jordan.miz



    begin

    reserve a,b,c,d,r,s for Real,

n for Element of NAT ,

p,p1,p2 for Point of ( TOP-REAL 2),

x,y for Point of ( TOP-REAL n),

C for Simple_closed_curve,

A,B,P for Subset of ( TOP-REAL 2),

U,V for Subset of (( TOP-REAL 2) | (C ` )),

D for compact with_the_max_arc Subset of ( TOP-REAL 2);

    set T2 = ( TOP-REAL 2);

    

     Lm1: for A,B,C,Z be set st A c= Z & B c= Z & C c= Z holds ((A \/ B) \/ C) c= Z

    proof

      let A,B,C,Z be set;

      assume that

       A1: A c= Z and

       A2: B c= Z;

      (A \/ B) c= Z by A1, A2, XBOOLE_1: 8;

      hence thesis by XBOOLE_1: 8;

    end;

    

     Lm2: for A,B,C,D,Z be set st A c= Z & B c= Z & C c= Z & D c= Z holds (((A \/ B) \/ C) \/ D) c= Z

    proof

      let A,B,C,D,Z be set;

      assume that

       A1: A c= Z and

       A2: B c= Z and

       A3: C c= Z;

      ((A \/ B) \/ C) c= Z by A1, A2, A3, Lm1;

      hence thesis by XBOOLE_1: 8;

    end;

    

     Lm3: for A,B,C,D,Z be set st A misses Z & B misses Z & C misses Z & D misses Z holds (((A \/ B) \/ C) \/ D) misses Z

    proof

      let A,B,C,D,Z be set;

      assume that

       A1: A misses Z and

       A2: B misses Z and

       A3: C misses Z;

      ((A \/ B) \/ C) misses Z by A1, A2, A3, XBOOLE_1: 114;

      hence thesis by XBOOLE_1: 70;

    end;

    registration

      let M be symmetric triangle Reflexive MetrStruct, x,y be Point of M;

      cluster ( dist (x,y)) -> non negative;

      coherence by METRIC_1: 5;

    end

    registration

      let n be Element of NAT , x,y be Point of ( TOP-REAL n);

      cluster ( dist (x,y)) -> non negative;

      coherence

      proof

        ex p,q be Point of ( Euclid n) st p = x & q = y & ( dist (x,y)) = ( dist (p,q)) by TOPREAL6:def 1;

        hence 0 <= ( dist (x,y));

      end;

    end

    theorem :: JORDAN:1

    

     Th1: for p1,p2 be Point of ( TOP-REAL n) st p1 <> p2 holds ((1 / 2) * (p1 + p2)) <> p1

    proof

      let p1,p2 be Point of ( TOP-REAL n);

      set r = (1 / 2);

      assume that

       A1: p1 <> p2 and

       A2: (r * (p1 + p2)) = p1;

      (r * (p1 + p2)) = ((r * p1) + (r * p2)) by RLVECT_1:def 5;

      

      then ( 0. ( TOP-REAL n)) = (p1 - ((r * p1) + (r * p2))) by A2, RLVECT_1: 5

      .= ((p1 - (r * p1)) - (r * p2)) by RLVECT_1: 27

      .= (((1 * p1) - (r * p1)) - (r * p2)) by RLVECT_1:def 8

      .= (((1 - r) * p1) - (r * p2)) by RLVECT_1: 35

      .= (r * (p1 - p2)) by RLVECT_1: 34;

      then (p1 - p2) = ( 0. ( TOP-REAL n)) by RLVECT_1: 11;

      hence thesis by A1, RLVECT_1: 21;

    end;

    theorem :: JORDAN:2

    

     Th2: (p1 `2 ) < (p2 `2 ) implies (p1 `2 ) < (((1 / 2) * (p1 + p2)) `2 )

    proof

      assume

       A1: (p1 `2 ) < (p2 `2 );

      (((1 / 2) * (p1 + p2)) `2 ) = ((1 / 2) * ((p1 + p2) `2 )) by TOPREAL3: 4

      .= ((1 / 2) * ((p1 `2 ) + (p2 `2 ))) by TOPREAL3: 2

      .= (((p1 `2 ) + (p2 `2 )) / 2);

      hence thesis by A1, XREAL_1: 226;

    end;

    theorem :: JORDAN:3

    

     Th3: (p1 `2 ) < (p2 `2 ) implies (((1 / 2) * (p1 + p2)) `2 ) < (p2 `2 )

    proof

      assume

       A1: (p1 `2 ) < (p2 `2 );

      (((1 / 2) * (p1 + p2)) `2 ) = ((1 / 2) * ((p1 + p2) `2 )) by TOPREAL3: 4

      .= ((1 / 2) * ((p1 `2 ) + (p2 `2 ))) by TOPREAL3: 2

      .= (((p1 `2 ) + (p2 `2 )) / 2);

      hence thesis by A1, XREAL_1: 226;

    end;

    theorem :: JORDAN:4

    

     Th4: for A be vertical Subset of ( TOP-REAL 2) holds (A /\ B) is vertical

    proof

      let A be vertical Subset of ( TOP-REAL 2);

      let p,q be Point of T2;

      assume that

       A1: p in (A /\ B) and

       A2: q in (A /\ B);

      

       A3: p in A by A1, XBOOLE_0:def 4;

      q in A by A2, XBOOLE_0:def 4;

      hence thesis by A3, SPPOL_1:def 3;

    end;

    theorem :: JORDAN:5

    for A be horizontal Subset of ( TOP-REAL 2) holds (A /\ B) is horizontal

    proof

      let A be horizontal Subset of ( TOP-REAL 2);

      let p,q be Point of T2;

      assume that

       A1: p in (A /\ B) and

       A2: q in (A /\ B);

      

       A3: p in A by A1, XBOOLE_0:def 4;

      q in A by A2, XBOOLE_0:def 4;

      hence thesis by A3, SPPOL_1:def 2;

    end;

    theorem :: JORDAN:6

    p in ( LSeg (p1,p2)) & ( LSeg (p1,p2)) is vertical implies ( LSeg (p,p2)) is vertical

    proof

      assume

       A1: p in ( LSeg (p1,p2));

      assume

       A2: ( LSeg (p1,p2)) is vertical;

      then

       A3: (p1 `1 ) = (p2 `1 ) by SPPOL_1: 16;

      p1 in ( LSeg (p1,p2)) by RLTOPSP1: 68;

      then (p `1 ) = (p1 `1 ) by A1, A2;

      hence thesis by A3, SPPOL_1: 16;

    end;

    theorem :: JORDAN:7

    p in ( LSeg (p1,p2)) & ( LSeg (p1,p2)) is horizontal implies ( LSeg (p,p2)) is horizontal

    proof

      assume

       A1: p in ( LSeg (p1,p2));

      assume

       A2: ( LSeg (p1,p2)) is horizontal;

      then

       A3: (p1 `2 ) = (p2 `2 ) by SPPOL_1: 15;

      p1 in ( LSeg (p1,p2)) by RLTOPSP1: 68;

      then (p `2 ) = (p1 `2 ) by A1, A2;

      hence thesis by A3, SPPOL_1: 15;

    end;

    registration

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

      cluster ( LSeg (( SW-corner P),( SE-corner P))) -> horizontal;

      coherence

      proof

        (( SW-corner P) `2 ) = ( S-bound P) by EUCLID: 52

        .= (( SE-corner P) `2 ) by EUCLID: 52;

        hence thesis by SPPOL_1: 15;

      end;

      cluster ( LSeg (( NW-corner P),( SW-corner P))) -> vertical;

      coherence

      proof

        (( NW-corner P) `1 ) = ( W-bound P) by EUCLID: 52

        .= (( SW-corner P) `1 ) by EUCLID: 52;

        hence thesis by SPPOL_1: 16;

      end;

      cluster ( LSeg (( NE-corner P),( SE-corner P))) -> vertical;

      coherence

      proof

        (( NE-corner P) `1 ) = ( E-bound P) by EUCLID: 52

        .= (( SE-corner P) `1 ) by EUCLID: 52;

        hence thesis by SPPOL_1: 16;

      end;

    end

    registration

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

      cluster ( LSeg (( SE-corner P),( SW-corner P))) -> horizontal;

      coherence ;

      cluster ( LSeg (( SW-corner P),( NW-corner P))) -> vertical;

      coherence ;

      cluster ( LSeg (( SE-corner P),( NE-corner P))) -> vertical;

      coherence ;

    end

    registration

      cluster vertical non empty compact -> with_the_max_arc for Subset of ( TOP-REAL 2);

      coherence

      proof

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

        assume

         A1: A is vertical non empty compact;

        then

         A2: ( W-bound A) = ( E-bound A) by SPRECT_1: 15;

        

         A3: ( E-min A) in A by A1, SPRECT_1: 14;

        (( E-min A) `1 ) = ( E-bound A) by EUCLID: 52;

        then ( E-min A) in ( Vertical_Line ((( W-bound A) + ( E-bound A)) / 2)) by A2, JORDAN6: 31;

        hence A meets ( Vertical_Line ((( W-bound A) + ( E-bound A)) / 2)) by A3, XBOOLE_0: 3;

      end;

    end

    theorem :: JORDAN:8

    

     Th8: (p1 `1 ) <= r & r <= (p2 `1 ) implies ( LSeg (p1,p2)) meets ( Vertical_Line r)

    proof

      assume that

       A1: (p1 `1 ) <= r and

       A2: r <= (p2 `1 );

      set a = (p1 `1 ), b = (p2 `1 );

      set l = ((r - a) / (b - a));

      set k = (((1 - l) * p1) + (l * p2));

      

       A3: (a - a) <= (r - a) by A1, XREAL_1: 9;

      

       A4: (r - a) <= (b - a) by A2, XREAL_1: 9;

      then l <= 1 by A3, XREAL_1: 183;

      then

       A5: k in ( LSeg (p1,p2)) by A3, A4;

      per cases ;

        suppose a <> b;

        then

         A6: (b - a) <> 0 ;

        (k `1 ) = (((1 - l) * a) + (l * b)) by TOPREAL9: 41

        .= (a + (l * (b - a)))

        .= (a + (r - a)) by A6, XCMPLX_1: 87;

        then k in ( Vertical_Line r) by JORDAN6: 31;

        hence thesis by A5, XBOOLE_0: 3;

      end;

        suppose

         A7: a = b;

        

         A8: p1 in ( LSeg (p1,p2)) by RLTOPSP1: 68;

        a = r by A1, A2, A7, XXREAL_0: 1;

        then p1 in ( Vertical_Line r) by JORDAN6: 31;

        hence thesis by A8, XBOOLE_0: 3;

      end;

    end;

    theorem :: JORDAN:9

    (p1 `2 ) <= r & r <= (p2 `2 ) implies ( LSeg (p1,p2)) meets ( Horizontal_Line r)

    proof

      assume that

       A1: (p1 `2 ) <= r and

       A2: r <= (p2 `2 );

      set a = (p1 `2 ), b = (p2 `2 );

      set l = ((r - a) / (b - a));

      set k = (((1 - l) * p1) + (l * p2));

      

       A3: (a - a) <= (r - a) by A1, XREAL_1: 9;

      

       A4: (r - a) <= (b - a) by A2, XREAL_1: 9;

      then l <= 1 by A3, XREAL_1: 183;

      then

       A5: k in ( LSeg (p1,p2)) by A3, A4;

      per cases ;

        suppose a <> b;

        then

         A6: (b - a) <> 0 ;

