrier of I[01] by A18, A29, TOPS_2:def 5;

            then

            reconsider g2 = f2 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A31, A27, A80, FUNCT_2: 2;

            

             A81: f2 is continuous by A29, TOPS_2:def 5;

            

             A82: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g2 " G) is open

            proof

              let G be Subset of ((( TOP-REAL 2) | P) | Qa);

              

               A83: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

              assume G is open;

              then

              consider G4 be Subset of ( TOP-REAL 2) such that

               A84: G4 is open and

               A85: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A83, TOPS_2: 24;

              reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q,q2))))) as Subset of (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q,q2)));

              

               A86: G5 is open by A84, TOPS_2: 24;

              

               A87: ( rng g2) = ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q,q2)))) by A29, TOPS_2:def 5

              .= ( Segment (Q1,q1,q2,q,q2)) by PRE_TOPC:def 5;

              

               A88: q2 in ( Segment (Q1,q1,q2,q,q2)) by A28, TOPREAL1: 1;

              

               A89: (f2 " G5) = (g2 " (G4 /\ ( Segment (Q1,q1,q2,q,q2)))) by PRE_TOPC:def 5

              .= ((g2 " G4) /\ (g2 " ( Segment (Q1,q1,q2,q,q2)))) by FUNCT_1: 68;

              (g2 " G) = ((g2 " G4) /\ (g2 " ( [#] (( TOP-REAL 2) | Qa9)))) by A85, FUNCT_1: 68

              .= ((g2 " G4) /\ (g2 " Qa9)) by PRE_TOPC:def 5

              .= ((g2 " G4) /\ (g2 " (( rng g2) /\ Qa9))) by RELAT_1: 133

              .= ((g2 " G4) /\ (g2 " ( Segment (Q1,q1,q2,q,q2)))) by A80, A87, XBOOLE_1: 28;

              hence thesis by A81, A88, A86, A89, TOPS_2: 43;

            end;

            ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

            then

             A90: g2 is continuous by A82, TOPS_2: 43;

            then (q9,q29) are_connected by A30, BORSUK_2:def 1;

            then

            reconsider G2 = g2 as Path of q9, q29 by A30, A90, BORSUK_2:def 2;

            

             A91: ((G1 + G2) . 1) = q29 by A55, A30, A90, BORSUK_2: 14;

            

             A92: (G1 + G2) is continuous & ((G1 + G2) . 0 ) = p19 by A55, A30, A90, BORSUK_2: 14;

            then

             A93: (((G1 + G2) + G3) . 1) = p29 by A91, A76, BORSUK_2: 14;

            ((G1 + G2) + G3) is continuous & (((G1 + G2) + G3) . 0 ) = p19 by A92, A91, A76, BORSUK_2: 14;

            hence contradiction by A1, A10, A8, A34, A93, Th18;

          end;

            case

             A94: q in ( Segment (P,p1,p2,q2,p2));

            

             A95: ( not p1 in {q2}) & not q1 in {q2} by A5, A11, TARSKI:def 1;

             not q1 in {q2} by A5, TARSKI:def 1;

            then

            reconsider Qa = (P \ {q2}) as non empty Subset of (( TOP-REAL 2) | P) by A10, A16, XBOOLE_0:def 5, XBOOLE_1: 36;

            

             A96: the carrier of ((( TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC: 8;

            reconsider Qa9 = Qa as Subset of ( TOP-REAL 2);

            

             A97: the carrier of ((( TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC: 8;

            

             A98: ( Segment (Q1,q1,q2,q1,q)) is_an_arc_of (q1,q) by A2, A20, A22, JORDAN16: 24;

            then

            consider f2 be Function of I[01] , (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q1,q))) such that

             A99: f2 is being_homeomorphism and

             A100: (f2 . 0 ) = q1 & (f2 . 1) = q by TOPREAL1:def 1;

            

             A101: ( rng f2) = ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q1,q)))) by A99, TOPS_2:def 5

            .= ( Segment (Q1,q1,q2,q1,q)) by PRE_TOPC:def 5;

            

             A102: not q in {q2} by A21, A19, TARSKI:def 1;

            q in { p3 where p3 be Point of ( TOP-REAL 2) : LE (q2,p3,P,p1,p2) & LE (p3,p2,P,p1,p2) } by A94, JORDAN6: 26;

            then

             A103: ex p3 be Point of ( TOP-REAL 2) st q = p3 & LE (q2,p3,P,p1,p2) & LE (p3,p2,P,p1,p2);

             A104:

            now

              assume

               A105: p2 = q2;

              then q = p2 by A1, A103, JORDAN5C: 12;

              hence contradiction by A6, A21, A105, TOPREAL1: 1;

            end;

            then not p2 in {q2} by TARSKI:def 1;

            then

            reconsider p19 = p1, q9 = q, q19 = q1, p29 = p2 as Point of ((( TOP-REAL 2) | P) | Qa) by A4, A10, A13, A20, A96, A95, A102, XBOOLE_0:def 5;

            now

              per cases ;

                case q <> p2;

                then

                 A106: ( Segment (P,p1,p2,q,p2)) is_an_arc_of (q,p2) by A1, A4, A20, Th21;

                then

                consider f1 be Function of I[01] , (( TOP-REAL 2) | ( Segment (P,p1,p2,q,p2))) such that

                 A107: f1 is being_homeomorphism and

                 A108: (f1 . 0 ) = q & (f1 . 1) = p2 by TOPREAL1:def 1;

                

                 A109: ( rng f1) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q,p2)))) by A107, TOPS_2:def 5

                .= ( Segment (P,p1,p2,q,p2)) by PRE_TOPC:def 5;

                { p where p be Point of ( TOP-REAL 2) : LE (q,p,P,p1,p2) & LE (p,p2,P,p1,p2) } c= Qa

                proof

                  let x be object;

                  assume x in { p where p be Point of ( TOP-REAL 2) : LE (q,p,P,p1,p2) & LE (p,p2,P,p1,p2) };

                  then

                   A110: ex p be Point of ( TOP-REAL 2) st x = p & LE (q,p,P,p1,p2) & LE (p,p2,P,p1,p2);

                  then x <> q2 by A1, A21, A19, A103, JORDAN5C: 12;

                  then

                   A111: not x in {q2} by TARSKI:def 1;

                  x in P by A110, JORDAN5C:def 3;

                  hence thesis by A111, XBOOLE_0:def 5;

                end;

                then

                 A112: ( Segment (P,p1,p2,q,p2)) c= Qa by JORDAN6: 26;

                ( dom f1) = the carrier of I[01] by A18, A107, TOPS_2:def 5;

                then

                reconsider g1 = f1 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A96, A109, A112, FUNCT_2: 2;

                

                 A113: f1 is continuous by A107, TOPS_2:def 5;

                

                 A114: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g1 " G) is open

                proof

                  let G be Subset of ((( TOP-REAL 2) | P) | Qa);

                  

                   A115: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

                  assume G is open;

                  then

                  consider G4 be Subset of ( TOP-REAL 2) such that

                   A116: G4 is open and

                   A117: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A115, TOPS_2: 24;

                  reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q,p2))))) as Subset of (( TOP-REAL 2) | ( Segment (P,p1,p2,q,p2)));

                  

                   A118: G5 is open by A116, TOPS_2: 24;

                  

                   A119: ( rng g1) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,q,p2)))) by A107, TOPS_2:def 5

                  .= ( Segment (P,p1,p2,q,p2)) by PRE_TOPC:def 5;

                  

                   A120: p2 in ( Segment (P,p1,p2,q,p2)) by A106, TOPREAL1: 1;

                  

                   A121: (f1 " G5) = (g1 " (G4 /\ ( Segment (P,p1,p2,q,p2)))) by PRE_TOPC:def 5

                  .= ((g1 " G4) /\ (g1 " ( Segment (P,p1,p2,q,p2)))) by FUNCT_1: 68;

                  (g1 " G) = ((g1 " G4) /\ (g1 " ( [#] (( TOP-REAL 2) | Qa9)))) by A117, FUNCT_1: 68

                  .= ((g1 " G4) /\ (g1 " Qa9)) by PRE_TOPC:def 5

                  .= ((g1 " G4) /\ (g1 " (( rng g1) /\ Qa9))) by RELAT_1: 133

                  .= ((g1 " G4) /\ (g1 " ( Segment (P,p1,p2,q,p2)))) by A112, A119, XBOOLE_1: 28;

                  hence thesis by A113, A120, A118, A121, TOPS_2: 43;

                end;

                ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

                then

                 A122: g1 is continuous by A114, TOPS_2: 43;

                then (q9,p29) are_connected by A108, BORSUK_2:def 1;

                then g1 is Path of q9, p29 by A108, A122, BORSUK_2:def 2;

                hence ex G1 be Path of q9, p29 st G1 is continuous & (G1 . 0 ) = q9 & (G1 . 1) = p29 by A108, A122;

              end;

                case

                 A123: q = p2;

                consider g01 be Function of I[01] , ((( TOP-REAL 2) | P) | Qa) such that

                 A124: g01 is continuous & (g01 . 0 ) = p29 & (g01 . 1) = p29 by BORSUK_2: 3;

                (p29,p29) are_connected ;

                then g01 is Path of p29, p29 by A124, BORSUK_2:def 2;

                hence ex G1 be Path of q9, p29 st G1 is continuous & (G1 . 0 ) = q9 & (G1 . 1) = p29 by A123, A124;

              end;

            end;

            then

            consider G1 be Path of q9, p29 such that

             A125: G1 is continuous & (G1 . 0 ) = q9 & (G1 . 1) = p29;

            now

              per cases ;

                case q1 <> p1;

                then

                 A126: ( Segment (P,p1,p2,p1,q1)) is_an_arc_of (p1,q1) by A1, A10, JORDAN16: 24;

                then

                consider f3 be Function of I[01] , (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q1))) such that

                 A127: f3 is being_homeomorphism and

                 A128: (f3 . 0 ) = p1 & (f3 . 1) = q1 by TOPREAL1:def 1;

                

                 A129: ( rng f3) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q1)))) by A127, TOPS_2:def 5

                .= ( Segment (P,p1,p2,p1,q1)) by PRE_TOPC:def 5;

                { p where p be Point of ( TOP-REAL 2) : LE (p1,p,P,p1,p2) & LE (p,q1,P,p1,p2) } c= Qa

                proof

                  let x be object;

                  assume x in { p where p be Point of ( TOP-REAL 2) : LE (p1,p,P,p1,p2) & LE (p,q1,P,p1,p2) };

                  then

                   A130: ex p be Point of ( TOP-REAL 2) st x = p & LE (p1,p,P,p1,p2) & LE (p,q1,P,p1,p2);

                  then x <> q2 by A1, A2, A3, JORDAN5C: 12, JORDAN6: 37;

                  then

                   A131: not x in {q2} by TARSKI:def 1;

                  x in P by A130, JORDAN5C:def 3;

                  hence thesis by A131, XBOOLE_0:def 5;

                end;

                then

                 A132: ( Segment (P,p1,p2,p1,q1)) c= Qa by JORDAN6: 26;

                

                 A133: the carrier of ((( TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC: 8;

                ( dom f3) = the carrier of I[01] by A18, A127, TOPS_2:def 5;

                then

                reconsider g3 = f3 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A129, A133, A132, FUNCT_2: 2;

                

                 A134: f3 is continuous by A127, TOPS_2:def 5;

                

                 A135: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g3 " G) is open

                proof

                  let G be Subset of ((( TOP-REAL 2) | P) | Qa);

                  

                   A136: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

                  assume G is open;

                  then

                  consider G4 be Subset of ( TOP-REAL 2) such that

                   A137: G4 is open and

                   A138: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A136, TOPS_2: 24;

                  reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q1))))) as Subset of (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q1)));

                  

                   A139: G5 is open by A137, TOPS_2: 24;

                  

                   A140: ( rng g3) = ( [#] (( TOP-REAL 2) | ( Segment (P,p1,p2,p1,q1)))) by A127, TOPS_2:def 5

                  .= ( Segment (P,p1,p2,p1,q1)) by PRE_TOPC:def 5;

                  

                   A141: p1 in ( Segment (P,p1,p2,p1,q1)) by A126, TOPREAL1: 1;

                  

                   A142: (f3 " G5) = (g3 " (G4 /\ ( Segment (P,p1,p2,p1,q1)))) by PRE_TOPC:def 5

                  .= ((g3 " G4) /\ (g3 " ( Segment (P,p1,p2,p1,q1)))) by FUNCT_1: 68;

                  (g3 " G) = ((g3 " G4) /\ (g3 " ( [#] (( TOP-REAL 2) | Qa9)))) by A138, FUNCT_1: 68

                  .= ((g3 " G4) /\ (g3 " Qa9)) by PRE_TOPC:def 5

                  .= ((g3 " G4) /\ (g3 " (( rng g3) /\ Qa9))) by RELAT_1: 133

                  .= ((g3 " G4) /\ (g3 " ( Segment (P,p1,p2,p1,q1)))) by A132, A140, XBOOLE_1: 28;

                  hence thesis by A134, A141, A139, A142, TOPS_2: 43;

                end;

                ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

                then

                 A143: g3 is continuous by A135, TOPS_2: 43;

                then (p19,q19) are_connected by A128, BORSUK_2:def 1;

                then g3 is Path of p19, q19 by A128, A143, BORSUK_2:def 2;

                hence ex G3 be Path of p19, q19 st G3 is continuous & (G3 . 0 ) = p19 & (G3 . 1) = q19 by A128, A143;

              end;

                case

                 A144: q1 = p1;

                consider g01 be Function of I[01] , ((( TOP-REAL 2) | P) | Qa) such that

                 A145: g01 is continuous & (g01 . 0 ) = q19 & (g01 . 1) = q19 by BORSUK_2: 3;

                (q19,q19) are_connected ;

                then g01 is Path of q19, q19 by A145, BORSUK_2:def 2;

                hence ex G3 be Path of p19, q19 st G3 is continuous & (G3 . 0 ) = p19 & (G3 . 1) = q19 by A144, A145;

              end;

            end;

            then

            consider G3 be Path of p19, q19 such that

             A146: G3 is continuous & (G3 . 0 ) = p19 & (G3 . 1) = q19;

            { p where p be Point of ( TOP-REAL 2) : LE (q1,p,Q1,q1,q2) & LE (p,q,Q1,q1,q2) } c= Qa

            proof

              let x be object;

              assume x in { p where p be Point of ( TOP-REAL 2) : LE (q1,p,Q1,q1,q2) & LE (p,q,Q1,q1,q2) };

              then

               A147: ex p be Point of ( TOP-REAL 2) st x = p & LE (q1,p,Q1,q1,q2) & LE (p,q,Q1,q1,q2);

              now

                assume

                 A148: x = q2;

                 LE (q,q2,Q1,q1,q2) by A2, A20, JORDAN5C: 10;

                hence contradiction by A2, A21, A19, A147, A148, JORDAN5C: 12;

              end;

              then

               A149: not x in {q2} by TARSKI:def 1;

              x in Q1 by A147, JORDAN5C:def 3;

              hence thesis by A4, A149, XBOOLE_0:def 5;

            end;

            then

             A150: ( Segment (Q1,q1,q2,q1,q)) c= Qa by JORDAN6: 26;

            ( dom f2) = the carrier of I[01] by A18, A99, TOPS_2:def 5;

            then

            reconsider g2 = f2 as Function of I[01] , ((( TOP-REAL 2) | P) | Qa) by A101, A97, A150, FUNCT_2: 2;

            

             A151: f2 is continuous by A99, TOPS_2:def 5;

            

             A152: for G be Subset of ((( TOP-REAL 2) | P) | Qa) st G is open holds (g2 " G) is open

            proof

              let G be Subset of ((( TOP-REAL 2) | P) | Qa);

              

               A153: ((( TOP-REAL 2) | P) | Qa) = (( TOP-REAL 2) | Qa9) by PRE_TOPC: 7, XBOOLE_1: 36;

              assume G is open;

              then

              consider G4 be Subset of ( TOP-REAL 2) such that

               A154: G4 is open and

               A155: G = (G4 /\ ( [#] (( TOP-REAL 2) | Qa9))) by A153, TOPS_2: 24;

              reconsider G5 = (G4 /\ ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q1,q))))) as Subset of (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q1,q)));

              

               A156: G5 is open by A154, TOPS_2: 24;

              

               A157: ( rng g2) = ( [#] (( TOP-REAL 2) | ( Segment (Q1,q1,q2,q1,q)))) by A99, TOPS_2:def 5

              .= ( Segment (Q1,q1,q2,q1,q)) by PRE_TOPC:def 5;

              

               A158: q1 in ( Segment (Q1,q1,q2,q1,q)) by A98, TOPREAL1: 1;

              

               A159: (f2 " G5) = (g2 " (G4 /\ ( Segment (Q1,q1,q2,q1,q)))) by PRE_TOPC:def 5

              .= ((g2 " G4) /\ (g2 " ( Segment (Q1,q1,q2,q1,q)))) by FUNCT_1: 68;

              (g2 " G) = ((g2 " G4) /\ (g2 " ( [#] (( TOP-REAL 2) | Qa9)))) by A155, FUNCT_1: 68

              .= ((g2 " G4) /\ (g2 " Qa9)) by PRE_TOPC:def 5

              .= ((g2 " G4) /\ (g2 " (( rng g2) /\ Qa9))) by RELAT_1: 133

              .= ((g2 " G4) /\ (g2 " ( Segment (Q1,q1,q2,q1,q)))) by A150, A157, XBOOLE_1: 28;

              hence thesis by A151, A158, A156, A159, TOPS_2: 43;

            end;

            ( [#] ((( TOP-REAL 2) | P) | Qa)) <> {} ;

            then

             A160: g2 is continuous by A152, TOPS_2: 43;

            then (q19,q9) are_connected by A100, BORSUK_2:def 1;

            then

            reconsider G2 = g2 as Path of q19, q9 by A100, A160, BORSUK_2:def 2;

            

             A161: ((G2 + G1) . 1) = p29 by A125, A100, A160, BORSUK_2: 14;

            

             A162: (G2 + G1) is continuous & ((G2 + G1) . 0 ) = q19 by A125, A100, A160, BORSUK_2: 14;

            then

             A163: ((G3 + (G2 + G1)) . 1) = p29 by A161, A146, BORSUK_2: 14;

            (G3 + (G2 + G1)) is continuous & ((G3 + (G2 + G1)) . 0 ) = p19 by A162, A161, A146, BORSUK_2: 14;

            hence contradiction by A1, A7, A11, A104, A163, Th18;

          end;

        end;

        hence contradiction;

      end;

      hence thesis by A2, A6, Th20;

    end;

    theorem :: JORDAN20:26

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real st q1 is_Lin (P,p1,p2,e) & (q2 `1 ) = e & ( LSeg (q1,q2)) c= P & p in ( LSeg (q1,q2)) holds p is_Lin (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: q1 is_Lin (P,p1,p2,e) and

       A2: (q2 `1 ) = e and

       A3: ( LSeg (q1,q2)) c= P and

       A4: p in ( LSeg (q1,q2));

      

       A5: q1 in P by A1;

      

       A6: q2 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      

       A7: (q1 `1 ) = e by A1;

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

       A8: (p4 `1 ) < e and

       A9: LE (p4,q1,P,p1,p2) and

       A10: for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,q1,P,p1,p2) holds (p5 `1 ) <= e by A1;

      

       A11: P is_an_arc_of (p1,p2) by A1;

      

       A12: p4 in P by A9, JORDAN5C:def 3;

      now

        per cases by A3, A11, A5, A6, Th19;

          case

           A13: LE (q1,q2,P,p1,p2);

           A14:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q1,q2)) is_an_arc_of (q1,q2) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A3, A11, A13, Th25;

            end;

              case

               A15: q1 = q2;

              then ( LSeg (q1,q2)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A11, A5, A15, Th1;

            end;

          end;

          ( Segment (P,p1,p2,q1,q2)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) } by JORDAN6: 26;

          then

           A16: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) by A4, A14;

          then

           A17: LE (p4,p,P,p1,p2) by A9, JORDAN5C: 13;

          

           A18: for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p,P,p1,p2) holds (p6 `1 ) <= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A19: LE (p4,p6,P,p1,p2) and

             A20: LE (p6,p,P,p1,p2);

            

             A21: p6 in P by A19, JORDAN5C:def 3;

            now

              per cases by A11, A5, A21, Th19;

                case LE (p6,q1,P,p1,p2);

                hence thesis by A10, A19;

              end;

                case

                 A22: LE (q1,p6,P,p1,p2);

                 LE (p6,q2,P,p1,p2) by A16, A20, JORDAN5C: 13;

                then p6 in { qq where qq be Point of ( TOP-REAL 2) : LE (q1,qq,P,p1,p2) & LE (qq,q2,P,p1,p2) } by A22;

                then p6 in ( LSeg (q1,q2)) by A14, JORDAN6: 26;

                hence thesis by A2, A7, GOBOARD7: 5;

              end;

            end;

            hence thesis;

          end;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A17, A18;

        end;

          case

           A23: LE (q2,q1,P,p1,p2);

           A24:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q2,q1)) is_an_arc_of (q2,q1) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A3, A11, A23, Th25;

            end;

              case

               A25: q1 = q2;

              then ( LSeg (q2,q1)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A11, A5, A25, Th1;

            end;

          end;

           A26:

          now

            assume LE (q2,p4,P,p1,p2);

            then p4 in { qq where qq be Point of ( TOP-REAL 2) : LE (q2,qq,P,p1,p2) & LE (qq,q1,P,p1,p2) } by A9;

            then p4 in ( Segment (P,p1,p2,q2,q1)) by JORDAN6: 26;

            hence contradiction by A2, A7, A8, A24, GOBOARD7: 5;

          end;

          ( Segment (P,p1,p2,q2,q1)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) } by JORDAN6: 26;

          then

           A27: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) by A4, A24;

          

           A28: for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p,P,p1,p2) holds (p6 `1 ) <= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A29: LE (p4,p6,P,p1,p2) and

             A30: LE (p6,p,P,p1,p2);

             LE (p6,q1,P,p1,p2) by A27, A30, JORDAN5C: 13;

            hence thesis by A10, A29;

          end;

           LE (q2,p4,P,p1,p2) or LE (p4,q2,P,p1,p2) by A3, A11, A6, A12, Th19;

          then

           A31: LE (p4,p,P,p1,p2) by A27, A26, JORDAN5C: 13;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A31, A28;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:27

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real st q1 is_Rin (P,p1,p2,e) & (q2 `1 ) = e & ( LSeg (q1,q2)) c= P & p in ( LSeg (q1,q2)) holds p is_Rin (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: q1 is_Rin (P,p1,p2,e) and

       A2: (q2 `1 ) = e and

       A3: ( LSeg (q1,q2)) c= P and

       A4: p in ( LSeg (q1,q2));

      

       A5: q1 in P by A1;

      

       A6: q2 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      

       A7: (q1 `1 ) = e by A1;

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

       A8: (p4 `1 ) > e and

       A9: LE (p4,q1,P,p1,p2) and

       A10: for p5 be Point of ( TOP-REAL 2) st LE (p4,p5,P,p1,p2) & LE (p5,q1,P,p1,p2) holds (p5 `1 ) >= e by A1;

      

       A11: P is_an_arc_of (p1,p2) by A1;

      

       A12: p4 in P by A9, JORDAN5C:def 3;

      now

        per cases by A3, A11, A5, A6, Th19;

          case

           A13: LE (q1,q2,P,p1,p2);

           A14:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q1,q2)) is_an_arc_of (q1,q2) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A3, A11, A13, Th25;

            end;

              case

               A15: q1 = q2;

              then ( LSeg (q1,q2)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A11, A5, A15, Th1;

            end;

          end;

          ( Segment (P,p1,p2,q1,q2)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) } by JORDAN6: 26;

          then

           A16: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) by A4, A14;

          then

           A17: LE (p4,p,P,p1,p2) by A9, JORDAN5C: 13;

          

           A18: for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p,P,p1,p2) holds (p6 `1 ) >= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A19: LE (p4,p6,P,p1,p2) and

             A20: LE (p6,p,P,p1,p2);

            

             A21: p6 in P by A19, JORDAN5C:def 3;

            now

              per cases by A11, A5, A21, Th19;

                case LE (p6,q1,P,p1,p2);

                hence thesis by A10, A19;

              end;

                case

                 A22: LE (q1,p6,P,p1,p2);

                 LE (p6,q2,P,p1,p2) by A16, A20, JORDAN5C: 13;

                then p6 in { qq where qq be Point of ( TOP-REAL 2) : LE (q1,qq,P,p1,p2) & LE (qq,q2,P,p1,p2) } by A22;

                then p6 in ( LSeg (q1,q2)) by A14, JORDAN6: 26;

                hence thesis by A2, A7, GOBOARD7: 5;

              end;

            end;

            hence thesis;

          end;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A17, A18;

        end;

          case

           A23: LE (q2,q1,P,p1,p2);

           A24:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q2,q1)) is_an_arc_of (q2,q1) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A3, A11, A23, Th25;

            end;

              case

               A25: q1 = q2;

              then ( LSeg (q2,q1)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A11, A5, A25, Th1;

            end;

          end;

           A26:

          now

            assume LE (q2,p4,P,p1,p2);

            then p4 in { qq where qq be Point of ( TOP-REAL 2) : LE (q2,qq,P,p1,p2) & LE (qq,q1,P,p1,p2) } by A9;

            then p4 in ( Segment (P,p1,p2,q2,q1)) by JORDAN6: 26;

            hence contradiction by A2, A7, A8, A24, GOBOARD7: 5;

          end;

          ( Segment (P,p1,p2,q2,q1)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) } by JORDAN6: 26;

          then

           A27: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) by A4, A24;

          

           A28: for p6 be Point of ( TOP-REAL 2) st LE (p4,p6,P,p1,p2) & LE (p6,p,P,p1,p2) holds (p6 `1 ) >= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A29: LE (p4,p6,P,p1,p2) and

             A30: LE (p6,p,P,p1,p2);

             LE (p6,q1,P,p1,p2) by A27, A30, JORDAN5C: 13;

            hence thesis by A10, A29;

          end;

           LE (q2,p4,P,p1,p2) or LE (p4,q2,P,p1,p2) by A3, A11, A6, A12, Th19;

          then

           A31: LE (p4,p,P,p1,p2) by A27, A26, JORDAN5C: 13;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A31, A28;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:28

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real st q1 is_Lout (P,p1,p2,e) & (q2 `1 ) = e & ( LSeg (q1,q2)) c= P & p in ( LSeg (q1,q2)) holds p is_Lout (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: q1 is_Lout (P,p1,p2,e) and

       A2: (q2 `1 ) = e and

       A3: ( LSeg (q1,q2)) c= P and

       A4: p in ( LSeg (q1,q2));

      

       A5: q1 in P by A1;

      

       A6: q2 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      

       A7: (q1 `1 ) = e by A1;