        (k `2 ) = (((1 - l) * a) + (l * b)) by TOPREAL9: 42

        .= (a + (l * (b - a)))

        .= (a + (r - a)) by A6, XCMPLX_1: 87;

        then k in ( Horizontal_Line r) by JORDAN6: 32;

        hence thesis by A5, XBOOLE_0: 3;

      end;

        suppose

         A7: a = b;

        

         A8: p1 in ( LSeg (p1,p2)) by RLTOPSP1: 68;

        a = r by A1, A2, A7, XXREAL_0: 1;

        then p1 in ( Horizontal_Line r) by JORDAN6: 32;

        hence thesis by A8, XBOOLE_0: 3;

      end;

    end;

    registration

      let n;

      cluster empty -> bounded for Subset of ( TOP-REAL n);

      coherence ;

      cluster non bounded -> non empty for Subset of ( TOP-REAL n);

      coherence ;

    end

    registration

      let n be non zero Nat;

      cluster open closed non bounded convex for Subset of ( TOP-REAL n);

      existence

      proof

        take ( [#] ( TOP-REAL n));

        reconsider n as Element of NAT by ORDINAL1:def 12;

        n >= 1 by NAT_1: 14;

        then not ( [#] ( TOP-REAL n)) is bounded by JORDAN2C: 35;

        hence thesis;

      end;

    end

    theorem :: JORDAN:10

    

     Th10: for C be compact Subset of ( TOP-REAL 2) holds (( north_halfline ( UMP C)) \ {( UMP C)}) misses C

    proof

      let C be compact Subset of ( TOP-REAL 2);

      set p = ( UMP C);

      set L = ( north_halfline p);

      set w = ((( W-bound C) + ( E-bound C)) / 2);

      assume (L \ {p}) meets C;

      then

      consider x be object such that

       A1: x in (L \ {p}) and

       A2: x in C by XBOOLE_0: 3;

      

       A3: x in L by A1, ZFMISC_1: 56;

      

       A4: x <> p by A1, ZFMISC_1: 56;

      reconsider x as Point of T2 by A1;

      

       A5: (x `1 ) = (p `1 ) by A3, TOPREAL1:def 10;

      

       A6: (x `2 ) >= (p `2 ) by A3, TOPREAL1:def 10;

      (x `2 ) <> (p `2 ) by A4, A5, TOPREAL3: 6;

      then

       A7: (x `2 ) > (p `2 ) by A6, XXREAL_0: 1;

      (x `1 ) = w by A5, EUCLID: 52;

      then x in ( Vertical_Line w) by JORDAN6: 31;

      then x in (C /\ ( Vertical_Line w)) by A2, XBOOLE_0:def 4;

      hence thesis by A7, JORDAN21: 28;

    end;

    theorem :: JORDAN:11

    

     Th11: for C be compact Subset of ( TOP-REAL 2) holds (( south_halfline ( LMP C)) \ {( LMP C)}) misses C

    proof

      let C be compact Subset of ( TOP-REAL 2);

      set p = ( LMP C);

      set L = ( south_halfline p);

      set w = ((( W-bound C) + ( E-bound C)) / 2);

      assume (L \ {p}) meets C;

      then

      consider x be object such that

       A1: x in (L \ {p}) and

       A2: x in C by XBOOLE_0: 3;

      

       A3: x in L by A1, ZFMISC_1: 56;

      

       A4: x <> p by A1, ZFMISC_1: 56;

      reconsider x as Point of T2 by A1;

      

       A5: (x `1 ) = (p `1 ) by A3, TOPREAL1:def 12;

      

       A6: (x `2 ) <= (p `2 ) by A3, TOPREAL1:def 12;

      (x `2 ) <> (p `2 ) by A4, A5, TOPREAL3: 6;

      then

       A7: (x `2 ) < (p `2 ) by A6, XXREAL_0: 1;

      (x `1 ) = w by A5, EUCLID: 52;

      then x in ( Vertical_Line w) by JORDAN6: 31;

      then x in (C /\ ( Vertical_Line w)) by A2, XBOOLE_0:def 4;

      hence thesis by A7, JORDAN21: 29;

    end;

    theorem :: JORDAN:12

    

     Th12: for C be compact Subset of ( TOP-REAL 2) holds (( north_halfline ( UMP C)) \ {( UMP C)}) c= ( UBD C)

    proof

      let C be compact Subset of ( TOP-REAL 2);

      set A = (( north_halfline ( UMP C)) \ {( UMP C)});

      reconsider A as non bounded Subset of T2 by JORDAN2C: 122, TOPREAL6: 90;

      A is convex by JORDAN21: 6;

      hence thesis by Th10, JORDAN2C: 125;

    end;

    theorem :: JORDAN:13

    

     Th13: for C be compact Subset of ( TOP-REAL 2) holds (( south_halfline ( LMP C)) \ {( LMP C)}) c= ( UBD C)

    proof

      let C be compact Subset of ( TOP-REAL 2);

      set A = (( south_halfline ( LMP C)) \ {( LMP C)});

      reconsider A as non bounded Subset of T2 by JORDAN2C: 123, TOPREAL6: 90;

      A is convex by JORDAN21: 7;

      hence thesis by Th11, JORDAN2C: 125;

    end;

    theorem :: JORDAN:14

    

     Th14: A is_inside_component_of B implies ( UBD B) misses A

    proof

      assume A is_inside_component_of B;

      then A c= ( BDD B) by JORDAN2C: 22;

      hence thesis by JORDAN2C: 24, XBOOLE_1: 63;

    end;

    theorem :: JORDAN:15

    A is_outside_component_of B implies ( BDD B) misses A

    proof

      assume

       A1: A is_outside_component_of B;

      ( BDD B) misses ( UBD B) by JORDAN2C: 24;

      hence thesis by A1, JORDAN2C: 23, XBOOLE_1: 63;

    end;

    

     Lm4: p in C implies {p} misses U

    proof

      assume

       A1: p in C;

      

       A2: U is Subset of T2 by PRE_TOPC: 11;

      the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      then U misses C by A2, SUBSET_1: 23;

      then not p in U by A1, XBOOLE_0: 3;

      hence thesis by ZFMISC_1: 50;

    end;

    set C0 = ( Closed-Interval-TSpace ( 0 ,1));

    set C1 = ( Closed-Interval-TSpace (( - 1),1));

    set l0 = ( (#) (( - 1),1));

    set l1 = ((( - 1),1) (#) );

    set h1 = ( L[01] (l0,l1));

    

     Lm5: the carrier of [:T2, T2:] = [:the carrier of T2, the carrier of T2:] by BORSUK_1:def 2;

     Lm6:

    now

      let T be non empty TopSpace;

      let a be Element of REAL ;

      set c = the carrier of T;

      set f = (c --> a);

      thus f is continuous

      proof

        

         A1: ( dom f) = c by FUNCT_2:def 1;

        

         A2: ( rng f) = {a} by FUNCOP_1: 8;

        let Y be Subset of REAL ;

        assume Y is closed;

        per cases ;

          suppose a in Y;

          then

           A3: ( rng f) c= Y by A2, ZFMISC_1: 31;

          (f " Y) = (f " (( rng f) /\ Y)) by RELAT_1: 133

          .= (f " ( rng f)) by A3, XBOOLE_1: 28

          .= ( [#] T) by A1, RELAT_1: 134;

          hence thesis;

        end;

          suppose not a in Y;

          then

           A4: ( rng f) misses Y by A2, ZFMISC_1: 50;

          (f " Y) = (f " (( rng f) /\ Y)) by RELAT_1: 133

          .= (f " {} ) by A4

          .= ( {} T);

          hence thesis;

        end;

      end;

    end;

    theorem :: JORDAN:16

    

     Th16: for n be Nat holds for r be positive Real holds for a be Point of ( TOP-REAL n) holds a in ( Ball (a,r))

    proof

      let n be Nat;

      let r be positive Real;

      let a be Point of ( TOP-REAL n);

       |.(a - a).| = 0 by TOPRNS_1: 28;

      hence thesis by TOPREAL9: 7;

    end;

    theorem :: JORDAN:17

    

     Th17: for r be non negative Real holds for p be Point of ( TOP-REAL n) holds p is Point of ( Tdisk (p,r))

    proof

      let r be non negative Real;

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

      

       A1: the carrier of ( Tdisk (p,r)) = ( cl_Ball (p,r)) by BROUWER: 3;

       |.(p - p).| = 0 by TOPRNS_1: 28;

      hence thesis by A1, TOPREAL9: 8;

    end;

    registration

      let r be positive Real;

      let n be non zero Element of NAT ;

      let p,q be Point of ( TOP-REAL n);

      cluster (( cl_Ball (p,r)) \ {q}) -> non empty;

      coherence

      proof

        

         A1: the carrier of ( Tcircle (p,r)) = ( Sphere (p,r)) by TOPREALB: 9;

        

         A2: the carrier of ( Tdisk (p,r)) = ( cl_Ball (p,r)) by BROUWER: 3;

        

         A3: ( Sphere (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 17;

        set a = the Point of ( Tcircle (p,r));

        

         A4: a in ( Sphere (p,r)) by A1;

        per cases ;

          suppose

           A5: a = q;

          

           A6: p is Point of ( Tdisk (p,r)) by Th17;

           |.(p - p).| <> r by TOPRNS_1: 28;

          then p <> q by A1, A5, TOPREAL9: 9;

          hence thesis by A2, A6, ZFMISC_1: 56;

        end;

          suppose a <> q;

          hence thesis by A3, A4, ZFMISC_1: 56;

        end;

      end;

    end

    theorem :: JORDAN:18

    

     Th18: r <= s implies ( Ball (x,r)) c= ( Ball (x,s))

    proof

      reconsider xe = x as Point of ( Euclid n) by TOPREAL3: 8;

      

       A1: ( Ball (x,r)) = ( Ball (xe,r)) by TOPREAL9: 13;

      ( Ball (x,s)) = ( Ball (xe,s)) by TOPREAL9: 13;

      hence thesis by A1, PCOMPS_1: 1;

    end;

    theorem :: JORDAN:19

    

     Th19: (( cl_Ball (x,r)) \ ( Ball (x,r))) = ( Sphere (x,r))

    proof

      thus (( cl_Ball (x,r)) \ ( Ball (x,r))) c= ( Sphere (x,r))

      proof

        let a be object;

        assume

         A1: a in (( cl_Ball (x,r)) \ ( Ball (x,r)));

        then

        reconsider a as Point of ( TOP-REAL n);

        

         A2: a in ( cl_Ball (x,r)) by A1, XBOOLE_0:def 5;

        

         A3: not a in ( Ball (x,r)) by A1, XBOOLE_0:def 5;

        

         A4: |.(a - x).| <= r by A2, TOPREAL9: 8;

         |.(a - x).| >= r by A3, TOPREAL9: 7;

        then |.(a - x).| = r by A4, XXREAL_0: 1;

        hence thesis by TOPREAL9: 9;

      end;

      let a be object;

      assume

       A5: a in ( Sphere (x,r));

      then

      reconsider a as Point of ( TOP-REAL n);

      

       A6: |.(a - x).| = r by A5, TOPREAL9: 9;

      then

       A7: a in ( cl_Ball (x,r)) by TOPREAL9: 8;

       not a in ( Ball (x,r)) by A6, TOPREAL9: 7;

      hence thesis by A7, XBOOLE_0:def 5;

    end;

    theorem :: JORDAN:20

    

     Th20: y in ( Sphere (x,r)) implies (( LSeg (x,y)) \ {x, y}) c= ( Ball (x,r))

    proof

      assume

       A1: y in ( Sphere (x,r));

      per cases ;

        suppose

         A2: r = 0 ;

        reconsider xe = x as Point of ( Euclid n) by TOPREAL3: 8;

        ( Sphere (x,r)) = ( Sphere (xe,r)) by TOPREAL9: 15;

        then ( Sphere (x,r)) = {x} by A2, TOPREAL6: 54;

        then

         A3: x = y by A1, TARSKI:def 1;

        

         A4: ( LSeg (x,x)) = {x} by RLTOPSP1: 70;

        

         A5: {x, x} = {x} by ENUMSET1: 29;

        ( {x} \ {x}) = {} by XBOOLE_1: 37;

        hence thesis by A3, A4, A5;

      end;

        suppose

         A6: r <> 0 ;

        let k be object;

        assume

         A7: k in (( LSeg (x,y)) \ {x, y});

        then k in ( LSeg (x,y)) by XBOOLE_0:def 5;

        then

        consider l be Real such that

         A8: k = (((1 - l) * x) + (l * y)) and

         A9: 0 <= l and

         A10: l <= 1;

        reconsider k as Point of ( TOP-REAL n) by A8;

         not k in {x, y} by A7, XBOOLE_0:def 5;

        then k <> y by TARSKI:def 2;

        then l <> 1 by A8, TOPREAL9: 4;

        then

         A11: l < 1 by A10, XXREAL_0: 1;

        (k - x) = ((((1 - l) * x) - x) + (l * y)) by A8, RLVECT_1:def 3

        .= ((((1 * x) - (l * x)) - x) + (l * y)) by RLVECT_1: 35

        .= (((x - (l * x)) - x) + (l * y)) by RLVECT_1:def 8

        .= (((x + ( - (l * x))) + ( - x)) + (l * y))

        .= (((x + ( - x)) + ( - (l * x))) + (l * y)) by RLVECT_1:def 3

        .= (((x - x) - (l * x)) + (l * y))

        .= ((( 0. ( TOP-REAL n)) - (l * x)) + (l * y)) by RLVECT_1: 5

        .= ((l * y) - (l * x)) by RLVECT_1: 4

        .= (l * (y - x)) by RLVECT_1: 34;

        

        then

         A12: |.(k - x).| = ( |.l.| * |.(y - x).|) by TOPRNS_1: 7

        .= (l * |.(y - x).|) by A9, ABSVALUE:def 1

        .= (l * r) by A1, TOPREAL9: 9;

         0 <= r by A1;

        then (l * r) < (1 * r) by A6, A11, XREAL_1: 68;

        hence thesis by A12, TOPREAL9: 7;

      end;

    end;

    theorem :: JORDAN:21

    

     Th21: r < s implies ( cl_Ball (x,r)) c= ( Ball (x,s))

    proof

      assume

       A1: r < s;

      let a be object;

      assume

       A2: a in ( cl_Ball (x,r));

      then

      reconsider a as Point of ( TOP-REAL n);

       |.(a - x).| <= r by A2, TOPREAL9: 8;

      then |.(a - x).| < s by A1, XXREAL_0: 2;

      hence thesis by TOPREAL9: 7;

    end;

    theorem :: JORDAN:22

    

     Th22: r < s implies ( Sphere (x,r)) c= ( Ball (x,s))

    proof

      assume r < s;

      then

       A1: ( cl_Ball (x,r)) c= ( Ball (x,s)) by Th21;

      ( Sphere (x,r)) c= ( cl_Ball (x,r)) by TOPREAL9: 17;

      hence thesis by A1;

    end;

    theorem :: JORDAN:23

    

     Th23: for r be non zero Real holds ( Cl ( Ball (x,r))) = ( cl_Ball (x,r))

    proof

      let r be non zero Real;

      thus ( Cl ( Ball (x,r))) c= ( cl_Ball (x,r)) by TOPREAL9: 16, TOPS_1: 5;

      per cases ;

        suppose ( Ball (x,r)) is empty;

        then r < 0 ;

        hence thesis;

      end;

        suppose

         A1: ( Ball (x,r)) is non empty;

        let a be object;

        assume

         A2: a in ( cl_Ball (x,r));

        then

        reconsider a as Point of ( TOP-REAL n);

        reconsider ae = a as Point of ( Euclid n) by TOPREAL3: 8;

        

         A3: 0 < r by A1;

        for s be Real st 0 < s & s < r holds ( Ball (ae,s)) meets ( Ball (x,r))

        proof

          let s be Real such that

           A4: 0 < s and

           A5: s < r;

          now

            

             A6: (( Ball (x,r)) \/ ( Sphere (x,r))) = ( cl_Ball (x,r)) by TOPREAL9: 18;

            per cases by A2, A6, XBOOLE_0:def 3;

              suppose

               A7: a in ( Ball (x,r));

               |.(a - a).| = 0 by TOPRNS_1: 28;

              then a in ( Ball (a,s)) by A4, TOPREAL9: 7;

              hence ( Ball (a,s)) meets ( Ball (x,r)) by A7, XBOOLE_0: 3;

            end;

              suppose

               A8: a in ( Sphere (x,r));

              then

               A9: |.(a - x).| = r by TOPREAL9: 9;

               |.(x - x).| = 0 by TOPRNS_1: 28;

              then

               A10: x in ( Ball (x,r)) by A3, TOPREAL9: 7;

              set z = (s / (2 * r));

              set q = (((1 - z) * a) + (z * x));

              (1 * r) < (2 * r) by A3, XREAL_1: 68;

              then s < (2 * r) by A5, XXREAL_0: 2;

              then

               A11: z < 1 by A4, XREAL_1: 189;

               0 < (2 * r) by A3, XREAL_1: 129;

              then

               A12: 0 < z by A4, XREAL_1: 139;

              

               A13: q in ( LSeg (a,x)) by A3, A4, A11;

              ( Ball (x,r)) misses ( Sphere (x,r)) by TOPREAL9: 19;

              then

               A14: a <> x by A8, A10, XBOOLE_0: 3;

              then

               A15: q <> a by A12, TOPREAL9: 4;

              q <> x by A11, A14, TOPREAL9: 4;

              then not q in {a, x} by A15, TARSKI:def 2;

              then

               A16: q in (( LSeg (a,x)) \ {a, x}) by A13, XBOOLE_0:def 5;

              

               A17: (( LSeg (a,x)) \ {a, x}) c= ( Ball (x,r)) by A8, Th20;

              (q - a) = ((((1 - z) * a) - a) + (z * x)) by RLVECT_1:def 3

              .= ((((1 * a) - (z * a)) - a) + (z * x)) by RLVECT_1: 35

              .= (((a - (z * a)) - a) + (z * x)) by RLVECT_1:def 8

              .= (((a + ( - (z * a))) + ( - a)) + (z * x))

              .= (((a + ( - a)) + ( - (z * a))) + (z * x)) by RLVECT_1:def 3

              .= (((a - a) - (z * a)) + (z * x))

              .= ((( 0. ( TOP-REAL n)) - (z * a)) + (z * x)) by RLVECT_1: 5

              .= ((z * x) - (z * a)) by RLVECT_1: 4

              .= (z * (x - a)) by RLVECT_1: 34;

              

              then |.(q - a).| = ( |.z.| * |.(x - a).|) by TOPRNS_1: 7

              .= (z * |.(x - a).|) by A3, A4, ABSVALUE:def 1

              .= (z * |.(a - x).|) by TOPRNS_1: 27

              .= (s / 2) by A9, XCMPLX_1: 92;

              then

               A18: q in ( Sphere (a,(s / 2))) by TOPREAL9: 9;

              (s / 2) < (s / 1) by A4, XREAL_1: 76;

              then ( Sphere (a,(s / 2))) c= ( Ball (a,s)) by Th22;

              hence ( Ball (a,s)) meets ( Ball (x,r)) by A16, A17, A18, XBOOLE_0: 3;

            end;

          end;

          hence thesis by TOPREAL9: 13;

        end;

        hence thesis by A3, GOBOARD6: 93;

      end;

    end;

    theorem :: JORDAN:24

    

     Th24: for r be non zero Real holds ( Fr ( Ball (x,r))) = ( Sphere (x,r))

    proof

      let r be non zero Real;

      set P = ( Ball (x,r));

      

      thus ( Fr P) = (( Cl P) \ P) by TOPS_1: 42

      .= (( cl_Ball (x,r)) \ P) by Th23

      .= ( Sphere (x,r)) by Th19;

    end;

    registration

      let n be non zero Element of NAT ;

      cluster bounded -> proper for Subset of ( TOP-REAL n);

      coherence

      proof

         not ( [#] ( TOP-REAL n)) is bounded by JORDAN2C: 35, NAT_1: 14;

        hence thesis by SUBSET_1:def 6;

      end;

    end

    registration

      let n;

      cluster non empty closed convex bounded for Subset of ( TOP-REAL n);

      existence

      proof

        take ( cl_Ball (( 0. ( TOP-REAL n)),1));

        thus thesis;

      end;

      cluster non empty open convex bounded for Subset of ( TOP-REAL n);

      existence

      proof

        take ( Ball (( 0. ( TOP-REAL n)),1));

        thus thesis;

      end;

    end

    registration

      let n be Element of NAT ;

      let A be bounded Subset of ( TOP-REAL n);

      cluster ( Cl A) -> bounded;

      coherence by TOPREAL6: 63;

    end

    registration

      let n be Element of NAT ;

      let A be bounded Subset of ( TOP-REAL n);

      cluster ( Fr A) -> bounded;

      coherence by TOPREAL6: 89;

    end

    theorem :: JORDAN:25

    

     Th25: for A be closed Subset of ( TOP-REAL n), p be Point of ( TOP-REAL n) st not p in A holds ex r be positive Real st ( Ball (p,r)) misses A

    proof

      let A be closed Subset of ( TOP-REAL n), p be Point of ( TOP-REAL n);

      assume not p in A;

      then

       A1: p in (A ` ) by SUBSET_1: 29;

      reconsider e = p as Point of ( Euclid n) by TOPREAL3: 8;

      

       A2: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider AA = (A ` ) as Subset of ( TopSpaceMetr ( Euclid n));

      AA is open by A2, PRE_TOPC: 30;

      then

      consider r be Real such that

       A3: r > 0 and

       A4: ( Ball (e,r)) c= (A ` ) by A1, TOPMETR: 15;

      reconsider r as positive Real by A3;

      take r;

      ( Ball (p,r)) = ( Ball (e,r)) by TOPREAL9: 13;

      hence thesis by A4, SUBSET_1: 23;

    end;

    theorem :: JORDAN:26

    