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

       A8: (p4 `1 ) < e and

       A9: LE (q1,p4,P,p1,p2) and

       A10: for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (q1,p5,P,p1,p2) holds (p5 `1 ) <= e by A1;

      

       A11: P is_an_arc_of (p1,p2) by A1;

      

       A12: p4 in P by A9, JORDAN5C:def 3;

      now

        per cases by A3, A11, A5, A6, Th19;

          case

           A13: LE (q2,q1,P,p1,p2);

           A14:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q2,q1)) is_an_arc_of (q2,q1) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A3, A11, A13, Th25;

            end;

              case

               A15: q1 = q2;

              then ( LSeg (q1,q2)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A11, A5, A15, Th1;

            end;

          end;

          ( Segment (P,p1,p2,q2,q1)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) } by JORDAN6: 26;

          then

           A16: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) by A4, A14;

          then

           A17: LE (p,p4,P,p1,p2) by A9, JORDAN5C: 13;

          

           A18: for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p,p6,P,p1,p2) holds (p6 `1 ) <= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A19: LE (p6,p4,P,p1,p2) and

             A20: LE (p,p6,P,p1,p2);

            

             A21: p6 in P by A19, JORDAN5C:def 3;

            now

              per cases by A11, A5, A21, Th19;

                case LE (q1,p6,P,p1,p2);

                hence thesis by A10, A19;

              end;

                case

                 A22: LE (p6,q1,P,p1,p2);

                 LE (q2,p6,P,p1,p2) by A16, A20, JORDAN5C: 13;

                then p6 in { qq where qq be Point of ( TOP-REAL 2) : LE (q2,qq,P,p1,p2) & LE (qq,q1,P,p1,p2) } by A22;

                then p6 in ( LSeg (q2,q1)) by A14, JORDAN6: 26;

                hence thesis by A2, A7, GOBOARD7: 5;

              end;

            end;

            hence thesis;

          end;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A17, A18;

        end;

          case

           A23: LE (q1,q2,P,p1,p2);

           A24:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q1,q2)) is_an_arc_of (q1,q2) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A3, A11, A23, Th25;

            end;

              case

               A25: q1 = q2;

              then ( LSeg (q2,q1)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A11, A5, A25, Th1;

            end;

          end;

           A26:

          now

            assume LE (p4,q2,P,p1,p2);

            then p4 in { qq where qq be Point of ( TOP-REAL 2) : LE (q1,qq,P,p1,p2) & LE (qq,q2,P,p1,p2) } by A9;

            then p4 in ( Segment (P,p1,p2,q1,q2)) by JORDAN6: 26;

            hence contradiction by A2, A7, A8, A24, GOBOARD7: 5;

          end;

          ( Segment (P,p1,p2,q1,q2)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) } by JORDAN6: 26;

          then

           A27: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) by A4, A24;

          

           A28: for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p,p6,P,p1,p2) holds (p6 `1 ) <= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A29: LE (p6,p4,P,p1,p2) and

             A30: LE (p,p6,P,p1,p2);

             LE (q1,p6,P,p1,p2) by A27, A30, JORDAN5C: 13;

            hence thesis by A10, A29;

          end;

           LE (q2,p4,P,p1,p2) or LE (p4,q2,P,p1,p2) by A3, A11, A6, A12, Th19;

          then

           A31: LE (p,p4,P,p1,p2) by A27, A26, JORDAN5C: 13;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A31, A28;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:29

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real st q1 is_Rout (P,p1,p2,e) & (q2 `1 ) = e & ( LSeg (q1,q2)) c= P & p in ( LSeg (q1,q2)) holds p is_Rout (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,q1,q2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: q1 is_Rout (P,p1,p2,e) and

       A2: (q2 `1 ) = e and

       A3: ( LSeg (q1,q2)) c= P and

       A4: p in ( LSeg (q1,q2));

      

       A5: q1 in P by A1;

      

       A6: q2 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      

       A7: (q1 `1 ) = e by A1;

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

       A8: (p4 `1 ) > e and

       A9: LE (q1,p4,P,p1,p2) and

       A10: for p5 be Point of ( TOP-REAL 2) st LE (p5,p4,P,p1,p2) & LE (q1,p5,P,p1,p2) holds (p5 `1 ) >= e by A1;

      

       A11: P is_an_arc_of (p1,p2) by A1;

      

       A12: p4 in P by A9, JORDAN5C:def 3;

      now

        per cases by A3, A11, A5, A6, Th19;

          case

           A13: LE (q2,q1,P,p1,p2);

           A14:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q2,q1)) is_an_arc_of (q2,q1) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A3, A11, A13, Th25;

            end;

              case

               A15: q1 = q2;

              then ( LSeg (q1,q2)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q2,q1)) = ( LSeg (q2,q1)) by A11, A5, A15, Th1;

            end;

          end;

          ( Segment (P,p1,p2,q2,q1)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) } by JORDAN6: 26;

          then

           A16: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q2,p3,P,p1,p2) & LE (p3,q1,P,p1,p2) by A4, A14;

          then

           A17: LE (p,p4,P,p1,p2) by A9, JORDAN5C: 13;

          

           A18: for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p,p6,P,p1,p2) holds (p6 `1 ) >= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A19: LE (p6,p4,P,p1,p2) and

             A20: LE (p,p6,P,p1,p2);

            

             A21: p6 in P by A19, JORDAN5C:def 3;

            now

              per cases by A11, A5, A21, Th19;

                case LE (q1,p6,P,p1,p2);

                hence thesis by A10, A19;

              end;

                case

                 A22: LE (p6,q1,P,p1,p2);

                 LE (q2,p6,P,p1,p2) by A16, A20, JORDAN5C: 13;

                then p6 in { qq where qq be Point of ( TOP-REAL 2) : LE (q2,qq,P,p1,p2) & LE (qq,q1,P,p1,p2) } by A22;

                then p6 in ( LSeg (q2,q1)) by A14, JORDAN6: 26;

                hence thesis by A2, A7, GOBOARD7: 5;

              end;

            end;

            hence thesis;

          end;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A17, A18;

        end;

          case

           A23: LE (q1,q2,P,p1,p2);

           A24:

          now

            per cases ;

              case q1 <> q2;

              then ( LSeg (q1,q2)) is_an_arc_of (q1,q2) by TOPREAL1: 9;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A3, A11, A23, Th25;

            end;

              case

               A25: q1 = q2;

              then ( LSeg (q2,q1)) = {q1} by RLTOPSP1: 70;

              hence ( Segment (P,p1,p2,q1,q2)) = ( LSeg (q1,q2)) by A11, A5, A25, Th1;

            end;

          end;

           A26:

          now

            assume LE (p4,q2,P,p1,p2);

            then p4 in { qq where qq be Point of ( TOP-REAL 2) : LE (q1,qq,P,p1,p2) & LE (qq,q2,P,p1,p2) } by A9;

            then p4 in ( Segment (P,p1,p2,q1,q2)) by JORDAN6: 26;

            hence contradiction by A2, A7, A8, A24, GOBOARD7: 5;

          end;

          ( Segment (P,p1,p2,q1,q2)) = { p3 where p3 be Point of ( TOP-REAL 2) : LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) } by JORDAN6: 26;

          then

           A27: ex p3 be Point of ( TOP-REAL 2) st p = p3 & LE (q1,p3,P,p1,p2) & LE (p3,q2,P,p1,p2) by A4, A24;

          

           A28: for p6 be Point of ( TOP-REAL 2) st LE (p6,p4,P,p1,p2) & LE (p,p6,P,p1,p2) holds (p6 `1 ) >= e

          proof

            let p6 be Point of ( TOP-REAL 2);

            assume that

             A29: LE (p6,p4,P,p1,p2) and

             A30: LE (p,p6,P,p1,p2);

             LE (q1,p6,P,p1,p2) by A27, A30, JORDAN5C: 13;

            hence thesis by A10, A29;

          end;

           LE (q2,p4,P,p1,p2) or LE (p4,q2,P,p1,p2) by A3, A11, A6, A12, Th19;

          then

           A31: LE (p,p4,P,p1,p2) by A27, A26, JORDAN5C: 13;

          (p `1 ) = e by A2, A4, A7, GOBOARD7: 5;

          hence thesis by A3, A4, A11, A8, A31, A28;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:30

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real st P is_S-P_arc_joining (p1,p2) & (p1 `1 ) < e & p in P & (p `1 ) = e holds p is_Lin (P,p1,p2,e) or p is_Rin (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: P is_S-P_arc_joining (p1,p2) and

       A2: (p1 `1 ) < e and

       A3: p in P and

       A4: (p `1 ) = e;

      consider f be FinSequence of ( TOP-REAL 2) such that

       A5: f is being_S-Seq and

       A6: P = ( L~ f) and

       A7: p1 = (f /. 1) and

       A8: p2 = (f /. ( len f)) by A1, TOPREAL4:def 1;

      

       A9: P is_an_arc_of (p1,p2) by A1, TOPREAL4: 2;

      ( len f) >= 2 by A5, TOPREAL1:def 8;

      then

       A10: ( len f) > 1 by XXREAL_0: 2;

      

       A11: ( L~ f) = ( union { ( LSeg (f,i)) where i be Nat : 1 <= i & (i + 1) <= ( len f) }) by TOPREAL1:def 4;

      then

      consider Y be set such that

       A12: p in Y and

       A13: Y in { ( LSeg (f,i)) where i be Nat : 1 <= i & (i + 1) <= ( len f) } by A3, A6, TARSKI:def 4;

      consider i be Nat such that

       A14: Y = ( LSeg (f,i)) and

       A15: 1 <= i and

       A16: (i + 1) <= ( len f) by A13;

      

       A17: (i - 1) >= 0 by A15, XREAL_1: 48;

      

       A18: 1 < (i + 1) by A15, NAT_1: 13;

      

       A19: Y c= ( L~ f) by A11, A13, TARSKI:def 4;

      defpred P[ Nat] means for p be Point of ( TOP-REAL 2) st p = (f . (i -' $1)) holds (p `1 ) <> e;

      

       A20: i < ( len f) by A16, NAT_1: 13;

      then

       A21: i in ( dom f) by A15, FINSEQ_3: 25;

      

       A22: 1 < ( len f) by A15, A20, XXREAL_0: 2;

      then 1 in ( dom f) by FINSEQ_3: 25;

      then (f /. 1) = (f . 1) by PARTFUN1:def 6;

      then

       A23: P[(i -' 1)] by A2, A7, A15, NAT_D: 58;

      then

       A24: ex k be Nat st P[k];

      ex k be Nat st P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5( A24);

      then

      consider k be Nat such that

       A25: P[k] and

       A26: for n be Nat st P[n] holds k <= n;

      k <= (i -' 1) by A23, A26;

      then k <= (i - 1) by A17, XREAL_0:def 2;

      then (k + 1) <= ((i - 1) + 1) by XREAL_1: 7;

      then

       A27: ((1 + k) - k) <= (i - k) by XREAL_1: 9;

      then

       A28: (i -' k) >= 1 by XREAL_0:def 2;

      (i -' k) <= i by NAT_D: 35;

      then

       A29: (i -' k) < ( len f) by A20, XXREAL_0: 2;

      then

       A30: (i -' k) in ( dom f) by A28, FINSEQ_3: 25;

      then

       A31: (f /. (i -' k)) = (f . (i -' k)) by PARTFUN1:def 6;

      then

      reconsider pk = (f . (i -' k)) as Point of ( TOP-REAL 2);

      

       A32: (i -' k) = (i - k) by A27, XREAL_0:def 2;

      now

        per cases by A25, XXREAL_0: 1;

          case

           A33: (pk `1 ) < e;

          now

            per cases ;

              case

               A34: k = 0 ;

              set p44 = (f /. i);

              

               A35: pk = (f . i) by A34, NAT_D: 40

              .= p44 by A21, PARTFUN1:def 6;

              reconsider ia = (i + 1) as Nat;

              reconsider g = ( mid (f,i,( len f))) as FinSequence of ( TOP-REAL 2);

              

               A36: i <= ( len f) by A16, NAT_1: 13;

              ia in ( Seg ( len f)) by A16, A18, FINSEQ_1: 1;

              then

               A37: (i + 1) in ( dom f) by FINSEQ_1:def 3;

              (1 + (1 + i)) <= (1 + ( len f)) by A16, XREAL_1: 7;

              then

               A38: (((1 + 1) + i) - i) <= ((( len f) + 1) - i) by XREAL_1: 9;

              then

               A39: 1 <= ((( len f) + 1) - i) by XXREAL_0: 2;

              

               A40: (( len f) - i) > 0 by A20, XREAL_1: 50;

              then (( len f) -' i) = (( len f) - i) by XREAL_0:def 2;

              then

               A41: ((( len f) -' i) + 1) > ( 0 + 1) by A40, XREAL_1: 8;

              

               A42: ( len g) = ((( len f) -' i) + 1) by A10, A15, A20, FINSEQ_6: 118;

              then

               A43: (1 + 1) <= ( len g) by A41, NAT_1: 13;

              then (1 + 1) in ( Seg ( len g)) by FINSEQ_1: 1;

              then (1 + 1) in ( dom g) by FINSEQ_1:def 3;

              

              then

               A44: (g /. (1 + 1)) = (g . (1 + 1)) by PARTFUN1:def 6

              .= (f . (((1 + 1) - 1) + i)) by A15, A20, A38, FINSEQ_6: 122

              .= (f /. (i + 1)) by A37, PARTFUN1:def 6;

              1 in ( dom g) by A42, A41, FINSEQ_3: 25;

              

              then

               A45: (g /. 1) = (g . 1) by PARTFUN1:def 6

              .= (f . ((1 - 1) + i)) by A15, A36, A39, FINSEQ_6: 122

              .= (f /. i) by A21, PARTFUN1:def 6;

              ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3

              .= ( LSeg (g,1)) by A43, A45, A44, TOPREAL1:def 3;

              then Y in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A14, A43;

              then p in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A12, TARSKI:def 4;

              then

               A46: p in ( L~ ( mid (f,i,( len f)))) by TOPREAL1:def 4;

              

               A47: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3;

              

               A48: for p5 be Point of ( TOP-REAL 2) st LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) <= e

              proof

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p44,p)) c= ( LSeg (f,i)) by A12, A14, A47, TOPREAL1: 6;

                then

                 A49: ( LSeg (p44,p)) c= P by A6, A19, A14;

                let p5 be Point of ( TOP-REAL 2);

                

                 A50: ( Segment (P,p1,p2,p44,p)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p44,p8,P,p1,p2) & LE (p8,p,P,p1,p2) } by JORDAN6: 26;

                assume LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2);

                then

                 A51: p5 in ( Segment (P,p1,p2,p44,p)) by A50;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p44,p)) is_an_arc_of (p44,p) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p44,p)) = ( LSeg (p44,p)) by A9, A5, A6, A7, A8, A15, A20, A46, A49, Th25, SPRECT_4: 3;

                    hence thesis by A4, A33, A35, A51, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    hence thesis by A4, A33, A35;

                  end;

                end;

                hence thesis;

              end;

               LE (p44,p,P,p1,p2) by A5, A6, A7, A8, A15, A20, A46, SPRECT_4: 3;

              hence thesis by A3, A4, A9, A33, A35, A48;

            end;

              case

               A52: k <> 0 ;

              reconsider ia = (i + 1) as Nat;

              reconsider g = ( mid (f,i,( len f))) as FinSequence of ( TOP-REAL 2);

              

               A53: i <= ( len f) by A16, NAT_1: 13;

              ia in ( Seg ( len f)) by A16, A18, FINSEQ_1: 1;

              then

               A54: (i + 1) in ( dom f) by FINSEQ_1:def 3;

              (1 + (1 + i)) <= (1 + ( len f)) by A16, XREAL_1: 7;

              then

               A55: (((1 + 1) + i) - i) <= ((( len f) + 1) - i) by XREAL_1: 9;

              then

               A56: 1 <= ((( len f) + 1) - i) by XXREAL_0: 2;

              

               A57: (( len f) - i) > 0 by A20, XREAL_1: 50;

              then (( len f) -' i) = (( len f) - i) by XREAL_0:def 2;

              then

               A58: ((( len f) -' i) + 1) > ( 0 + 1) by A57, XREAL_1: 8;

              

               A59: ( len g) = ((( len f) -' i) + 1) by A10, A15, A20, FINSEQ_6: 118;

              then

               A60: (1 + 1) <= ( len g) by A58, NAT_1: 13;

              then (1 + 1) in ( Seg ( len g)) by FINSEQ_1: 1;

              then (1 + 1) in ( dom g) by FINSEQ_1:def 3;

              

              then

               A61: (g /. (1 + 1)) = (g . (1 + 1)) by PARTFUN1:def 6

              .= (f . (((1 + 1) - 1) + i)) by A15, A20, A55, FINSEQ_6: 122

              .= (f /. (i + 1)) by A54, PARTFUN1:def 6;

              1 in ( dom g) by A59, A58, FINSEQ_3: 25;

              

              then

               A62: (g /. 1) = (g . 1) by PARTFUN1:def 6

              .= (f . ((1 - 1) + i)) by A15, A53, A56, FINSEQ_6: 122

              .= (f /. i) by A21, PARTFUN1:def 6;

              ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3

              .= ( LSeg (g,1)) by A60, A62, A61, TOPREAL1:def 3;

              then Y in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A14, A60;

              then p in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A12, TARSKI:def 4;

              then

               A63: p in ( L~ ( mid (f,i,( len f)))) by TOPREAL1:def 4;

              reconsider g = ( mid (f,1,i)) as FinSequence of ( TOP-REAL 2);

              set p44 = (f /. i);

              

               A64: i <= ( len f) & 1 <= (i -' k) by A16, A27, NAT_1: 13, XREAL_0:def 2;

              

               A65: k >= ( 0 + 1) by A52, NAT_1: 13;

              then

               A66: (i -' k) <= ((i + 1) - 1) by A28, NAT_D: 51;

              

               A67: i > (i -' k) by A28, A65, NAT_D: 51;

              then

               A68: i > 1 by A28, XXREAL_0: 2;

              then (i - 1) > 0 by XREAL_1: 50;

              then

               A69: (i -' 1) = (i - 1) by XREAL_0:def 2;

               A70:

              now

                assume

                 A71: ((f /. i) `1 ) <> e;

                (f . i) = (f /. i) by A21, PARTFUN1:def 6;

                then for p9 be Point of ( TOP-REAL 2) st p9 = (f . (i -' 0 )) holds (p9 `1 ) <> e by A71, NAT_D: 40;

                hence contradiction by A26, A52;

              end;

               A72:

              now

                assume not for p51 be Point of ( TOP-REAL 2) st LE (pk,p51,P,p1,p2) & LE (p51,p44,P,p1,p2) holds (p51 `1 ) <= e;

                then

                consider p51 be Point of ( TOP-REAL 2) such that

                 A73: LE (pk,p51,P,p1,p2) and

                 A74: LE (p51,p44,P,p1,p2) and

                 A75: (p51 `1 ) > e;

                p51 in P by A73, JORDAN5C:def 3;

                then

                consider Y3 be set such that

                 A76: p51 in Y3 and

                 A77: Y3 in { ( LSeg (f,i5)) where i5 be Nat : 1 <= i5 & (i5 + 1) <= ( len f) } by A6, A11, TARSKI:def 4;

                consider kk be Nat such that

                 A78: Y3 = ( LSeg (f,kk)) and

                 A79: 1 <= kk and

                 A80: (kk + 1) <= ( len f) by A77;

                

                 A81: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A79, A80, TOPREAL1:def 3;

                1 < (kk + 1) by A79, NAT_1: 13;

                then (kk + 1) in ( Seg ( len f)) by A80, FINSEQ_1: 1;

                then

                 A82: (kk + 1) in ( dom f) by FINSEQ_1:def 3;

                

                 A83: kk < ( len f) by A80, NAT_1: 13;

                then kk in ( Seg ( len f)) by A79, FINSEQ_1: 1;

                then

                 A84: kk in ( dom f) by FINSEQ_1:def 3;

                

                 A85: LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A76, A78, A79, A80, JORDAN5C: 26;

                now

                  per cases by A75, A76, A78, A81, Th2;

                    case

                     A86: ((f /. kk) `1 ) > e;

                    

                     A87: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A88: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A79, A80, A81;

                      hence thesis by A11, A88, TARSKI:def 4;

                    end;

                    f is special by A5, TOPREAL1:def 8;

                    then

                     A89: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A79, A80, TOPREAL1:def 5;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A90: (f . kk) <> (f . (kk + 1)) by A84, A82, FUNCT_1:def 4;

                    

                     A91: LE ((f /. (i -' k)),p51,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A30, A73, PARTFUN1:def 6;

                    

                     A92: LE ((f /. (i -' k)),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A73, A85, JORDAN5C: 13;

                    set k2 = (i -' kk);

                     LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A76, A78, A79, A80, JORDAN5C: 25;

                    then

                     A93: LE ((f /. kk),p44,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A74, JORDAN5C: 13;

                    now

                      assume (i - kk) <= 0 ;

                      then ((i - kk) + kk) <= ( 0 + kk) by XREAL_1: 7;

                      then LE ((f /. i),(f /. kk),( L~ f),(f /. 1),(f /. ( len f))) by A5, A68, A83, JORDAN5C: 24;

                      hence contradiction by A1, A6, A7, A8, A70, A86, A93, JORDAN5C: 12, TOPREAL4: 2;

                    end;

                    then

                     A94: (i -' kk) = (i - kk) by XREAL_0:def 2;

                    then

                     A95: (i - k2) = (i -' k2) by XREAL_0:def 2;

                    (i - k2) > 0 by A79, A94;

                    then (i -' k2) > 0 by XREAL_0:def 2;

                    then (i -' k2) >= ( 0 + 1) by NAT_1: 13;

                    then P[k2] by A20, A86, A94, A95, FINSEQ_4: 15, NAT_D: 50;

                    then k2 >= k by A26;

                    then (i - k2) <= (i - k) by XREAL_1: 10;

                    then

                     A96: LE ((f /. (i -' k2)),(f /. (i -' k)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A29, A32, A79, A94, A95, JORDAN5C: 24;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A84, A82, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A90, TOPREAL1: 9;

                    then

                     A97: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A6, A7, A8, A94, A95, A96, A92, A87, Th25, JORDAN5C: 13;

                    ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } by JORDAN6: 26;

                    then

                     A98: (f /. (i -' k)) in ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) by A94, A95, A96, A92;

                    then ((f /. (kk + 1)) `1 ) < e by A31, A33, A86, A97, Th3;

                    then ((f /. kk) `1 ) > ((f /. (kk + 1)) `1 ) by A86, XXREAL_0: 2;

                    then ((f /. (i -' k)) `1 ) >= (p51 `1 ) by A5, A76, A78, A79, A83, A81, A91, A98, A97, A89, Th6;

                    hence contradiction by A31, A33, A75, XXREAL_0: 2;

                  end;

                    case

                     A99: ((f /. (kk + 1)) `1 ) > e & ((f /. kk) `1 ) <= e;

                    set k2 = ((i -' kk) -' 1);

                    

                     A100: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A101: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A79, A80, A81;

                      hence thesis by A11, A101, TARSKI:def 4;

                    end;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A102: (f . kk) <> (f . (kk + 1)) by A84, A82, FUNCT_1:def 4;

                    

                     A103: ((f /. kk) `1 ) < ((f /. (kk + 1)) `1 ) by A99, XXREAL_0: 2;

                     LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A76, A78, A79, A80, JORDAN5C: 25;

                    then

                     A104: LE ((f /. kk),p44,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A74, JORDAN5C: 13;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A84, A82, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A102, TOPREAL1: 9;

                    then

                     A105: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } & ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A5, A6, A7, A8, A79, A80, A100, Th25, JORDAN5C: 23, JORDAN6: 26;

                     A106:

                    now

                      assume ((i - kk) - 1) <= 0 ;

                      then ((i - (kk + 1)) + (kk + 1)) <= ( 0 + (kk + 1)) by XREAL_1: 7;

                      then LE ((f /. i),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A68, A80, JORDAN5C: 24;

                      then

                       A107: (f /. i) in ( LSeg ((f /. kk),(f /. (kk + 1)))) by A105, A104;

                      f is special by A5, TOPREAL1:def 8;

                      then

                       A108: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A79, A80, TOPREAL1:def 5;

                      ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A79, A80, TOPREAL1:def 3;

                      hence contradiction by A5, A6, A7, A8, A70, A74, A75, A76, A78, A79, A83, A103, A107, A108, Th7;

                    end;

                    then (((i - kk) - 1) + 1) >= ( 0 + 1) by XREAL_1: 7;

                    then (i -' kk) = (i - kk) by XREAL_0:def 2;

                    

                    then

                     A109: (i - k2) = (i - ((i - kk) - 1)) by A106, XREAL_0:def 2

                    .= (kk + 1);

                    then (i -' k2) > 0 by XREAL_0:def 2;

                    then

                     A110: (i -' k2) >= ( 0 + 1) by NAT_1: 13;

                    

                     A111: (i - k2) = (i -' k2) by A109, XREAL_0:def 2;

                    then P[k2] by A20, A99, A109, A110, FINSEQ_4: 15, NAT_D: 50;

                    then k2 >= k by A26;

                    then (i - k2) <= (i - k) by XREAL_1: 10;

                    then

                     A112: LE ((f /. (kk + 1)),(f /. (i -' k)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A29, A32, A109, A111, A110, JORDAN5C: 24;

                     LE ((f /. (i -' k)),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A73, A85, JORDAN5C: 13;

                    hence contradiction by A1, A6, A7, A8, A31, A33, A99, A112, JORDAN5C: 12, TOPREAL4: 2;

                  end;

                end;

                hence contradiction;

              end;

              

               A113: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3;

              

               A114: for p5 be Point of ( TOP-REAL 2) st LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                

                 A115: ( Segment (P,p1,p2,p44,p)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p44,p8,P,p1,p2) & LE (p8,p,P,p1,p2) } by JORDAN6: 26;

                assume LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2);

                then

                 A116: p5 in ( Segment (P,p1,p2,p44,p)) by A115;

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p44,p)) c= ( LSeg (f,i)) by A12, A14, A113, TOPREAL1: 6;

                then

                 A117: ( LSeg (p44,p)) c= P by A6, A19, A14;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p44,p)) is_an_arc_of (p44,p) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p44,p)) = ( LSeg (p44,p)) by A9, A5, A6, A7, A8, A15, A20, A63, A117, Th25, SPRECT_4: 3;

                    hence thesis by A4, A70, A116, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    then ( Segment (P,p1,p2,p44,p)) = {p44} by A1, A3, Th1, TOPREAL4: 2;

                    hence thesis by A70, A116, TARSKI:def 1;

                  end;

                end;

                hence thesis;

              end;

              

               A118: ( len g) = ((i -' 1) + 1) by A15, A20, A22, FINSEQ_6: 118;

              then ((i -' k) + 1) <= ( len g) by A67, A69, NAT_1: 13;

              then

               A119: ( LSeg (g,(i -' k))) = ( LSeg ((g /. (i -' k)),(g /. ((i -' k) + 1)))) by A28, TOPREAL1:def 3;