     Th26: for A be bounded Subset of ( TOP-REAL n), a be Point of ( TOP-REAL n) holds ex r be positive Real st A c= ( Ball (a,r))

    proof

      let A be bounded Subset of ( TOP-REAL n);

      let a be Point of ( TOP-REAL n);

      reconsider C = A as bounded Subset of ( Euclid n) by JORDAN2C: 11;

      consider r be Real, x be Element of ( Euclid n) such that

       A1: 0 < r and

       A2: C c= ( Ball (x,r)) by METRIC_6:def 3;

      reconsider r as positive Real by A1;

      reconsider x1 = x as Point of ( TOP-REAL n) by TOPREAL3: 8;

      take s = (r + |.(x1 - a).|);

      let p be object;

      assume

       A3: p in A;

      then

      reconsider p1 = p as Point of ( TOP-REAL n);

      p = p1;

      then

      reconsider p as Point of ( Euclid n) by TOPREAL3: 8;

      

       A4: ( dist (p,x)) < r by A2, A3, METRIC_1: 11;

      

       A5: |.(p1 - x1).| = ( dist (p,x)) by SPPOL_1: 39;

      

       A6: |.(p1 - a).| <= ( |.(p1 - x1).| + |.(x1 - a).|) by TOPRNS_1: 34;

      ( |.(p1 - x1).| + |.(x1 - a).|) < s by A4, A5, XREAL_1: 6;

      then |.(p1 - a).| < s by A6, XXREAL_0: 2;

      hence thesis by TOPREAL9: 7;

    end;

    theorem :: JORDAN:27

    for S,T be TopStruct, f be Function of S, T st f is being_homeomorphism holds f is onto;

    registration

      let T be non empty T_2 TopSpace;

      cluster -> T_2 for non empty SubSpace of T;

      coherence ;

    end

    registration

      let p, r;

      cluster ( Tdisk (p,r)) -> closed;

      coherence

      proof

        let A be Subset of T2;

        assume A = the carrier of ( Tdisk (p,r));

        then A = ( cl_Ball (p,r)) by BROUWER: 3;

        hence thesis;

      end;

    end

    registration

      let p, r;

      cluster ( Tdisk (p,r)) -> compact;

      coherence

      proof

        set D = ( Tdisk (p,r));

        reconsider Q = ( [#] D) as Subset of T2 by TSEP_1: 1;

        ( [#] D) = ( cl_Ball (p,r)) by BROUWER: 3;

        then Q is compact by TOPREAL6: 79;

        then ( [#] D) is compact by COMPTS_1: 2;

        hence thesis by COMPTS_1: 1;

      end;

    end

    begin

    theorem :: JORDAN:28

    for T be non empty TopSpace, a,b be Point of T holds for f be Path of a, b st (a,b) are_connected holds ( rng f) is connected

    proof

      let T be non empty TopSpace, a,b be Point of T;

      let f be Path of a, b such that

       A1: (a,b) are_connected ;

      

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

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

      reconsider B = A as Subset of I[01] by BORSUK_1: 40;

      

       A3: B is connected by CONNSP_1: 23;

      

       A4: f is continuous by A1, BORSUK_2:def 2;

      (f .: B) = ( rng f) by A2, BORSUK_1: 40, RELAT_1: 113;

      hence thesis by A3, A4, TOPS_2: 61;

    end;

    theorem :: JORDAN:29

    

     Th29: for X be non empty TopSpace, Y be non empty SubSpace of X, x1,x2 be Point of X, y1,y2 be Point of Y, f be Path of x1, x2 st x1 = y1 & x2 = y2 & (x1,x2) are_connected & ( rng f) c= the carrier of Y holds (y1,y2) are_connected & f is Path of y1, y2

    proof

      let X be non empty TopSpace, Y be non empty SubSpace of X, x1,x2 be Point of X, y1,y2 be Point of Y, f be Path of x1, x2 such that

       A1: x1 = y1 and

       A2: x2 = y2 and

       A3: (x1,x2) are_connected ;

      assume ( rng f) c= the carrier of Y;

      then

      reconsider g = f as Function of I[01] , Y by FUNCT_2: 6;

      

       A4: f is continuous by A3, BORSUK_2:def 2;

      

       A5: (f . 0 ) = y1 & (f . 1) = y2 by A1, A2, A3, BORSUK_2:def 2;

      

       A6: g is continuous by A4, PRE_TOPC: 27;

      thus ex f be Function of I[01] , Y st f is continuous & (f . 0 ) = y1 & (f . 1) = y2

      proof

        take g;

        thus g is continuous by A4, PRE_TOPC: 27;

        thus thesis by A1, A2, A3, BORSUK_2:def 2;

      end;

      (y1,y2) are_connected by A5, A6;

      hence thesis by A5, A6, BORSUK_2:def 2;

    end;

    theorem :: JORDAN:30

    

     Th30: for X be pathwise_connected non empty TopSpace, Y be non empty SubSpace of X, x1,x2 be Point of X, y1,y2 be Point of Y, f be Path of x1, x2 st x1 = y1 & x2 = y2 & ( rng f) c= the carrier of Y holds (y1,y2) are_connected & f is Path of y1, y2

    proof

      let X be pathwise_connected non empty TopSpace, Y be non empty SubSpace of X, x1,x2 be Point of X, y1,y2 be Point of Y;

      (x1,x2) are_connected by BORSUK_2:def 3;

      hence thesis by Th29;

    end;

    

     Lm7: for T be non empty TopSpace, a,b be Point of T holds for f be Path of a, b st (a,b) are_connected holds ( rng f) c= ( rng ( - f))

    proof

      let T be non empty TopSpace;

      let a,b be Point of T;

      let f be Path of a, b;

      assume

       A1: (a,b) are_connected ;

      let y be object;

      assume y in ( rng f);

      then

      consider x be object such that

       A2: x in ( dom f) and

       A3: (f . x) = y by FUNCT_1:def 3;

      reconsider x as Point of I[01] by A2;

      

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

      

       A5: (1 - x) is Point of I[01] by JORDAN5B: 4;

      then (( - f) . (1 - x)) = (f . (1 - (1 - x))) by A1, BORSUK_2:def 6;

      hence thesis by A3, A4, A5, FUNCT_1:def 3;

    end;

    theorem :: JORDAN:31

    

     Th31: for T be non empty TopSpace, a,b be Point of T holds for f be Path of a, b st (a,b) are_connected holds ( rng f) = ( rng ( - f))

    proof

      let T be non empty TopSpace;

      let a,b be Point of T;

      let f be Path of a, b;

      assume

       A1: (a,b) are_connected ;

      hence ( rng f) c= ( rng ( - f)) by Lm7;

      f = ( - ( - f)) by A1, BORSUK_6: 43;

      hence thesis by A1, Lm7;

    end;

    theorem :: JORDAN:32

    

     Th32: for T be pathwise_connected non empty TopSpace, a,b be Point of T holds for f be Path of a, b holds ( rng f) = ( rng ( - f)) by Th31, BORSUK_2:def 3;

    theorem :: JORDAN:33

    

     Th33: for T be non empty TopSpace, a,b,c be Point of T holds for f be Path of a, b, g be Path of b, c st (a,b) are_connected & (b,c) are_connected holds ( rng f) c= ( rng (f + g))

    proof

      let T be non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected ;

      let y be object;

      assume y in ( rng f);

      then

      consider x be object such that

       A3: x in ( dom f) and

       A4: (f . x) = y by FUNCT_1:def 3;

      

       A5: ( dom (f + g)) = the carrier of I[01] by FUNCT_2:def 1;

      reconsider x as Point of I[01] by A3;

      ((1 / 2) * x) = (x / 2);

      then

       A6: (x / 2) is Point of I[01] by BORSUK_6: 6;

      x <= 1 by BORSUK_1: 43;

      then (x / 2) <= (1 / 2) by XREAL_1: 72;

      then ((f + g) . (x / 2)) = (f . (2 * (x / 2))) by A1, A2, A6, BORSUK_2:def 5;

      hence thesis by A4, A5, A6, FUNCT_1:def 3;

    end;

    theorem :: JORDAN:34

    for T be pathwise_connected non empty TopSpace, a,b,c be Point of T holds for f be Path of a, b, g be Path of b, c holds ( rng f) c= ( rng (f + g))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      (b,c) are_connected by BORSUK_2:def 3;

      hence thesis by A1, Th33;

    end;

    theorem :: JORDAN:35

    

     Th35: for T be non empty TopSpace, a,b,c be Point of T holds for f be Path of b, c, g be Path of a, b st (a,b) are_connected & (b,c) are_connected holds ( rng f) c= ( rng (g + f))

    proof

      let T be non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of b, c;

      let g be Path of a, b;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected ;

      let y be object;

      assume y in ( rng f);

      then

      consider x be object such that

       A3: x in ( dom f) and

       A4: (f . x) = y by FUNCT_1:def 3;

      

       A5: ( dom (g + f)) = the carrier of I[01] by FUNCT_2:def 1;

      reconsider x as Point of I[01] by A3;

      

       A6: 0 <= x by BORSUK_1: 43;

      then

       A7: ( 0 + (1 / 2)) <= ((x / 2) + (1 / 2)) by XREAL_1: 6;

      x <= 1 by BORSUK_1: 43;

      then (x + 1) <= (1 + 1) by XREAL_1: 6;

      then ((x + 1) / 2) <= (2 / 2) by XREAL_1: 72;

      then

       A8: ((x / 2) + (1 / 2)) is Point of I[01] by A6, BORSUK_1: 43;

      then ((g + f) . ((x / 2) + (1 / 2))) = (f . ((2 * ((x / 2) + (1 / 2))) - 1)) by A1, A2, A7, BORSUK_2:def 5;

      hence thesis by A4, A5, A8, FUNCT_1:def 3;

    end;

    theorem :: JORDAN:36

    for T be pathwise_connected non empty TopSpace, a,b,c be Point of T holds for f be Path of b, c, g be Path of a, b holds ( rng f) c= ( rng (g + f))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of b, c;

      let g be Path of a, b;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      (b,c) are_connected by BORSUK_2:def 3;

      hence thesis by A1, Th35;

    end;

    theorem :: JORDAN:37

    

     Th37: for T be non empty TopSpace, a,b,c be Point of T holds for f be Path of a, b, g be Path of b, c st (a,b) are_connected & (b,c) are_connected holds ( rng (f + g)) = (( rng f) \/ ( rng g))

    proof

      let T be non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected ;

      thus ( rng (f + g)) c= (( rng f) \/ ( rng g))

      proof

        let y be object;

        assume y in ( rng (f + g));

        then

        consider x be object such that

         A3: x in ( dom (f + g)) and

         A4: y = ((f + g) . x) by FUNCT_1:def 3;

        reconsider x as Point of I[01] by A3;

        per cases ;

          suppose

           A5: x <= (1 / 2);

          then

           A6: ((f + g) . x) = (f . (2 * x)) by A1, A2, BORSUK_2:def 5;

          

           A7: ( rng f) c= (( rng f) \/ ( rng g)) by XBOOLE_1: 7;

          

           A8: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

          (2 * x) is Point of I[01] by A5, BORSUK_6: 3;

          then y in ( rng f) by A4, A6, A8, FUNCT_1:def 3;

          hence thesis by A7;

        end;

          suppose

           A9: (1 / 2) <= x;

          then

           A10: ((f + g) . x) = (g . ((2 * x) - 1)) by A1, A2, BORSUK_2:def 5;

          

           A11: ( rng g) c= (( rng f) \/ ( rng g)) by XBOOLE_1: 7;

          

           A12: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

          ((2 * x) - 1) is Point of I[01] by A9, BORSUK_6: 4;

          then y in ( rng g) by A4, A10, A12, FUNCT_1:def 3;

          hence thesis by A11;

        end;

      end;

      

       A13: ( rng f) c= ( rng (f + g)) by A1, A2, Th33;

      ( rng g) c= ( rng (f + g)) by A1, A2, Th35;

      hence thesis by A13, XBOOLE_1: 8;

    end;

    theorem :: JORDAN:38

    for T be pathwise_connected non empty TopSpace, a,b,c be Point of T holds for f be Path of a, b, g be Path of b, c holds ( rng (f + g)) = (( rng f) \/ ( rng g))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      (b,c) are_connected by BORSUK_2:def 3;

      hence thesis by A1, Th37;

    end;

    theorem :: JORDAN:39

    

     Th39: for T be non empty TopSpace, a,b,c,d be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d st (a,b) are_connected & (b,c) are_connected & (c,d) are_connected holds ( rng ((f + g) + h)) = ((( rng f) \/ ( rng g)) \/ ( rng h))

    proof

      let T be non empty TopSpace;

      let a,b,c,d be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected and

       A3: (c,d) are_connected ;

      (a,c) are_connected by A1, A2, BORSUK_6: 42;

      

      hence ( rng ((f + g) + h)) = (( rng (f + g)) \/ ( rng h)) by A3, Th37

      .= ((( rng f) \/ ( rng g)) \/ ( rng h)) by A1, A2, Th37;

    end;

    theorem :: JORDAN:40

    

     Th40: for T be pathwise_connected non empty TopSpace, a,b,c,d be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d holds ( rng ((f + g) + h)) = ((( rng f) \/ ( rng g)) \/ ( rng h))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c,d be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      

       A2: (b,c) are_connected by BORSUK_2:def 3;

      (c,d) are_connected by BORSUK_2:def 3;

      hence thesis by A1, A2, Th39;

    end;

    

     Lm8: for T be non empty TopSpace, a,b,c,d,e be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d, i be Path of d, e st (a,b) are_connected & (b,c) are_connected & (c,d) are_connected & (d,e) are_connected holds ( rng (((f + g) + h) + i)) = (((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i))

    proof

      let T be non empty TopSpace;

      let a,b,c,d,e be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      let i be Path of d, e;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected and

       A3: (c,d) are_connected and

       A4: (d,e) are_connected ;

      (a,c) are_connected by A1, A2, BORSUK_6: 42;

      then (a,d) are_connected by A3, BORSUK_6: 42;

      

      hence ( rng (((f + g) + h) + i)) = (( rng ((f + g) + h)) \/ ( rng i)) by A4, Th37

      .= (((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i)) by A1, A2, A3, Th39;

    end;

    

     Lm9: for T be pathwise_connected non empty TopSpace, a,b,c,d,e be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d, i be Path of d, e holds ( rng (((f + g) + h) + i)) = (((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c,d,e be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      let i be Path of d, e;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      

       A2: (b,c) are_connected by BORSUK_2:def 3;

      

       A3: (c,d) are_connected by BORSUK_2:def 3;

      (d,e) are_connected by BORSUK_2:def 3;

      hence thesis by A1, A2, A3, Lm8;

    end;

    

     Lm10: for T be non empty TopSpace, a,b,c,d,e,z be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d, i be Path of d, e, j be Path of e, z st (a,b) are_connected & (b,c) are_connected & (c,d) are_connected & (d,e) are_connected & (e,z) are_connected holds ( rng ((((f + g) + h) + i) + j)) = ((((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i)) \/ ( rng j))

    proof

      let T be non empty TopSpace;

      let a,b,c,d,e,z be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      let i be Path of d, e;

      let j be Path of e, z;

      assume that

       A1: (a,b) are_connected and

       A2: (b,c) are_connected and

       A3: (c,d) are_connected and

       A4: (d,e) are_connected and

       A5: (e,z) are_connected ;

      (a,c) are_connected by A1, A2, BORSUK_6: 42;

      then (a,d) are_connected by A3, BORSUK_6: 42;

      then (a,e) are_connected by A4, BORSUK_6: 42;

      

      hence ( rng ((((f + g) + h) + i) + j)) = (( rng (((f + g) + h) + i)) \/ ( rng j)) by A5, Th37

      .= ((((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i)) \/ ( rng j)) by A1, A2, A3, A4, Lm8;

    end;

    

     Lm11: for T be pathwise_connected non empty TopSpace, a,b,c,d,e,z be Point of T holds for f be Path of a, b, g be Path of b, c, h be Path of c, d, i be Path of d, e, j be Path of e, z holds ( rng ((((f + g) + h) + i) + j)) = ((((( rng f) \/ ( rng g)) \/ ( rng h)) \/ ( rng i)) \/ ( rng j))

    proof

      let T be pathwise_connected non empty TopSpace;

      let a,b,c,d,e,z be Point of T;

      let f be Path of a, b;

      let g be Path of b, c;

      let h be Path of c, d;

      let i be Path of d, e;

      let j be Path of e, z;

      

       A1: (a,b) are_connected by BORSUK_2:def 3;

      

       A2: (b,c) are_connected by BORSUK_2:def 3;

      

       A3: (c,d) are_connected by BORSUK_2:def 3;

      

       A4: (d,e) are_connected by BORSUK_2:def 3;

      (e,z) are_connected by BORSUK_2:def 3;

      hence thesis by A1, A2, A3, A4, Lm10;

    end;

    theorem :: JORDAN:41

    

     Th41: for T be non empty TopSpace, a be Point of T holds ( I[01] --> a) is Path of a, a

    proof

      let T be non empty TopSpace, a be Point of T;

      thus (a,a) are_connected ;

      thus thesis by BORSUK_1:def 14, BORSUK_1:def 15, TOPALG_3: 4;

    end;

    theorem :: JORDAN:42

    

     Th42: for p1,p2 be Point of ( TOP-REAL n), P be Subset of ( TOP-REAL n) holds P is_an_arc_of (p1,p2) implies ex F be Path of p1, p2, f be Function of I[01] , (( TOP-REAL n) | P) st ( rng f) = P & F = f

    proof

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

      assume

       A1: P is_an_arc_of (p1,p2);

      then

      reconsider P1 = P as non empty Subset of ( TOP-REAL n) by TOPREAL1: 1;

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

       A2: h is being_homeomorphism and

       A3: (h . 0 ) = p1 and

       A4: (h . 1) = p2 by A1, TOPREAL1:def 1;

      h is Function of I[01] , (( TOP-REAL n) | P1);

      then

      reconsider h1 = h as Function of I[01] , ( TOP-REAL n) by TOPREALA: 7;

      h1 is continuous by A2, PRE_TOPC: 26;

      then

      reconsider f = h as Path of p1, p2 by A3, A4, BORSUK_2:def 4;

      take f, h;

      

      thus ( rng h) = ( [#] (( TOP-REAL n) | P)) by A2, TOPS_2:def 5

      .= P by PRE_TOPC: 8;

      thus thesis;

    end;

    theorem :: JORDAN:43

    

     Th43: for p1,p2 be Point of ( TOP-REAL n) holds ex F be Path of p1, p2, f be Function of I[01] , (( TOP-REAL n) | ( LSeg (p1,p2))) st ( rng f) = ( LSeg (p1,p2)) & F = f

    proof

      let p1,p2 be Point of ( TOP-REAL n);

      per cases ;

        suppose

         A1: p1 = p2;

        then

        reconsider g = ( I[01] --> p1) as Path of p1, p2 by Th41;

        take g;

        

         A2: ( LSeg (p1,p2)) = {p1} by A1, RLTOPSP1: 70;

        

         A3: ( rng g) = {p1} by FUNCOP_1: 8;

        the carrier of (( TOP-REAL n) | ( LSeg (p1,p2))) = ( LSeg (p1,p2)) by PRE_TOPC: 8;

        then

        reconsider f = g as Function of I[01] , (( TOP-REAL n) | ( LSeg (p1,p2))) by A2, A3, FUNCT_2: 6;

        take f;

        thus thesis by A1, A3, RLTOPSP1: 70;

      end;

        suppose p1 <> p2;

        hence thesis by Th42, TOPREAL1: 9;

      end;

    end;

    theorem :: JORDAN:44

    

     Th44: for p1,p2,q1,q2 be Point of ( TOP-REAL 2) st P is_an_arc_of (p1,p2) & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 holds ex f be Path of q1, q2 st ( rng f) c= P & ( rng f) misses {p1, p2}

    proof

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2) such that

       A1: P is_an_arc_of (p1,p2) and

       A2: q1 in P and

       A3: q2 in P and

       A4: q1 <> p1 and

       A5: q1 <> p2 and

       A6: q2 <> p1 and

       A7: q2 <> p2;

      per cases ;

        suppose q1 = q2;

        then

        reconsider f = ( I[01] --> q1) as Path of q1, q2 by Th41;

        take f;

        

         A8: ( rng f) = {q1} by FUNCOP_1: 8;

        thus ( rng f) c= P by A2, A8, TARSKI:def 1;

        

         A9: not p1 in {q1} by A4, TARSKI:def 1;

         not p2 in {q1} by A5, TARSKI:def 1;

        hence thesis by A8, A9, ZFMISC_1: 51;

      end;

        suppose q1 <> q2;

        then

        consider Q be non empty Subset of T2 such that

         A10: Q is_an_arc_of (q1,q2) and

         A11: Q c= P and

         A12: Q misses {p1, p2} by A1, A2, A3, A4, A5, A6, A7, JORDAN16: 23;

        consider g be Path of q1, q2, f be Function of I[01] , (T2 | Q) such that

         A13: ( rng f) = Q and

         A14: g = f by A10, Th42;

        reconsider h = f as Function of I[01] , T2 by TOPREALA: 7;

        the carrier of (T2 | Q) = Q by PRE_TOPC: 8;

        then

        reconsider z1 = q1, z2 = q2 as Point of (T2 | Q) by A10, TOPREAL1: 1;

        

         A15: (z1,z2) are_connected

        proof

          take f;

          thus f is continuous by A14, PRE_TOPC: 27;

          thus thesis by A14, BORSUK_2:def 4;

        end;

        

         A16: f is continuous by A14, PRE_TOPC: 27;

        (f . 0 ) = z1 & (f . 1) = z2 by A14, BORSUK_2:def 4;

        then f is Path of z1, z2 by A15, A16, BORSUK_2:def 2;

        then

        reconsider h as Path of q1, q2 by A15, TOPALG_2: 1;

        take h;

        thus thesis by A11, A12, A13;

      end;

    end;

    begin

    theorem :: JORDAN:45

    

     Th45: a <= b & c <= d implies ( rectangle (a,b,c,d)) c= ( closed_inside_of_rectangle (a,b,c,d))

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      let x be object;

      assume x in ( rectangle (a,b,c,d));

      then x in { p : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c } by A1, A2, SPPOL_2: 54;

      then ex p st x = p & ((p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c);

      hence thesis by A1, A2;

    end;

    theorem :: JORDAN:46

    

     Th46: ( inside_of_rectangle (a,b,c,d)) c= ( closed_inside_of_rectangle (a,b,c,d))

    proof

      let x be object;

      assume x in ( inside_of_rectangle (a,b,c,d));

      then ex p st x = p & a < (p `1 ) & (p `1 ) < b & c < (p `2 ) & (p `2 ) < d;

      hence thesis;

    end;

    theorem :: JORDAN:47

    