              (i -' k) < i by A28, A65, NAT_D: 51;

              then (i -' k) in ( Seg ( len g)) by A28, A118, A69, FINSEQ_1: 1;

              then (i -' k) in ( dom g) by FINSEQ_1:def 3;

              

              then (g /. (i -' k)) = (g . (i -' k)) by PARTFUN1:def 6

              .= (f . (((i -' k) - 1) + 1)) by A15, A64, A66, FINSEQ_6: 122

              .= (f /. (i -' k)) by A30, PARTFUN1:def 6;

              then

               A120: pk in ( LSeg (g,(i -' k))) by A31, A119, RLTOPSP1: 68;

              

               A121: ((i -' k) + 1) <= i by A67, NAT_1: 13;

              1 <= (i -' k) by A27, XREAL_0:def 2;

              then ( LSeg (g,(i -' k))) in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A118, A69, A121;

              then pk in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A120, TARSKI:def 4;

              then pk in ( L~ ( mid (f,1,i))) by TOPREAL1:def 4;

              then

               A122: LE (pk,p44,P,p1,p2) by A5, A6, A7, A8, A20, A68, SPRECT_3: 17;

              then

               A123: p44 in P by JORDAN5C:def 3;

              

               A124: for p5 be Point of ( TOP-REAL 2) st LE (pk,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A125: LE (pk,p5,P,p1,p2) and

                 A126: LE (p5,p,P,p1,p2);

                

                 A127: p5 in P by A125, JORDAN5C:def 3;

                now

                  per cases by A1, A123, A127, Th19, TOPREAL4: 2;

                    case LE (p5,p44,P,p1,p2);

                    hence thesis by A72, A125;

                  end;

                    case LE (p44,p5,P,p1,p2);

                    hence thesis by A114, A126;

                  end;

                end;

                hence thesis;

              end;

               LE (p44,p,P,p1,p2) by A5, A6, A7, A8, A15, A20, A63, SPRECT_4: 3;

              then LE (pk,p,P,p1,p2) by A122, JORDAN5C: 13;

              hence thesis by A3, A4, A9, A33, A124;

            end;

          end;

          hence thesis;

        end;

          case

           A128: (pk `1 ) > e;

          now

            per cases ;

              case

               A129: k = 0 ;

              set p44 = (f /. i);

              

               A130: pk = (f . i) by A129, NAT_D: 40

              .= p44 by A21, PARTFUN1:def 6;

              reconsider ia = (i + 1) as Nat;

              reconsider g = ( mid (f,i,( len f))) as FinSequence of ( TOP-REAL 2);

              

               A131: i <= ( len f) by A16, NAT_1: 13;

              ia in ( Seg ( len f)) by A16, A18, FINSEQ_1: 1;

              then

               A132: (i + 1) in ( dom f) by FINSEQ_1:def 3;

              (1 + (1 + i)) <= (1 + ( len f)) by A16, XREAL_1: 7;

              then

               A133: (((1 + 1) + i) - i) <= ((( len f) + 1) - i) by XREAL_1: 9;

              then

               A134: 1 <= ((( len f) + 1) - i) by XXREAL_0: 2;

              

               A135: (( len f) - i) > 0 by A20, XREAL_1: 50;

              then (( len f) -' i) = (( len f) - i) by XREAL_0:def 2;

              then

               A136: ((( len f) -' i) + 1) > ( 0 + 1) by A135, XREAL_1: 8;

              

               A137: ( len g) = ((( len f) -' i) + 1) by A10, A15, A20, FINSEQ_6: 118;

              then

               A138: (1 + 1) <= ( len g) by A136, NAT_1: 13;

              then (1 + 1) in ( Seg ( len g)) by FINSEQ_1: 1;

              then (1 + 1) in ( dom g) by FINSEQ_1:def 3;

              

              then

               A139: (g /. (1 + 1)) = (g . (1 + 1)) by PARTFUN1:def 6

              .= (f . (((1 + 1) - 1) + i)) by A15, A20, A133, FINSEQ_6: 122

              .= (f /. (i + 1)) by A132, PARTFUN1:def 6;

              1 in ( Seg ( len g)) by A137, A136, FINSEQ_1: 1;

              then 1 in ( dom g) by FINSEQ_1:def 3;

              

              then

               A140: (g /. 1) = (g . 1) by PARTFUN1:def 6

              .= (f . ((1 - 1) + i)) by A15, A131, A134, FINSEQ_6: 122

              .= (f /. i) by A21, PARTFUN1:def 6;

              ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3

              .= ( LSeg (g,1)) by A138, A140, A139, TOPREAL1:def 3;

              then Y in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A14, A138;

              then p in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A12, TARSKI:def 4;

              then

               A141: p in ( L~ ( mid (f,i,( len f)))) by TOPREAL1:def 4;

              

               A142: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3;

              

               A143: for p5 be Point of ( TOP-REAL 2) st LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) >= e

              proof

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p44,p)) c= ( LSeg (f,i)) by A12, A14, A142, TOPREAL1: 6;

                then

                 A144: ( LSeg (p44,p)) c= P by A6, A19, A14;

                let p5 be Point of ( TOP-REAL 2);

                

                 A145: ( Segment (P,p1,p2,p44,p)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p44,p8,P,p1,p2) & LE (p8,p,P,p1,p2) } by JORDAN6: 26;

                assume LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2);

                then

                 A146: p5 in ( Segment (P,p1,p2,p44,p)) by A145;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p44,p)) is_an_arc_of (p44,p) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p44,p)) = ( LSeg (p44,p)) by A9, A5, A6, A7, A8, A15, A20, A141, A144, Th25, SPRECT_4: 3;

                    hence thesis by A4, A128, A130, A146, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    hence thesis by A4, A128, A130;

                  end;

                end;

                hence thesis;

              end;

               LE (p44,p,P,p1,p2) by A5, A6, A7, A8, A15, A20, A141, SPRECT_4: 3;

              hence thesis by A3, A4, A9, A128, A130, A143;

            end;

              case

               A147: k <> 0 ;

              reconsider ia = (i + 1) as Nat;

              reconsider g = ( mid (f,i,( len f))) as FinSequence of ( TOP-REAL 2);

              

               A148: i <= ( len f) by A16, NAT_1: 13;

              ia in ( Seg ( len f)) by A16, A18, FINSEQ_1: 1;

              then

               A149: (i + 1) in ( dom f) by FINSEQ_1:def 3;

              (1 + (1 + i)) <= (1 + ( len f)) by A16, XREAL_1: 7;

              then

               A150: (((1 + 1) + i) - i) <= ((( len f) + 1) - i) by XREAL_1: 9;

              then

               A151: 1 <= ((( len f) + 1) - i) by XXREAL_0: 2;

              

               A152: (( len f) - i) > 0 by A20, XREAL_1: 50;

              then (( len f) -' i) = (( len f) - i) by XREAL_0:def 2;

              then

               A153: ((( len f) -' i) + 1) > ( 0 + 1) by A152, XREAL_1: 8;

              

               A154: ( len g) = ((( len f) -' i) + 1) by A10, A15, A20, FINSEQ_6: 118;

              then

               A155: (1 + 1) <= ( len g) by A153, NAT_1: 13;

              then (1 + 1) in ( Seg ( len g)) by FINSEQ_1: 1;

              then (1 + 1) in ( dom g) by FINSEQ_1:def 3;

              

              then

               A156: (g /. (1 + 1)) = (g . (1 + 1)) by PARTFUN1:def 6

              .= (f . (((1 + 1) - 1) + i)) by A15, A20, A150, FINSEQ_6: 122

              .= (f /. (i + 1)) by A149, PARTFUN1:def 6;

              1 in ( Seg ( len g)) by A154, A153, FINSEQ_1: 1;

              then 1 in ( dom g) by FINSEQ_1:def 3;

              

              then

               A157: (g /. 1) = (g . 1) by PARTFUN1:def 6

              .= (f . ((1 - 1) + i)) by A15, A148, A151, FINSEQ_6: 122

              .= (f /. i) by A21, PARTFUN1:def 6;

              ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3

              .= ( LSeg (g,1)) by A155, A157, A156, TOPREAL1:def 3;

              then Y in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A14, A155;

              then p in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A12, TARSKI:def 4;

              then

               A158: p in ( L~ ( mid (f,i,( len f)))) by TOPREAL1:def 4;

              reconsider g = ( mid (f,1,i)) as FinSequence of ( TOP-REAL 2);

              set p44 = (f /. i);

              

               A159: i <= ( len f) & 1 <= (i -' k) by A16, A27, NAT_1: 13, XREAL_0:def 2;

              

               A160: k >= ( 0 + 1) by A147, NAT_1: 13;

              then

               A161: (i -' k) <= ((i + 1) - 1) by A28, NAT_D: 51;

              

               A162: i > (i -' k) by A28, A160, NAT_D: 51;

              then

               A163: i > 1 by A28, XXREAL_0: 2;

              then (i - 1) > 0 by XREAL_1: 50;

              then

               A164: (i -' 1) = (i - 1) by XREAL_0:def 2;

               A165:

              now

                assume

                 A166: ((f /. i) `1 ) <> e;

                (f . i) = (f /. i) by A21, PARTFUN1:def 6;

                then for p9 be Point of ( TOP-REAL 2) st p9 = (f . (i -' 0 )) holds (p9 `1 ) <> e by A166, NAT_D: 40;

                hence contradiction by A26, A147;

              end;

               A167:

              now

                assume not for p51 be Point of ( TOP-REAL 2) st LE (pk,p51,P,p1,p2) & LE (p51,p44,P,p1,p2) holds (p51 `1 ) >= e;

                then

                consider p51 be Point of ( TOP-REAL 2) such that

                 A168: LE (pk,p51,P,p1,p2) and

                 A169: LE (p51,p44,P,p1,p2) and

                 A170: (p51 `1 ) < e;

                p51 in P by A168, JORDAN5C:def 3;

                then

                consider Y3 be set such that

                 A171: p51 in Y3 and

                 A172: Y3 in { ( LSeg (f,i5)) where i5 be Nat : 1 <= i5 & (i5 + 1) <= ( len f) } by A6, A11, TARSKI:def 4;

                consider kk be Nat such that

                 A173: Y3 = ( LSeg (f,kk)) and

                 A174: 1 <= kk and

                 A175: (kk + 1) <= ( len f) by A172;

                

                 A176: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A174, A175, TOPREAL1:def 3;

                1 < (kk + 1) by A174, NAT_1: 13;

                then (kk + 1) in ( Seg ( len f)) by A175, FINSEQ_1: 1;

                then

                 A177: (kk + 1) in ( dom f) by FINSEQ_1:def 3;

                

                 A178: kk < ( len f) by A175, NAT_1: 13;

                then kk in ( Seg ( len f)) by A174, FINSEQ_1: 1;

                then

                 A179: kk in ( dom f) by FINSEQ_1:def 3;

                

                 A180: LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A171, A173, A174, A175, JORDAN5C: 26;

                now

                  per cases by A170, A171, A173, A176, Th3;

                    case

                     A181: ((f /. kk) `1 ) < e;

                    set k2 = (i -' kk);

                     LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A171, A173, A174, A175, JORDAN5C: 25;

                    then

                     A182: LE ((f /. kk),p44,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A169, JORDAN5C: 13;

                    now

                      assume (i - kk) <= 0 ;

                      then ((i - kk) + kk) <= ( 0 + kk) by XREAL_1: 7;

                      then LE ((f /. i),(f /. kk),( L~ f),(f /. 1),(f /. ( len f))) by A5, A163, A178, JORDAN5C: 24;

                      hence contradiction by A1, A6, A7, A8, A165, A181, A182, JORDAN5C: 12, TOPREAL4: 2;

                    end;

                    

                    then

                     A183: (i - k2) = (i - (i - kk)) by XREAL_0:def 2

                    .= kk;

                    then

                     A184: (i - k2) = (i -' k2) by XREAL_0:def 2;

                    then P[k2] by A20, A174, A181, A183, FINSEQ_4: 15, NAT_D: 50;

                    then k2 >= k by A26;

                    then (i - k2) <= (i - k) by XREAL_1: 10;

                    then

                     A185: LE ((f /. (i -' k2)),(f /. (i -' k)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A29, A32, A174, A183, A184, JORDAN5C: 24;

                    

                     A186: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A187: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A174, A175, A176;

                      hence thesis by A11, A187, TARSKI:def 4;

                    end;

                    f is special by A5, TOPREAL1:def 8;

                    then

                     A188: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A174, A175, TOPREAL1:def 5;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A189: (f . kk) <> (f . (kk + 1)) by A179, A177, FUNCT_1:def 4;

                    

                     A190: LE ((f /. (i -' k)),p51,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A30, A168, PARTFUN1:def 6;

                    

                     A191: LE ((f /. (i -' k)),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A168, A180, JORDAN5C: 13;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A179, A177, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A189, TOPREAL1: 9;

                    then

                     A192: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A6, A7, A8, A183, A184, A185, A191, A186, Th25, JORDAN5C: 13;

                    ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } by JORDAN6: 26;

                    then

                     A193: (f /. (i -' k)) in ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) by A183, A184, A185, A191;

                    then ((f /. (kk + 1)) `1 ) > e by A31, A128, A181, A192, Th2;

                    then ((f /. kk) `1 ) < ((f /. (kk + 1)) `1 ) by A181, XXREAL_0: 2;

                    then ((f /. (i -' k)) `1 ) <= (p51 `1 ) by A5, A171, A173, A174, A178, A176, A190, A193, A192, A188, Th7;

                    hence contradiction by A31, A128, A170, XXREAL_0: 2;

                  end;

                    case

                     A194: ((f /. (kk + 1)) `1 ) < e & ((f /. kk) `1 ) >= e;

                    set k2 = ((i -' kk) -' 1);

                    

                     A195: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A196: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A174, A175, A176;

                      hence thesis by A11, A196, TARSKI:def 4;

                    end;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A197: (f . kk) <> (f . (kk + 1)) by A179, A177, FUNCT_1:def 4;

                    

                     A198: ((f /. kk) `1 ) > ((f /. (kk + 1)) `1 ) by A194, XXREAL_0: 2;

                     LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A171, A173, A174, A175, JORDAN5C: 25;

                    then

                     A199: LE ((f /. kk),p44,( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A169, JORDAN5C: 13;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A179, A177, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A197, TOPREAL1: 9;

                    then

                     A200: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } & ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A5, A6, A7, A8, A174, A175, A195, Th25, JORDAN5C: 23, JORDAN6: 26;

                     A201:

                    now

                      assume ((i - kk) - 1) <= 0 ;

                      then ((i - (kk + 1)) + (kk + 1)) <= ( 0 + (kk + 1)) by XREAL_1: 7;

                      then LE ((f /. i),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A163, A175, JORDAN5C: 24;

                      then

                       A202: (f /. i) in ( LSeg ((f /. kk),(f /. (kk + 1)))) by A200, A199;

                      f is special by A5, TOPREAL1:def 8;

                      then

                       A203: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A174, A175, TOPREAL1:def 5;

                      ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A174, A175, TOPREAL1:def 3;

                      hence contradiction by A5, A6, A7, A8, A165, A169, A170, A171, A173, A174, A178, A198, A202, A203, Th6;

                    end;

                    then (((i - kk) - 1) + 1) >= ( 0 + 1) by XREAL_1: 7;

                    then (i -' kk) = (i - kk) by XREAL_0:def 2;

                    

                    then

                     A204: (i - k2) = (i - ((i - kk) - 1)) by A201, XREAL_0:def 2

                    .= (kk + 1);

                    then (i -' k2) > 0 by XREAL_0:def 2;

                    then

                     A205: (i -' k2) >= ( 0 + 1) by NAT_1: 13;

                    

                     A206: (i - k2) = (i -' k2) by A204, XREAL_0:def 2;

                    then P[k2] by A20, A194, A204, A205, FINSEQ_4: 15, NAT_D: 50;

                    then k2 >= k by A26;

                    then (i - k2) <= (i - k) by XREAL_1: 10;

                    then

                     A207: LE ((f /. (kk + 1)),(f /. (i -' k)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A29, A32, A204, A206, A205, JORDAN5C: 24;

                     LE ((f /. (i -' k)),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A168, A180, JORDAN5C: 13;

                    hence contradiction by A1, A6, A7, A8, A31, A128, A194, A207, JORDAN5C: 12, TOPREAL4: 2;

                  end;

                end;

                hence contradiction;

              end;

              

               A208: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A15, A16, TOPREAL1:def 3;

              

               A209: for p5 be Point of ( TOP-REAL 2) st LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                

                 A210: ( Segment (P,p1,p2,p44,p)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p44,p8,P,p1,p2) & LE (p8,p,P,p1,p2) } by JORDAN6: 26;

                assume LE (p44,p5,P,p1,p2) & LE (p5,p,P,p1,p2);

                then

                 A211: p5 in ( Segment (P,p1,p2,p44,p)) by A210;

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p44,p)) c= ( LSeg (f,i)) by A12, A14, A208, TOPREAL1: 6;

                then

                 A212: ( LSeg (p44,p)) c= P by A6, A19, A14;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p44,p)) is_an_arc_of (p44,p) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p44,p)) = ( LSeg (p44,p)) by A9, A5, A6, A7, A8, A15, A20, A158, A212, Th25, SPRECT_4: 3;

                    hence thesis by A4, A165, A211, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    then ( Segment (P,p1,p2,p44,p)) = {p44} by A1, A3, Th1, TOPREAL4: 2;

                    hence thesis by A165, A211, TARSKI:def 1;

                  end;

                end;

                hence thesis;

              end;

              

               A213: ( len g) = ((i -' 1) + 1) by A15, A20, A22, FINSEQ_6: 118;

              then ((i -' k) + 1) <= ( len g) by A162, A164, NAT_1: 13;

              then

               A214: ( LSeg (g,(i -' k))) = ( LSeg ((g /. (i -' k)),(g /. ((i -' k) + 1)))) by A28, TOPREAL1:def 3;

              (i -' k) < i by A28, A160, NAT_D: 51;

              then (i -' k) in ( Seg ( len g)) by A28, A213, A164, FINSEQ_1: 1;

              then (i -' k) in ( dom g) by FINSEQ_1:def 3;

              

              then (g /. (i -' k)) = (g . (i -' k)) by PARTFUN1:def 6

              .= (f . (((i -' k) - 1) + 1)) by A15, A159, A161, FINSEQ_6: 122

              .= (f /. (i -' k)) by A30, PARTFUN1:def 6;

              then

               A215: pk in ( LSeg (g,(i -' k))) by A31, A214, RLTOPSP1: 68;

              

               A216: ((i -' k) + 1) <= i by A162, NAT_1: 13;

              1 <= (i -' k) by A27, XREAL_0:def 2;

              then ( LSeg (g,(i -' k))) in { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) } by A213, A164, A216;

              then pk in ( union { ( LSeg (g,i2)) where i2 be Nat : 1 <= i2 & (i2 + 1) <= ( len g) }) by A215, TARSKI:def 4;

              then pk in ( L~ ( mid (f,1,i))) by TOPREAL1:def 4;

              then

               A217: LE (pk,p44,P,p1,p2) by A5, A6, A7, A8, A20, A163, SPRECT_3: 17;

              then

               A218: p44 in P by JORDAN5C:def 3;

              

               A219: for p5 be Point of ( TOP-REAL 2) st LE (pk,p5,P,p1,p2) & LE (p5,p,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A220: LE (pk,p5,P,p1,p2) and

                 A221: LE (p5,p,P,p1,p2);

                

                 A222: p5 in P by A220, JORDAN5C:def 3;

                now

                  per cases by A1, A218, A222, Th19, TOPREAL4: 2;

                    case LE (p5,p44,P,p1,p2);

                    hence thesis by A167, A220;

                  end;

                    case LE (p44,p5,P,p1,p2);

                    hence thesis by A209, A221;

                  end;

                end;

                hence thesis;

              end;

               LE (p44,p,P,p1,p2) by A5, A6, A7, A8, A15, A20, A158, SPRECT_4: 3;

              then LE (pk,p,P,p1,p2) by A217, JORDAN5C: 13;

              hence thesis by A3, A4, A9, A128, A219;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN20:31

    for P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real st P is_S-P_arc_joining (p1,p2) & (p2 `1 ) > e & p in P & (p `1 ) = e holds p is_Lout (P,p1,p2,e) or p is_Rout (P,p1,p2,e)

    proof

      let P be non empty Subset of ( TOP-REAL 2), p1,p2,p be Point of ( TOP-REAL 2), e be Real;

      assume that

       A1: P is_S-P_arc_joining (p1,p2) and

       A2: (p2 `1 ) > e and

       A3: p in P and

       A4: (p `1 ) = e;

      consider f be FinSequence of ( TOP-REAL 2) such that

       A5: f is being_S-Seq and

       A6: P = ( L~ f) and

       A7: p1 = (f /. 1) and

       A8: p2 = (f /. ( len f)) by A1, TOPREAL4:def 1;

      

       A9: P is_an_arc_of (p1,p2) by A1, TOPREAL4: 2;

      

       A10: ( L~ f) = ( union { ( LSeg (f,i)) where i be Nat : 1 <= i & (i + 1) <= ( len f) }) by TOPREAL1:def 4;

      then

      consider Y be set such that

       A11: p in Y and

       A12: Y in { ( LSeg (f,i)) where i be Nat : 1 <= i & (i + 1) <= ( len f) } by A3, A6, TARSKI:def 4;

      consider i be Nat such that

       A13: Y = ( LSeg (f,i)) and

       A14: 1 <= i and

       A15: (i + 1) <= ( len f) by A12;

      

       A16: 1 < (i + 1) by A14, NAT_1: 13;

      

       A17: 1 < (i + 1) by A14, NAT_1: 13;

      then (i + 1) in ( Seg ( len f)) by A15, FINSEQ_1: 1;

      then

       A18: (i + 1) in ( dom f) by FINSEQ_1:def 3;

      

       A19: Y c= ( L~ f) by A10, A12, TARSKI:def 4;

      defpred P[ Nat] means for p be Point of ( TOP-REAL 2) st p = (f . ((i + 1) + $1)) holds (p `1 ) <> e;

      

       A20: (( len f) - (i + 1)) >= 0 by A15, XREAL_1: 48;

      

      then

       A21: ((i + 1) + (( len f) -' (i + 1))) = ((i + 1) + (( len f) - (i + 1))) by XREAL_0:def 2

      .= ( len f);

      

       A22: (( len f) -' (i + 1)) = (( len f) - (i + 1)) by A20, XREAL_0:def 2;

      

       A23: i < ( len f) by A15, NAT_1: 13;

      then 1 < ( len f) by A14, XXREAL_0: 2;

      then ( len f) in ( Seg ( len f)) by FINSEQ_1: 1;

      then ( len f) in ( dom f) by FINSEQ_1:def 3;

      then

       A24: P[(( len f) -' (i + 1))] by A2, A8, A21, PARTFUN1:def 6;

      then

       A25: ex k be Nat st P[k];

      ex k be Nat st P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5( A25);

      then

      consider k be Nat such that

       A26: P[k] and

       A27: for n be Nat st P[n] holds k <= n;

      k <= (( len f) -' (i + 1)) by A24, A27;

      then

       A28: (k + (i + 1)) <= ((( len f) - (i + 1)) + (i + 1)) by A22, XREAL_1: 7;

      (i + k) >= i by NAT_1: 11;

      then

       A29: ((i + k) + 1) >= (i + 1) by XREAL_1: 7;

      then

       A30: ((i + 1) + k) > 1 by A16, XXREAL_0: 2;

      1 <= ((i + 1) + k) by A17, NAT_1: 12;

      then ((i + 1) + k) in ( Seg ( len f)) by A28, FINSEQ_1: 1;

      then

       A31: ((i + 1) + k) in ( dom f) by FINSEQ_1:def 3;

      then

       A32: (f /. ((i + 1) + k)) = (f . ((i + 1) + k)) by PARTFUN1:def 6;

      then

      reconsider pk = (f . ((i + 1) + k)) as Point of ( TOP-REAL 2);

      

       A33: ((k + i) + 1) > 1 by A16, A29, XXREAL_0: 2;

      now

        per cases by A26, XXREAL_0: 1;

          case

           A34: (pk `1 ) < e;

          now

            per cases ;

              case

               A35: k = 0 ;

              set p44 = (f /. (i + 1));

              

               A36: pk = p44 by A18, A35, PARTFUN1:def 6;

              

               A37: p44 in ( LSeg (p,(f /. (i + 1)))) by RLTOPSP1: 68;

              

               A38: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, TOPREAL1:def 3;

              

               A39: for p5 be Point of ( TOP-REAL 2) st LE (p5,p44,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) <= e

              proof

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p,p44)) c= ( LSeg (f,i)) by A11, A13, A38, TOPREAL1: 6;

                then

                 A40: ( LSeg (p,p44)) c= P by A6, A19, A13;

                let p5 be Point of ( TOP-REAL 2);

                

                 A41: ( Segment (P,p1,p2,p,p44)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p,p8,P,p1,p2) & LE (p8,p44,P,p1,p2) } by JORDAN6: 26;

                assume LE (p5,p44,P,p1,p2) & LE (p,p5,P,p1,p2);

                then

                 A42: p5 in ( Segment (P,p1,p2,p,p44)) by A41;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p,p44)) is_an_arc_of (p,p44) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p,p44)) = ( LSeg (p,p44)) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A37, A40, Th25, SPRECT_4: 4;

                    hence thesis by A4, A34, A36, A42, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    hence thesis by A4, A18, A34, A35, PARTFUN1:def 6;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p44,P,p1,p2) by A5, A6, A7, A8, A11, A13, A14, A23, A37, SPRECT_4: 4;

              hence thesis by A3, A4, A9, A34, A36, A39;

            end;

              case

               A43: k <> 0 ;

              set p44 = (f /. (i + 1));

               A44:

              now

                assume ((f /. (i + 1)) `1 ) <> e;

                then for p9 be Point of ( TOP-REAL 2) st p9 = (f . ((i + 1) + 0 )) holds (p9 `1 ) <> e by A18, PARTFUN1:def 6;

                hence contradiction by A27, A43;

              end;

              

               A45: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, TOPREAL1:def 3;

               A46:

              now

                assume not for p51 be Point of ( TOP-REAL 2) st LE (p44,p51,P,p1,p2) & LE (p51,pk,P,p1,p2) holds (p51 `1 ) <= e;

                then

                consider p51 be Point of ( TOP-REAL 2) such that

                 A47: LE (p44,p51,P,p1,p2) and

                 A48: LE (p51,pk,P,p1,p2) and

                 A49: (p51 `1 ) > e;

                p51 in P by A47, JORDAN5C:def 3;

                then

                consider Y3 be set such that

                 A50: p51 in Y3 and

                 A51: Y3 in { ( LSeg (f,i5)) where i5 be Nat : 1 <= i5 & (i5 + 1) <= ( len f) } by A6, A10, TARSKI:def 4;

                consider kk be Nat such that

                 A52: Y3 = ( LSeg (f,kk)) and

                 A53: 1 <= kk and

                 A54: (kk + 1) <= ( len f) by A51;