     Th47: ( closed_inside_of_rectangle (a,b,c,d)) = (( outside_of_rectangle (a,b,c,d)) ` )

    proof

      set R = ( closed_inside_of_rectangle (a,b,c,d));

      set O = ( outside_of_rectangle (a,b,c,d));

      thus R c= (O ` )

      proof

        let x be object;

        assume x in R;

        then

        consider p such that

         A1: x = p and

         A2: a <= (p `1 ) and

         A3: (p `1 ) <= b and

         A4: c <= (p `2 ) and

         A5: (p `2 ) <= d;

        now

          assume p in O;

          then ex p1 st p1 = p & not (a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p1 `2 ) & (p1 `2 ) <= d);

          hence contradiction by A2, A3, A4, A5;

        end;

        hence thesis by A1, SUBSET_1: 29;

      end;

      let x be object;

      assume

       A6: x in (O ` );

      then

       A7: not x in O by XBOOLE_0:def 5;

      reconsider x as Point of T2 by A6;

      

       A8: a <= (x `1 ) by A7;

      

       A9: (x `1 ) <= b by A7;

      

       A10: c <= (x `2 ) by A7;

      (x `2 ) <= d by A7;

      hence thesis by A8, A9, A10;

    end;

    registration

      let a,b,c,d be Real;

      cluster ( closed_inside_of_rectangle (a,b,c,d)) -> closed;

      coherence

      proof

        set P2 = ( outside_of_rectangle (a,b,c,d));

        reconsider P2 as open Subset of T2 by JORDAN1: 34;

        (P2 ` ) is closed;

        hence thesis by Th47;

      end;

    end

    theorem :: JORDAN:48

    

     Th48: ( closed_inside_of_rectangle (a,b,c,d)) misses ( outside_of_rectangle (a,b,c,d))

    proof

      set R = ( closed_inside_of_rectangle (a,b,c,d));

      set P2 = ( outside_of_rectangle (a,b,c,d));

      assume R meets P2;

      then

      consider x be object such that

       A1: x in R and

       A2: x in P2 by XBOOLE_0: 3;

      

       A3: ex p st x = p & a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d by A1;

      ex p st x = p & not (a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d) by A2;

      hence thesis by A3;

    end;

    theorem :: JORDAN:49

    

     Th49: (( closed_inside_of_rectangle (a,b,c,d)) /\ ( inside_of_rectangle (a,b,c,d))) = ( inside_of_rectangle (a,b,c,d))

    proof

      set R = ( closed_inside_of_rectangle (a,b,c,d));

      set P1 = ( inside_of_rectangle (a,b,c,d));

      thus (R /\ P1) c= P1 by XBOOLE_1: 17;

      (P1 /\ P1) c= (P1 /\ R) by Th46, XBOOLE_1: 26;

      hence thesis;

    end;

    theorem :: JORDAN:50

    

     Th50: a < b & c < d implies ( Int ( closed_inside_of_rectangle (a,b,c,d))) = ( inside_of_rectangle (a,b,c,d))

    proof

      assume that

       A1: a < b and

       A2: c < d;

      set P = ( rectangle (a,b,c,d));

      set R = ( closed_inside_of_rectangle (a,b,c,d));

      set P1 = ( inside_of_rectangle (a,b,c,d));

      set P2 = ( outside_of_rectangle (a,b,c,d));

      

       A3: P = { p where p be Point of T2 : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c } by A1, A2, SPPOL_2: 54;

      

       A4: R misses P2 by Th48;

      

      thus ( Int R) = (( Cl ((P2 ` ) ` )) ` ) by Th47

      .= ((P2 \/ P) ` ) by A1, A2, A3, JORDAN1: 44

      .= ((P2 ` ) /\ (P ` )) by XBOOLE_1: 53

      .= (R /\ (P ` )) by Th47

      .= (R /\ (P1 \/ P2)) by A1, A2, A3, JORDAN1: 36

      .= ((R /\ P1) \/ (R /\ P2)) by XBOOLE_1: 23

      .= ((R /\ P1) \/ {} ) by A4

      .= P1 by Th49;

    end;

    theorem :: JORDAN:51

    

     Th51: a <= b & c <= d implies (( closed_inside_of_rectangle (a,b,c,d)) \ ( inside_of_rectangle (a,b,c,d))) = ( rectangle (a,b,c,d))

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      set R = ( rectangle (a,b,c,d));

      set P = ( closed_inside_of_rectangle (a,b,c,d));

      set P1 = ( inside_of_rectangle (a,b,c,d));

      

       A3: R = { p where p be Point of T2 : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c } by A1, A2, SPPOL_2: 54;

      thus (P \ P1) c= R

      proof

        let x be object;

        assume

         A4: x in (P \ P1);

        then

         A5: not x in P1 by XBOOLE_0:def 5;

        x in P by A4, XBOOLE_0:def 5;

        then

        consider p such that

         A6: x = p and

         A7: a <= (p `1 ) and

         A8: (p `1 ) <= b and

         A9: c <= (p `2 ) and

         A10: (p `2 ) <= d;

         not (a < (p `1 ) & (p `1 ) < b & c < (p `2 ) & (p `2 ) < d) by A5, A6;

        then (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c by A7, A8, A9, A10, XXREAL_0: 1;

        hence thesis by A3, A6;

      end;

      let x be object;

      assume

       A11: x in R;

      then

       A12: ex p st p = x & ((p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c) by A3;

      

       A13: R c= P by A1, A2, Th45;

      now

        assume x in P1;

        then ex p st x = p & a < (p `1 ) & (p `1 ) < b & c < (p `2 ) & (p `2 ) < d;

        hence contradiction by A12;

      end;

      hence thesis by A11, A13, XBOOLE_0:def 5;

    end;

    theorem :: JORDAN:52

    

     Th52: a < b & c < d implies ( Fr ( closed_inside_of_rectangle (a,b,c,d))) = ( rectangle (a,b,c,d))

    proof

      assume that

       A1: a < b and

       A2: c < d;

      set P = ( closed_inside_of_rectangle (a,b,c,d));

      

      thus ( Fr P) = (P \ ( Int P)) by TOPS_1: 43

      .= (P \ ( inside_of_rectangle (a,b,c,d))) by A1, A2, Th50

      .= ( rectangle (a,b,c,d)) by A1, A2, Th51;

    end;

    theorem :: JORDAN:53

    a <= b & c <= d implies ( W-bound ( closed_inside_of_rectangle (a,b,c,d))) = a

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      set X = ( closed_inside_of_rectangle (a,b,c,d));

      reconsider Z = (( proj1 | X) .: the carrier of (T2 | X)) as Subset of REAL ;

      

       A3: X = the carrier of (T2 | X) by PRE_TOPC: 8;

      

       A4: |[a, c]| in X by A1, A2, TOPREALA: 31;

      

       A5: for p be Real st p in Z holds p >= a

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A6: p0 in the carrier of (T2 | X) and p0 in the carrier of (T2 | X) and

         A7: p = (( proj1 | X) . p0) by FUNCT_2: 64;

        ex p1 st p0 = p1 & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p1 `2 ) & (p1 `2 ) <= d by A3, A6;

        hence thesis by A3, A6, A7, PSCOMP_1: 22;

      end;

      for q be Real st for p be Real st p in Z holds p >= q holds a >= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p >= q;

        

         A9: ( |[a, c]| `1 ) = a by EUCLID: 52;

        (( proj1 | X) . |[a, c]|) = ( |[a, c]| `1 ) by A1, A2, PSCOMP_1: 22, TOPREALA: 31;

        hence thesis by A3, A4, A8, A9, FUNCT_2: 35;

      end;

      hence thesis by A4, A5, SEQ_4: 44;

    end;

    theorem :: JORDAN:54

    a <= b & c <= d implies ( S-bound ( closed_inside_of_rectangle (a,b,c,d))) = c

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      set X = ( closed_inside_of_rectangle (a,b,c,d));

      reconsider Z = (( proj2 | X) .: the carrier of (T2 | X)) as Subset of REAL ;

      

       A3: X = the carrier of (T2 | X) by PRE_TOPC: 8;

      

       A4: |[a, c]| in X by A1, A2, TOPREALA: 31;

      

       A5: for p be Real st p in Z holds p >= c

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A6: p0 in the carrier of (T2 | X) and p0 in the carrier of (T2 | X) and

         A7: p = (( proj2 | X) . p0) by FUNCT_2: 64;

        ex p1 st p0 = p1 & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p1 `2 ) & (p1 `2 ) <= d by A3, A6;

        hence thesis by A3, A6, A7, PSCOMP_1: 23;

      end;

      for q be Real st for p be Real st p in Z holds p >= q holds c >= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p >= q;

        

         A9: ( |[a, c]| `2 ) = c by EUCLID: 52;

        (( proj2 | X) . |[a, c]|) = ( |[a, c]| `2 ) by A1, A2, PSCOMP_1: 23, TOPREALA: 31;

        hence thesis by A3, A4, A8, A9, FUNCT_2: 35;

      end;

      hence thesis by A4, A5, SEQ_4: 44;

    end;

    theorem :: JORDAN:55

    a <= b & c <= d implies ( E-bound ( closed_inside_of_rectangle (a,b,c,d))) = b

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      set X = ( closed_inside_of_rectangle (a,b,c,d));

      reconsider Z = (( proj1 | X) .: the carrier of (T2 | X)) as Subset of REAL ;

      

       A3: X = the carrier of (T2 | X) by PRE_TOPC: 8;

      

       A4: for p be Real st p in Z holds p <= b

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A5: p0 in the carrier of (T2 | X) and p0 in the carrier of (T2 | X) and

         A6: p = (( proj1 | X) . p0) by FUNCT_2: 64;

        ex p1 st p0 = p1 & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p1 `2 ) & (p1 `2 ) <= d by A3, A5;

        hence thesis by A3, A5, A6, PSCOMP_1: 22;

      end;

      

       A7: for q be Real st for p be Real st p in Z holds p <= q holds b <= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p <= q;

        

         A9: ( |[b, d]| `1 ) = b by EUCLID: 52;

        ( |[b, d]| `2 ) = d by EUCLID: 52;

        then

         A10: |[b, d]| in X by A1, A2, A9;

        then (( proj1 | X) . |[b, d]|) = ( |[b, d]| `1 ) by PSCOMP_1: 22;

        hence thesis by A3, A8, A9, A10, FUNCT_2: 35;

      end;

       |[a, c]| in X by A1, A2, TOPREALA: 31;

      hence thesis by A4, A7, SEQ_4: 46;

    end;

    theorem :: JORDAN:56

    a <= b & c <= d implies ( N-bound ( closed_inside_of_rectangle (a,b,c,d))) = d

    proof

      assume that

       A1: a <= b and

       A2: c <= d;

      set X = ( closed_inside_of_rectangle (a,b,c,d));

      reconsider Z = (( proj2 | X) .: the carrier of (T2 | X)) as Subset of REAL ;

      

       A3: X = the carrier of (T2 | X) by PRE_TOPC: 8;

      

       A4: for p be Real st p in Z holds p <= d

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A5: p0 in the carrier of (T2 | X) and p0 in the carrier of (T2 | X) and

         A6: p = (( proj2 | X) . p0) by FUNCT_2: 64;

        ex p1 st p0 = p1 & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p1 `2 ) & (p1 `2 ) <= d by A3, A5;

        hence thesis by A3, A5, A6, PSCOMP_1: 23;

      end;