                

                 A55: LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A50, A52, A53, A54, JORDAN5C: 26;

                

                 A56: LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A50, A52, A53, A54, JORDAN5C: 25;

                

                 A57: (kk - 1) >= 0 by A53, XREAL_1: 48;

                

                 A58: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A53, A54, TOPREAL1:def 3;

                

                 A59: kk < ( len f) by A54, NAT_1: 13;

                then

                 A60: kk in ( dom f) by A53, FINSEQ_3: 25;

                then

                 A61: (f /. kk) = (f . kk) by PARTFUN1:def 6;

                

                 A62: 1 < (kk + 1) by A53, NAT_1: 13;

                then

                 A63: (kk + 1) in ( dom f) by A54, FINSEQ_3: 25;

                f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                then

                 A64: (f . kk) <> (f . (kk + 1)) by A60, A63, FUNCT_1:def 4;

                now

                  per cases by A49, A50, A52, A58, Th2;

                    case

                     A65: ((f /. (kk + 1)) `1 ) > e;

                    set k2 = (kk -' i);

                    

                     A66: LE (p44,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A47, A55, JORDAN5C: 13;

                    now

                      assume (kk - i) < 0 ;

                      then ((kk - i) + i) < ( 0 + i) by XREAL_1: 6;

                      then LE ((f /. (kk + 1)),(f /. (i + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A15, A62, JORDAN5C: 24, XREAL_1: 7;

                      hence contradiction by A1, A6, A7, A8, A44, A65, A66, JORDAN5C: 12, TOPREAL4: 2;

                    end;

                    

                    then

                     A67: ((i + 1) + k2) = ((1 + i) + (kk - i)) by XREAL_0:def 2

                    .= (kk + 1);

                    

                     A68: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A69: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A53, A54, A58;

                      hence thesis by A10, A69, TARSKI:def 4;

                    end;

                    f is special by A5, TOPREAL1:def 8;

                    then

                     A70: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A53, A54, TOPREAL1:def 5;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A71: (f . kk) <> (f . (kk + 1)) by A60, A63, FUNCT_1:def 4;

                    

                     A72: LE (p51,(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A48, PARTFUN1:def 6;

                    

                     A73: LE ((f /. kk),(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A32, A48, A56, JORDAN5C: 13;

                    1 < (kk + 1) by A53, NAT_1: 13;

                    then P[k2] by A54, A65, A67, FINSEQ_4: 15;

                    then k2 >= k by A27;

                    then

                     A74: LE ((f /. ((i + 1) + k)),(f /. ((i + 1) + k2)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A33, A54, A67, JORDAN5C: 24, XREAL_1: 7;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A60, A63, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A71, TOPREAL1: 9;

                    then

                     A75: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A6, A7, A8, A67, A74, A73, A68, Th25, JORDAN5C: 13;

                    ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } by JORDAN6: 26;

                    then

                     A76: (f /. ((i + 1) + k)) in ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) by A67, A74, A73;

                    then ((f /. kk) `1 ) < e by A32, A34, A65, A75, Th3;

                    then ((f /. kk) `1 ) < ((f /. (kk + 1)) `1 ) by A65, XXREAL_0: 2;

                    then ((f /. ((i + 1) + k)) `1 ) >= (p51 `1 ) by A5, A50, A52, A53, A59, A58, A72, A76, A75, A70, Th7;

                    hence contradiction by A32, A34, A49, XXREAL_0: 2;

                  end;

                    case

                     A77: ((f /. kk) `1 ) > e & ((f /. (kk + 1)) `1 ) <= e;

                    set k2 = (kk -' (i + 1));

                    

                     A78: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A79: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A53, A54, A58;

                      hence thesis by A10, A79, TARSKI:def 4;

                    end;

                     LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A50, A52, A53, A54, JORDAN5C: 26;

                    then

                     A80: LE (p44,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A47, JORDAN5C: 13;

                    (f /. (kk + 1)) = (f . (kk + 1)) by A63, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A64, A61, TOPREAL1: 9;

                    then

                     A81: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } & ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A5, A6, A7, A8, A53, A54, A78, Th25, JORDAN5C: 23, JORDAN6: 26;

                     A82:

                    now

                      assume (kk - (i + 1)) < 0 ;

                      then ((kk - (i + 1)) + (i + 1)) < ( 0 + (i + 1)) by XREAL_1: 6;

                      then kk <= i by NAT_1: 13;

                      then

                       A83: LE ((f /. kk),(f /. i),( L~ f),(f /. 1),(f /. ( len f))) by A5, A23, A53, JORDAN5C: 24;

                      

                       A84: (f /. i) in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                       LE ((f /. i),p44,( L~ f),(f /. 1),(f /. ( len f))) by A5, A14, A15, JORDAN5C: 23;

                      then LE ((f /. i),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A80, JORDAN5C: 13;

                      then (f /. i) in ( LSeg ((f /. kk),(f /. (kk + 1)))) by A81, A83;

                      then (( LSeg ((f /. kk),(f /. (kk + 1)))) /\ ( LSeg ((f /. i),(f /. (i + 1))))) <> {} by A84, XBOOLE_0:def 4;

                      then

                       A85: not ( LSeg ((f /. kk),(f /. (kk + 1)))) misses ( LSeg ((f /. i),(f /. (i + 1)))) by XBOOLE_0:def 7;

                      

                       A86: (kk - 1) = (kk -' 1) by A57, XREAL_0:def 2;

                       A87:

                      now

                        assume

                         A88: i = ((kk -' 1) + 2);

                        then (kk + 1) < (i + 1) by A86, NAT_1: 13;

                        then LE ((f /. (kk + 1)),p44,( L~ f),(f /. 1),(f /. ( len f))) by A5, A15, A62, JORDAN5C: 24;

                        then p44 = (f /. (kk + 1)) by A1, A6, A7, A8, A80, JORDAN5C: 12, TOPREAL4: 2;

                        then (f . (i + 1)) = (f /. (kk + 1)) by A18, PARTFUN1:def 6;

                        then

                         A89: (f . (i + 1)) = (f . (kk + 1)) by A63, PARTFUN1:def 6;

                        f is one-to-one by A5, TOPREAL1:def 8;

                        then (i + 1) = (kk + 1) by A18, A63, A89, FUNCT_1:def 4;

                        hence contradiction by A86, A88;

                      end;

                      

                       A90: f is s.n.c. by A5, TOPREAL1:def 8;

                      

                       A91: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) & ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, A53, A54, TOPREAL1:def 3;

                      then (i + 1) >= kk by A85, A90, TOPREAL1:def 7;

                      then

                       A92: ((i + 1) - 1) >= (kk - 1) by XREAL_1: 9;

                      (kk + 1) >= i by A85, A91, A90, TOPREAL1:def 7;

                      then

                       A93: i = ((kk -' 1) + 0 ) or ... or i = ((kk -' 1) + 2) by A86, A92, NAT_1: 62;

                       A94:

                      now

                        per cases by A86, A93, A87;

                          case i = kk;

                          hence p44 in ( LSeg (f,kk)) by A45, RLTOPSP1: 68;

                        end;

                          case i = (kk - 1);

                          hence p44 in ( LSeg (f,kk)) by A58, RLTOPSP1: 68;

                        end;

                      end;

                      f is special by A5, TOPREAL1:def 8;

                      then ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A53, A54, TOPREAL1:def 5;

                      hence contradiction by A5, A6, A7, A8, A44, A47, A49, A50, A52, A53, A59, A77, A94, Th6;

                    end;

                    

                    then ((i + 1) + k2) = ((i + 1) + (kk - (i + 1))) by XREAL_0:def 2

                    .= kk;

                    then P[k2] by A53, A59, A77, FINSEQ_4: 15;

                    then

                     A95: k2 >= k by A27;

                    (kk -' (i + 1)) = (kk - (i + 1)) by A82, XREAL_0:def 2;

                    then ((kk - (i + 1)) + (i + 1)) >= (k + (i + 1)) by A95, XREAL_1: 7;

                    then

                     A96: LE ((f /. ((i + 1) + k)),(f /. kk),( L~ f),(f /. 1),(f /. ( len f))) by A5, A30, A59, JORDAN5C: 24;

                     LE ((f /. kk),(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A32, A48, A56, JORDAN5C: 13;

                    hence contradiction by A1, A6, A7, A8, A32, A34, A77, A96, JORDAN5C: 12, TOPREAL4: 2;

                  end;

                end;

                hence contradiction;

              end;

              

               A97: p44 in ( LSeg (p,(f /. (i + 1)))) by RLTOPSP1: 68;

              

               A98: for p5 be Point of ( TOP-REAL 2) st LE (p,p5,P,p1,p2) & LE (p5,p44,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                

                 A99: ( Segment (P,p1,p2,p,p44)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p,p8,P,p1,p2) & LE (p8,p44,P,p1,p2) } by JORDAN6: 26;

                assume LE (p,p5,P,p1,p2) & LE (p5,p44,P,p1,p2);

                then

                 A100: p5 in ( Segment (P,p1,p2,p,p44)) by A99;

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p,p44)) c= ( LSeg (f,i)) by A11, A13, A45, TOPREAL1: 6;

                then

                 A101: ( LSeg (p,p44)) c= P by A6, A19, A13;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p,p44)) is_an_arc_of (p,p44) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p,p44)) = ( LSeg (p,p44)) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A97, A101, Th25, SPRECT_4: 4;

                    hence thesis by A4, A44, A100, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    then ( Segment (P,p1,p2,p,p44)) = {p44} by A1, A3, Th1, TOPREAL4: 2;

                    hence thesis by A44, A100, TARSKI:def 1;

                  end;

                end;

                hence thesis;

              end;

              (i + 1) <= ((i + 1) + k) by NAT_1: 11;

              then

               A102: LE (p44,pk,P,p1,p2) by A5, A6, A7, A8, A17, A28, A32, JORDAN5C: 24;

              then

               A103: p44 in P by JORDAN5C:def 3;

              

               A104: for p5 be Point of ( TOP-REAL 2) st LE (p5,pk,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) <= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A105: LE (p5,pk,P,p1,p2) and

                 A106: LE (p,p5,P,p1,p2);

                

                 A107: p5 in P by A105, JORDAN5C:def 3;

                now

                  per cases by A1, A103, A107, Th19, TOPREAL4: 2;

                    case LE (p44,p5,P,p1,p2);

                    hence thesis by A46, A105;

                  end;

                    case LE (p5,p44,P,p1,p2);

                    hence thesis by A98, A106;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p44,P,p1,p2) by A5, A6, A7, A8, A11, A13, A14, A23, A97, SPRECT_4: 4;

              then LE (p,pk,P,p1,p2) by A102, JORDAN5C: 13;

              hence thesis by A3, A4, A9, A34, A104;

            end;

          end;

          hence thesis;

        end;

          case

           A108: (pk `1 ) > e;

          now

            per cases ;

              case

               A109: k = 0 ;

              set p44 = (f /. (i + 1));

              

               A110: pk = p44 by A18, A109, PARTFUN1:def 6;

              

               A111: p44 in ( LSeg (p,(f /. (i + 1)))) by RLTOPSP1: 68;

              

               A112: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, TOPREAL1:def 3;

              

               A113: for p5 be Point of ( TOP-REAL 2) st LE (p5,p44,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) >= e

              proof

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p,p44)) c= ( LSeg (f,i)) by A11, A13, A112, TOPREAL1: 6;

                then

                 A114: ( LSeg (p,p44)) c= P by A6, A19, A13;

                let p5 be Point of ( TOP-REAL 2);

                

                 A115: ( Segment (P,p1,p2,p,p44)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p,p8,P,p1,p2) & LE (p8,p44,P,p1,p2) } by JORDAN6: 26;

                assume LE (p5,p44,P,p1,p2) & LE (p,p5,P,p1,p2);

                then

                 A116: p5 in ( Segment (P,p1,p2,p,p44)) by A115;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p,p44)) is_an_arc_of (p,p44) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p,p44)) = ( LSeg (p,p44)) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A111, A114, Th25, SPRECT_4: 4;

                    hence thesis by A4, A108, A110, A116, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    hence thesis by A4, A18, A108, A109, PARTFUN1:def 6;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p44,P,p1,p2) by A5, A6, A7, A8, A11, A13, A14, A23, A111, SPRECT_4: 4;

              hence thesis by A3, A4, A9, A108, A110, A113;

            end;

              case

               A117: k <> 0 ;

              set p44 = (f /. (i + 1));

               A118:

              now

                assume ((f /. (i + 1)) `1 ) <> e;

                then for p9 be Point of ( TOP-REAL 2) st p9 = (f . ((i + 1) + 0 )) holds (p9 `1 ) <> e by A18, PARTFUN1:def 6;

                hence contradiction by A27, A117;

              end;

              

               A119: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, TOPREAL1:def 3;

               A120:

              now

                assume not for p51 be Point of ( TOP-REAL 2) st LE (p44,p51,P,p1,p2) & LE (p51,pk,P,p1,p2) holds (p51 `1 ) >= e;

                then

                consider p51 be Point of ( TOP-REAL 2) such that

                 A121: LE (p44,p51,P,p1,p2) and

                 A122: LE (p51,pk,P,p1,p2) and

                 A123: (p51 `1 ) < e;

                p51 in P by A121, JORDAN5C:def 3;

                then

                consider Y3 be set such that

                 A124: p51 in Y3 and

                 A125: Y3 in { ( LSeg (f,i5)) where i5 be Nat : 1 <= i5 & (i5 + 1) <= ( len f) } by A6, A10, TARSKI:def 4;

                consider kk be Nat such that

                 A126: Y3 = ( LSeg (f,kk)) and

                 A127: 1 <= kk and

                 A128: (kk + 1) <= ( len f) by A125;