      

       A7: for q be Real st for p be Real st p in Z holds p <= q holds d <= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p <= q;

        

         A9: ( |[b, d]| `1 ) = b by EUCLID: 52;

        

         A10: ( |[b, d]| `2 ) = d by EUCLID: 52;

        then

         A11: |[b, d]| in X by A1, A2, A9;

        then (( proj2 | X) . |[b, d]|) = ( |[b, d]| `2 ) by PSCOMP_1: 23;

        hence thesis by A3, A8, A10, A11, FUNCT_2: 35;

      end;

       |[a, c]| in X by A1, A2, TOPREALA: 31;

      hence thesis by A4, A7, SEQ_4: 46;

    end;

    theorem :: JORDAN:57

    

     Th57: a < b & c < d & p1 in ( closed_inside_of_rectangle (a,b,c,d)) & not p2 in ( closed_inside_of_rectangle (a,b,c,d)) & P is_an_arc_of (p1,p2) implies ( Segment (P,p1,p2,p1,( First_Point (P,p1,p2,( rectangle (a,b,c,d)))))) c= ( closed_inside_of_rectangle (a,b,c,d))

    proof

      set R = ( closed_inside_of_rectangle (a,b,c,d));

      set dR = ( rectangle (a,b,c,d));

      set n = ( First_Point (P,p1,p2,dR));

      assume that

       A1: a < b and

       A2: c < d and

       A3: p1 in R and

       A4: not p2 in R and

       A5: P is_an_arc_of (p1,p2);

      let x be object;

      assume that

       A6: x in ( Segment (P,p1,p2,p1,n)) and

       A7: not x in R;

      reconsider x as Point of T2 by A6;

      

       A8: ( Fr R) = dR by A1, A2, Th52;

      p1 in P by A5, TOPREAL1: 1;

      then

       A9: P meets R by A3, XBOOLE_0: 3;

      p2 in P by A5, TOPREAL1: 1;

      then (P \ R) <> ( {} T2) by A4, XBOOLE_0:def 5;

      then

       A10: P meets dR by A5, A8, A9, CONNSP_1: 22, JORDAN6: 10;

      

       A11: P is closed by A5, JORDAN6: 11;

      then

       A12: (P /\ dR) is closed;

      

       A13: n in (P /\ dR) by A5, A10, A11, JORDAN5C:def 1;

      per cases ;

        suppose x = n;

        then

         A14: x in dR by A13, XBOOLE_0:def 4;

        dR c= R by A1, A2, Th45;

        hence thesis by A7, A14;

      end;

        suppose

         A15: x <> n;

        reconsider P as non empty Subset of T2 by A5, TOPREAL1: 1;

        consider f be Function of I[01] , (T2 | P) such that

         A16: f is being_homeomorphism and

         A17: (f . 0 ) = p1 and

         A18: (f . 1) = p2 by A5, TOPREAL1:def 1;

        

         A19: ( rng f) = ( [#] (T2 | P)) by A16, TOPS_2:def 5

        .= P by PRE_TOPC:def 5;

        n in P by A13, XBOOLE_0:def 4;

        then

        consider na be object such that

         A20: na in ( dom f) and

         A21: (f . na) = n by A19, FUNCT_1:def 3;

        reconsider na as Real by A20;

        

         A22: 0 <= na by A20, BORSUK_1: 43;

        

         A23: na <= 1 by A20, BORSUK_1: 43;

        

         A24: ( Segment (P,p1,p2,p1,n)) c= P by JORDAN16: 2;

        then

        consider xa be object such that

         A25: xa in ( dom f) and

         A26: (f . xa) = x by A6, A19, FUNCT_1:def 3;

        reconsider xa as Real by A25;

        

         A27: 0 <= xa by A25, BORSUK_1: 43;

        

         A28: xa <= 1 by A25, BORSUK_1: 43;

        

         A29: ( Segment (P,p1,p2,p1,x)) is_an_arc_of (p1,x) by A3, A5, A6, A7, A24, JORDAN16: 24;

        then p1 in ( Segment (P,p1,p2,p1,x)) by TOPREAL1: 1;

        then

         A30: ( Segment (P,p1,p2,p1,x)) meets R by A3, XBOOLE_0: 3;

        x in ( Segment (P,p1,p2,p1,x)) by A29, TOPREAL1: 1;

        then (( Segment (P,p1,p2,p1,x)) \ R) <> ( {} T2) by A7, XBOOLE_0:def 5;

        then ( Segment (P,p1,p2,p1,x)) meets ( Fr R) by A29, A30, CONNSP_1: 22, JORDAN6: 10;

        then

        consider z be object such that

         A31: z in ( Segment (P,p1,p2,p1,x)) and

         A32: z in dR by A8, XBOOLE_0: 3;

        reconsider z as Point of T2 by A31;

        ( Segment (P,p1,p2,p1,x)) = { p : LE (p1,p,P,p1,p2) & LE (p,x,P,p1,p2) } by JORDAN6: 26;

        then

         A33: ex zz be Point of T2 st (zz = z) & ( LE (p1,zz,P,p1,p2)) & ( LE (zz,x,P,p1,p2)) by A31;

        ( Segment (P,p1,p2,p1,x)) c= P by JORDAN16: 2;

        then

        consider za be object such that

         A34: za in ( dom f) and

         A35: (f . za) = z by A19, A31, FUNCT_1:def 3;

        reconsider za as Real by A34;

        

         A36: 0 <= za by A34, BORSUK_1: 43;

        

         A37: za <= 1 by A34, BORSUK_1: 43;

        

         A38: na <= za by A5, A10, A12, A16, A17, A18, A21, A23, A32, A35, A36, JORDAN5C:def 1;

        

         A39: za <= xa by A16, A17, A18, A26, A27, A28, A33, A35, A37, JORDAN5C:def 3;

        ( Segment (P,p1,p2,p1,n)) = { p : LE (p1,p,P,p1,p2) & LE (p,n,P,p1,p2) } by JORDAN6: 26;

        then ex xx be Point of T2 st (xx = x) & ( LE (p1,xx,P,p1,p2)) & ( LE (xx,n,P,p1,p2)) by A6;

        then xa <= na by A16, A17, A18, A21, A22, A23, A26, A28, JORDAN5C:def 3;

        then xa < na by A15, A21, A26, XXREAL_0: 1;

        hence thesis by A38, A39, XXREAL_0: 2;

      end;

    end;

    begin

    definition

      let S,T be non empty TopSpace, x be Point of [:S, T:];

      :: original: `1

      redefine

      func x `1 -> Element of S ;

      coherence

      proof

        the carrier of [:S, T:] = [:the carrier of S, the carrier of T:] by BORSUK_1:def 2;

        hence thesis by MCART_1: 10;

      end;

      :: original: `2

      redefine

      func x `2 -> Element of T ;

      coherence

      proof

        the carrier of [:S, T:] = [:the carrier of S, the carrier of T:] by BORSUK_1:def 2;

        hence thesis by MCART_1: 10;

      end;

    end

    definition

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

      :: JORDAN:def1

      func diffX2_1 (o) -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def1: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = (((x `2 ) `1 ) - (o `1 ));

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In (((($1 `2 ) `1 ) - (o `1 )), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A1: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A1;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A2: for x be Point of [:T2, T2:] holds (f . x) = (((x `2 ) `1 ) - (o `1 )) and

         A3: for x be Point of [:T2, T2:] holds (g . x) = (((x `2 ) `1 ) - (o `1 ));

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = (((x `2 ) `1 ) - (o `1 )) by A2

          .= (g . x) by A3;

        end;

        hence thesis by FUNCT_2: 63;

      end;

      :: JORDAN:def2

      func diffX2_2 (o) -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def2: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = (((x `2 ) `2 ) - (o `2 ));

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In (((($1 `2 ) `2 ) - (o `2 )), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A4: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A4;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A5: for x be Point of [:T2, T2:] holds (f . x) = (((x `2 ) `2 ) - (o `2 )) and

         A6: for x be Point of [:T2, T2:] holds (g . x) = (((x `2 ) `2 ) - (o `2 ));

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = (((x `2 ) `2 ) - (o `2 )) by A5

          .= (g . x) by A6;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    definition

      :: JORDAN:def3

      func diffX1_X2_1 -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def3: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = (((x `1 ) `1 ) - ((x `2 ) `1 ));

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In (((($1 `1 ) `1 ) - (($1 `2 ) `1 )), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A1: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A1;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A2: for x be Point of [:T2, T2:] holds (f . x) = (((x `1 ) `1 ) - ((x `2 ) `1 )) and

         A3: for x be Point of [:T2, T2:] holds (g . x) = (((x `1 ) `1 ) - ((x `2 ) `1 ));

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = (((x `1 ) `1 ) - ((x `2 ) `1 )) by A2

          .= (g . x) by A3;

        end;

        hence thesis by FUNCT_2: 63;

      end;

      :: JORDAN:def4

      func diffX1_X2_2 -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def4: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = (((x `1 ) `2 ) - ((x `2 ) `2 ));

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In (((($1 `1 ) `2 ) - (($1 `2 ) `2 )), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A4: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A4;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A5: for x be Point of [:T2, T2:] holds (f . x) = (((x `1 ) `2 ) - ((x `2 ) `2 )) and

         A6: for x be Point of [:T2, T2:] holds (g . x) = (((x `1 ) `2 ) - ((x `2 ) `2 ));

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = (((x `1 ) `2 ) - ((x `2 ) `2 )) by A5

          .= (g . x) by A6;

        end;

        hence thesis by FUNCT_2: 63;

      end;

      :: JORDAN:def5

      func Proj2_1 -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def5: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = ((x `2 ) `1 );

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In ((($1 `2 ) `1 ), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A7: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A7;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A8: for x be Point of [:T2, T2:] holds (f . x) = ((x `2 ) `1 ) and

         A9: for x be Point of [:T2, T2:] holds (g . x) = ((x `2 ) `1 );

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = ((x `2 ) `1 ) by A8

          .= (g . x) by A9;

        end;

        hence thesis by FUNCT_2: 63;

      end;

      :: JORDAN:def6

      func Proj2_2 -> RealMap of [:( TOP-REAL 2), ( TOP-REAL 2):] means

      : Def6: for x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):] holds (it . x) = ((x `2 ) `2 );

      existence

      proof

        deffunc F( Point of [:T2, T2:]) = ( In ((($1 `2 ) `2 ), REAL ));

        consider xo be RealMap of [:T2, T2:] such that

         A10: for x be Point of [:T2, T2:] holds (xo . x) = F(x) from FUNCT_2:sch 4;

        take xo;

        let x be Point of [:( TOP-REAL 2), ( TOP-REAL 2):];

        (xo . x) = F(x) by A10;

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be RealMap of [:T2, T2:] such that

         A11: for x be Point of [:T2, T2:] holds (f . x) = ((x `2 ) `2 ) and

         A12: for x be Point of [:T2, T2:] holds (g . x) = ((x `2 ) `2 );

        now

          let x be Point of [:T2, T2:];

          

          thus (f . x) = ((x `2 ) `2 ) by A11

          .= (g . x) by A12;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    theorem :: JORDAN:58

    

     Th58: for o be Point of ( TOP-REAL 2) holds ( diffX2_1 o) is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

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

      reconsider Xo = ( diffX2_1 o) as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (Xo . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (Xo .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (Xo . p) in V and