                

                 A129: LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A124, A126, A127, A128, JORDAN5C: 26;

                

                 A130: LE ((f /. kk),p51,( L~ f),(f /. 1),(f /. ( len f))) by A5, A124, A126, A127, A128, JORDAN5C: 25;

                

                 A131: (kk - 1) >= 0 by A127, XREAL_1: 48;

                

                 A132: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A127, A128, TOPREAL1:def 3;

                

                 A133: kk < ( len f) by A128, NAT_1: 13;

                then kk in ( Seg ( len f)) by A127, FINSEQ_1: 1;

                then

                 A134: kk in ( dom f) by FINSEQ_1:def 3;

                then

                 A135: (f /. kk) = (f . kk) by PARTFUN1:def 6;

                

                 A136: 1 < (kk + 1) by A127, NAT_1: 13;

                then

                 A137: (kk + 1) in ( dom f) by A128, FINSEQ_3: 25;

                f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                then

                 A138: (f . kk) <> (f . (kk + 1)) by A134, A137, FUNCT_1:def 4;

                now

                  per cases by A123, A124, A126, A132, Th3;

                    case

                     A139: ((f /. (kk + 1)) `1 ) < e;

                    set k2 = (kk -' i);

                    

                     A140: LE (p44,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A121, A129, JORDAN5C: 13;

                    now

                      assume (kk - i) < 0 ;

                      then ((kk - i) + i) < ( 0 + i) by XREAL_1: 6;

                      then LE ((f /. (kk + 1)),(f /. (i + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A15, A136, JORDAN5C: 24, XREAL_1: 7;

                      hence contradiction by A1, A6, A7, A8, A118, A139, A140, JORDAN5C: 12, TOPREAL4: 2;

                    end;

                    

                    then

                     A141: ((i + 1) + k2) = ((1 + i) + (kk - i)) by XREAL_0:def 2

                    .= (kk + 1);

                    

                     A142: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A143: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A127, A128, A132;

                      hence thesis by A10, A143, TARSKI:def 4;

                    end;

                    f is special by A5, TOPREAL1:def 8;

                    then

                     A144: ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A127, A128, TOPREAL1:def 5;

                    f is one-to-one & kk < (kk + 1) by A5, NAT_1: 13, TOPREAL1:def 8;

                    then

                     A145: (f . kk) <> (f . (kk + 1)) by A134, A137, FUNCT_1:def 4;

                    

                     A146: LE (p51,(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A31, A122, PARTFUN1:def 6;

                    

                     A147: LE ((f /. kk),(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A32, A122, A130, JORDAN5C: 13;

                    1 < (kk + 1) by A127, NAT_1: 13;

                    then P[k2] by A128, A139, A141, FINSEQ_4: 15;

                    then k2 >= k by A27;

                    then

                     A148: LE ((f /. ((i + 1) + k)),(f /. ((i + 1) + k2)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A33, A128, A141, JORDAN5C: 24, XREAL_1: 7;

                    (f /. kk) = (f . kk) & (f /. (kk + 1)) = (f . (kk + 1)) by A134, A137, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A145, TOPREAL1: 9;

                    then

                     A149: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A6, A7, A8, A141, A148, A147, A142, Th25, JORDAN5C: 13;

                    ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } by JORDAN6: 26;

                    then

                     A150: (f /. ((i + 1) + k)) in ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) by A141, A148, A147;

                    then ((f /. kk) `1 ) > e by A32, A108, A139, A149, Th2;

                    then ((f /. kk) `1 ) > ((f /. (kk + 1)) `1 ) by A139, XXREAL_0: 2;

                    then ((f /. ((i + 1) + k)) `1 ) <= (p51 `1 ) by A5, A124, A126, A127, A133, A132, A146, A150, A149, A144, Th6;

                    hence contradiction by A32, A108, A123, XXREAL_0: 2;

                  end;

                    case

                     A151: ((f /. kk) `1 ) < e & ((f /. (kk + 1)) `1 ) >= e;

                    set k2 = (kk -' (i + 1));

                    

                     A152: ( LSeg ((f /. kk),(f /. (kk + 1)))) c= ( L~ f)

                    proof

                      let z be object;

                      assume

                       A153: z in ( LSeg ((f /. kk),(f /. (kk + 1))));

                      ( LSeg ((f /. kk),(f /. (kk + 1)))) in { ( LSeg (f,i7)) where i7 be Nat : 1 <= i7 & (i7 + 1) <= ( len f) } by A127, A128, A132;

                      hence thesis by A10, A153, TARSKI:def 4;

                    end;

                     LE (p51,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A5, A124, A126, A127, A128, JORDAN5C: 26;

                    then

                     A154: LE (p44,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A121, JORDAN5C: 13;

                    (f /. (kk + 1)) = (f . (kk + 1)) by A137, PARTFUN1:def 6;

                    then ( LSeg ((f /. kk),(f /. (kk + 1)))) is_an_arc_of ((f /. kk),(f /. (kk + 1))) by A138, A135, TOPREAL1: 9;

                    then

                     A155: ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = { p8 where p8 be Point of ( TOP-REAL 2) : LE ((f /. kk),p8,( L~ f),(f /. 1),(f /. ( len f))) & LE (p8,(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) } & ( Segment (( L~ f),(f /. 1),(f /. ( len f)),(f /. kk),(f /. (kk + 1)))) = ( LSeg ((f /. kk),(f /. (kk + 1)))) by A9, A5, A6, A7, A8, A127, A128, A152, Th25, JORDAN5C: 23, JORDAN6: 26;

                     A156:

                    now

                      assume (kk - (i + 1)) < 0 ;

                      then ((kk - (i + 1)) + (i + 1)) < ( 0 + (i + 1)) by XREAL_1: 6;

                      then kk <= i by NAT_1: 13;

                      then

                       A157: LE ((f /. kk),(f /. i),( L~ f),(f /. 1),(f /. ( len f))) by A5, A23, A127, JORDAN5C: 24;

                      

                       A158: (f /. i) in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                       LE ((f /. i),p44,( L~ f),(f /. 1),(f /. ( len f))) by A5, A14, A15, JORDAN5C: 23;

                      then LE ((f /. i),(f /. (kk + 1)),( L~ f),(f /. 1),(f /. ( len f))) by A154, JORDAN5C: 13;

                      then (f /. i) in ( LSeg ((f /. kk),(f /. (kk + 1)))) by A155, A157;

                      then (( LSeg ((f /. kk),(f /. (kk + 1)))) /\ ( LSeg ((f /. i),(f /. (i + 1))))) <> {} by A158, XBOOLE_0:def 4;

                      then

                       A159: not ( LSeg ((f /. kk),(f /. (kk + 1)))) misses ( LSeg ((f /. i),(f /. (i + 1)))) by XBOOLE_0:def 7;

                      

                       A160: (kk - 1) = (kk -' 1) by A131, XREAL_0:def 2;

                       A161:

                      now

                        assume

                         A162: i = ((kk -' 1) + 2);

                        then (kk + 1) < (i + 1) by A160, NAT_1: 13;

                        then LE ((f /. (kk + 1)),p44,( L~ f),(f /. 1),(f /. ( len f))) by A5, A15, A136, JORDAN5C: 24;

                        then p44 = (f /. (kk + 1)) by A1, A6, A7, A8, A154, JORDAN5C: 12, TOPREAL4: 2;

                        then (f . (i + 1)) = (f /. (kk + 1)) by A18, PARTFUN1:def 6;

                        then

                         A163: (f . (i + 1)) = (f . (kk + 1)) by A137, PARTFUN1:def 6;

                        f is one-to-one by A5, TOPREAL1:def 8;

                        then (i + 1) = (kk + 1) by A18, A137, A163, FUNCT_1:def 4;

                        hence contradiction by A160, A162;

                      end;

                      

                       A164: f is s.n.c. by A5, TOPREAL1:def 8;

                      

                       A165: ( LSeg (f,kk)) = ( LSeg ((f /. kk),(f /. (kk + 1)))) & ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A14, A15, A127, A128, TOPREAL1:def 3;

                      then (i + 1) >= kk by A159, A164, TOPREAL1:def 7;

                      then

                       A166: ((i + 1) - 1) >= (kk - 1) by XREAL_1: 9;

                      (kk + 1) >= i by A159, A165, A164, TOPREAL1:def 7;

                      then

                       A167: i = ((kk -' 1) + 0 ) or ... or i = ((kk -' 1) + 2) by A160, A166, NAT_1: 62;

                       A168:

                      now

                        per cases by A160, A167, A161;

                          case i = kk;

                          hence p44 in ( LSeg (f,kk)) by A119, RLTOPSP1: 68;

                        end;

                          case i = (kk - 1);

                          hence p44 in ( LSeg (f,kk)) by A132, RLTOPSP1: 68;

                        end;

                      end;

                      f is special by A5, TOPREAL1:def 8;

                      then ((f /. kk) `1 ) = ((f /. (kk + 1)) `1 ) or ((f /. kk) `2 ) = ((f /. (kk + 1)) `2 ) by A127, A128, TOPREAL1:def 5;

                      hence contradiction by A5, A6, A7, A8, A118, A121, A123, A124, A126, A127, A133, A151, A168, Th7;

                    end;

                    

                    then ((i + 1) + k2) = ((i + 1) + (kk - (i + 1))) by XREAL_0:def 2

                    .= kk;

                    then P[k2] by A127, A133, A151, FINSEQ_4: 15;

                    then

                     A169: k2 >= k by A27;

                    (kk -' (i + 1)) = (kk - (i + 1)) by A156, XREAL_0:def 2;

                    then ((kk - (i + 1)) + (i + 1)) >= (k + (i + 1)) by A169, XREAL_1: 7;

                    then

                     A170: LE ((f /. ((i + 1) + k)),(f /. kk),( L~ f),(f /. 1),(f /. ( len f))) by A5, A30, A133, JORDAN5C: 24;

                     LE ((f /. kk),(f /. ((i + 1) + k)),( L~ f),(f /. 1),(f /. ( len f))) by A6, A7, A8, A32, A122, A130, JORDAN5C: 13;

                    hence contradiction by A1, A6, A7, A8, A32, A108, A151, A170, JORDAN5C: 12, TOPREAL4: 2;

                  end;

                end;

                hence contradiction;

              end;

              

               A171: p44 in ( LSeg (p,(f /. (i + 1)))) by RLTOPSP1: 68;

              

               A172: for p5 be Point of ( TOP-REAL 2) st LE (p,p5,P,p1,p2) & LE (p5,p44,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                

                 A173: ( Segment (P,p1,p2,p,p44)) = { p8 where p8 be Point of ( TOP-REAL 2) : LE (p,p8,P,p1,p2) & LE (p8,p44,P,p1,p2) } by JORDAN6: 26;

                assume LE (p,p5,P,p1,p2) & LE (p5,p44,P,p1,p2);

                then

                 A174: p5 in ( Segment (P,p1,p2,p,p44)) by A173;

                p44 in ( LSeg ((f /. i),(f /. (i + 1)))) by RLTOPSP1: 68;

                then ( LSeg (p,p44)) c= ( LSeg (f,i)) by A11, A13, A119, TOPREAL1: 6;

                then

                 A175: ( LSeg (p,p44)) c= P by A6, A19, A13;

                now

                  per cases ;

                    case p44 <> p;

                    then ( LSeg (p,p44)) is_an_arc_of (p,p44) by TOPREAL1: 9;

                    then ( Segment (P,p1,p2,p,p44)) = ( LSeg (p,p44)) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A171, A175, Th25, SPRECT_4: 4;

                    hence thesis by A4, A118, A174, TOPREAL1: 3;

                  end;

                    case p44 = p;

                    then ( Segment (P,p1,p2,p,p44)) = {p44} by A1, A3, Th1, TOPREAL4: 2;

                    hence thesis by A118, A174, TARSKI:def 1;

                  end;

                end;

                hence thesis;

              end;

              (i + 1) <= ((i + 1) + k) by NAT_1: 11;

              then

               A176: LE (p44,pk,P,p1,p2) by A5, A6, A7, A8, A17, A28, A32, JORDAN5C: 24;

              then

               A177: p44 in P by JORDAN5C:def 3;

              

               A178: for p5 be Point of ( TOP-REAL 2) st LE (p5,pk,P,p1,p2) & LE (p,p5,P,p1,p2) holds (p5 `1 ) >= e

              proof

                let p5 be Point of ( TOP-REAL 2);

                assume that

                 A179: LE (p5,pk,P,p1,p2) and

                 A180: LE (p,p5,P,p1,p2);

                

                 A181: p5 in P by A179, JORDAN5C:def 3;

                now

                  per cases by A1, A177, A181, Th19, TOPREAL4: 2;

                    case LE (p44,p5,P,p1,p2);

                    hence thesis by A120, A179;

                  end;

                    case LE (p5,p44,P,p1,p2);

                    hence thesis by A172, A180;

                  end;

                end;

                hence thesis;

              end;

               LE (p,p44,P,p1,p2) by A5, A6, A7, A8, A11, A13, A14, A23, A171, SPRECT_4: 4;

              then LE (p,pk,P,p1,p2) by A176, JORDAN5C: 13;

              hence thesis by A3, A4, A9, A108, A178;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;