         A2: V is open;

        

         A3: (Xo . p) = (((p `2 ) `1 ) - (o `1 )) by Def1;

        set r = (((p `2 ) `1 ) - (o `1 ));

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A4: 0 < g and

         A5: ].(r - g), (r + g).[ c= V1 by A1, A3, RCOMP_1: 19;

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

        set W2 = { |[x, y]| where x,y be Real : (((p `2 ) `1 ) - g) < x & x < (((p `2 ) `1 ) + g) };

        W2 c= the carrier of T2

        proof

          let a be object;

          assume a in W2;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `1 ) - g) < x & x < (((p `2 ) `1 ) + g);

          hence thesis;

        end;

        then

        reconsider W2 as Subset of T2;

        take [:( [#] T2), W2:];

        

         A6: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A7: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A8: (((p `2 ) `1 ) - g) < (((p `2 ) `1 ) - 0 ) by A4, XREAL_1: 15;

        (((p `2 ) `1 ) + 0 ) < (((p `2 ) `1 ) + g) by A4, XREAL_1: 6;

        then (p `2 ) in W2 by A6, A8;

        hence p in [:( [#] T2), W2:] by A7, ZFMISC_1:def 2;

        W2 is open by PSCOMP_1: 19;

        hence [:( [#] T2), W2:] is open by BORSUK_1: 6;

        let b be object;

        assume b in (Xo .: [:( [#] T2), W2:]);

        then

        consider a be Point of [:T2, T2:] such that

         A9: a in [:( [#] T2), W2:] and

         A10: (Xo . a) = b by FUNCT_2: 65;

        

         A11: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A12: (( diffX2_1 o) . a) = (((a `2 ) `1 ) - (o `1 )) by Def1;

        (a `2 ) in W2 by A9, A11, ZFMISC_1: 87;

        then

        consider x2,y2 be Real such that

         A13: (a `2 ) = |[x2, y2]| and

         A14: (((p `2 ) `1 ) - g) < x2 and

         A15: x2 < (((p `2 ) `1 ) + g);

        

         A16: ((a `2 ) `1 ) = x2 by A13, EUCLID: 52;

        then

         A17: ((((p `2 ) `1 ) - g) - (o `1 )) < (((a `2 ) `1 ) - (o `1 )) by A14, XREAL_1: 9;

        (((a `2 ) `1 ) - (o `1 )) < ((((p `2 ) `1 ) + g) - (o `1 )) by A15, A16, XREAL_1: 9;

        then (((a `2 ) `1 ) - (o `1 )) in ].(r - g), (r + g).[ by A17, XXREAL_1: 4;

        hence thesis by A5, A10, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:59

    

     Th59: for o be Point of ( TOP-REAL 2) holds ( diffX2_2 o) is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

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

      reconsider Yo = ( diffX2_2 o) as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (Yo . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (Yo .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (Yo . p) in V and

         A2: V is open;

        

         A3: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A4: (Yo . p) = (((p `2 ) `2 ) - (o `2 )) by Def2;

        set r = (((p `2 ) `2 ) - (o `2 ));

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A5: 0 < g and

         A6: ].(r - g), (r + g).[ c= V1 by A1, A4, RCOMP_1: 19;

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

        set W2 = { |[x, y]| where x,y be Real : (((p `2 ) `2 ) - g) < y & y < (((p `2 ) `2 ) + g) };

        W2 c= the carrier of T2

        proof

          let a be object;

          assume a in W2;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `2 ) - g) < y & y < (((p `2 ) `2 ) + g);

          hence thesis;

        end;

        then

        reconsider W2 as Subset of T2;

        take [:( [#] T2), W2:];

        

         A7: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A8: (((p `2 ) `2 ) - g) < (((p `2 ) `2 ) - 0 ) by A5, XREAL_1: 15;

        (((p `2 ) `2 ) + 0 ) < (((p `2 ) `2 ) + g) by A5, XREAL_1: 6;

        then (p `2 ) in W2 by A7, A8;

        hence p in [:( [#] T2), W2:] by A3, ZFMISC_1:def 2;

        W2 is open by PSCOMP_1: 21;

        hence [:( [#] T2), W2:] is open by BORSUK_1: 6;

        let b be object;

        assume b in (Yo .: [:( [#] T2), W2:]);

        then

        consider a be Point of [:T2, T2:] such that

         A9: a in [:( [#] T2), W2:] and

         A10: (Yo . a) = b by FUNCT_2: 65;

        

         A11: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A12: (( diffX2_2 o) . a) = (((a `2 ) `2 ) - (o `2 )) by Def2;

        (a `2 ) in W2 by A9, A11, ZFMISC_1: 87;

        then

        consider x2,y2 be Real such that

         A13: (a `2 ) = |[x2, y2]| and

         A14: (((p `2 ) `2 ) - g) < y2 and

         A15: y2 < (((p `2 ) `2 ) + g);

        

         A16: ((a `2 ) `2 ) = y2 by A13, EUCLID: 52;

        then

         A17: ((((p `2 ) `2 ) - g) - (o `2 )) < (((a `2 ) `2 ) - (o `2 )) by A14, XREAL_1: 9;

        (((a `2 ) `2 ) - (o `2 )) < ((((p `2 ) `2 ) + g) - (o `2 )) by A15, A16, XREAL_1: 9;

        then (((a `2 ) `2 ) - (o `2 )) in ].(r - g), (r + g).[ by A17, XXREAL_1: 4;

        hence thesis by A6, A10, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:60

    

     Th60: diffX1_X2_1 is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

      reconsider Dx = diffX1_X2_1 as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (Dx . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (Dx .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (Dx . p) in V and

         A2: V is open;

        

         A3: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A4: ( diffX1_X2_1 . p) = (((p `1 ) `1 ) - ((p `2 ) `1 )) by Def3;

        set r = (((p `1 ) `1 ) - ((p `2 ) `1 ));

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A5: 0 < g and

         A6: ].(r - g), (r + g).[ c= V1 by A1, A4, RCOMP_1: 19;

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

        set W1 = { |[x, y]| where x,y be Real : (((p `1 ) `1 ) - (g / 2)) < x & x < (((p `1 ) `1 ) + (g / 2)) };

        set W2 = { |[x, y]| where x,y be Real : (((p `2 ) `1 ) - (g / 2)) < x & x < (((p `2 ) `1 ) + (g / 2)) };

        W1 c= the carrier of T2

        proof

          let a be object;

          assume a in W1;

          then ex x,y be Real st a = |[x, y]| & (((p `1 ) `1 ) - (g / 2)) < x & x < (((p `1 ) `1 ) + (g / 2));

          hence thesis;

        end;

        then

        reconsider W1 as Subset of T2;

        W2 c= the carrier of T2

        proof

          let a be object;

          assume a in W2;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `1 ) - (g / 2)) < x & x < (((p `2 ) `1 ) + (g / 2));

          hence thesis;

        end;

        then

        reconsider W2 as Subset of T2;

        take [:W1, W2:];

        

         A7: (p `1 ) = |[((p `1 ) `1 ), ((p `1 ) `2 )]| by EUCLID: 53;

        

         A8: ( 0 / 2) < (g / 2) by A5, XREAL_1: 74;

        then

         A9: (((p `1 ) `1 ) - (g / 2)) < (((p `1 ) `1 ) - 0 ) by XREAL_1: 15;

        (((p `1 ) `1 ) + 0 ) < (((p `1 ) `1 ) + (g / 2)) by A8, XREAL_1: 6;

        then

         A10: (p `1 ) in W1 by A7, A9;

        

         A11: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A12: (((p `2 ) `1 ) - (g / 2)) < (((p `2 ) `1 ) - 0 ) by A8, XREAL_1: 15;

        (((p `2 ) `1 ) + 0 ) < (((p `2 ) `1 ) + (g / 2)) by A8, XREAL_1: 6;

        then (p `2 ) in W2 by A11, A12;

        hence p in [:W1, W2:] by A3, A10, ZFMISC_1:def 2;

        

         A13: W1 is open by PSCOMP_1: 19;

        W2 is open by PSCOMP_1: 19;

        hence [:W1, W2:] is open by A13, BORSUK_1: 6;

        let b be object;

        assume b in (Dx .: [:W1, W2:]);

        then

        consider a be Point of [:T2, T2:] such that

         A14: a in [:W1, W2:] and

         A15: (Dx . a) = b by FUNCT_2: 65;

        

         A16: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A17: ( diffX1_X2_1 . a) = (((a `1 ) `1 ) - ((a `2 ) `1 )) by Def3;

        (a `1 ) in W1 by A14, A16, ZFMISC_1: 87;

        then

        consider x1,y1 be Real such that

         A18: (a `1 ) = |[x1, y1]| and

         A19: (((p `1 ) `1 ) - (g / 2)) < x1 and

         A20: x1 < (((p `1 ) `1 ) + (g / 2));

        

         A21: ((a `1 ) `1 ) = x1 by A18, EUCLID: 52;

        

         A22: ((((p `1 ) `1 ) - (g / 2)) + (g / 2)) < (x1 + (g / 2)) by A19, XREAL_1: 6;

        

         A23: (((p `1 ) `1 ) - x1) > (((p `1 ) `1 ) - (((p `1 ) `1 ) + (g / 2))) by A20, XREAL_1: 15;

        

         A24: (((p `1 ) `1 ) - x1) < ((x1 + (g / 2)) - x1) by A22, XREAL_1: 9;

        (((p `1 ) `1 ) - x1) > ( - (g / 2)) by A23;

        then

         A25: |.(((p `1 ) `1 ) - x1).| < (g / 2) by A24, SEQ_2: 1;

        (a `2 ) in W2 by A14, A16, ZFMISC_1: 87;

        then

        consider x2,y2 be Real such that

         A26: (a `2 ) = |[x2, y2]| and

         A27: (((p `2 ) `1 ) - (g / 2)) < x2 and

         A28: x2 < (((p `2 ) `1 ) + (g / 2));

        

         A29: ((a `2 ) `1 ) = x2 by A26, EUCLID: 52;

        

         A30: ((((p `2 ) `1 ) - (g / 2)) + (g / 2)) < (x2 + (g / 2)) by A27, XREAL_1: 6;

        

         A31: (((p `2 ) `1 ) - x2) > (((p `2 ) `1 ) - (((p `2 ) `1 ) + (g / 2))) by A28, XREAL_1: 15;

        

         A32: (((p `2 ) `1 ) - x2) < ((x2 + (g / 2)) - x2) by A30, XREAL_1: 9;

        (((p `2 ) `1 ) - x2) > ( - (g / 2)) by A31;

        then |.(((p `2 ) `1 ) - x2).| < (g / 2) by A32, SEQ_2: 1;

        then

         A33: ( |.(((p `1 ) `1 ) - x1).| + |.(((p `2 ) `1 ) - x2).|) < ((g / 2) + (g / 2)) by A25, XREAL_1: 8;

         |.((((p `1 ) `1 ) - x1) - (((p `2 ) `1 ) - x2)).| <= ( |.(((p `1 ) `1 ) - x1).| + |.(((p `2 ) `1 ) - x2).|) by COMPLEX1: 57;

        then |.( - ((((p `1 ) `1 ) - x1) - (((p `2 ) `1 ) - x2))).| <= ( |.(((p `1 ) `1 ) - x1).| + |.(((p `2 ) `1 ) - x2).|) by COMPLEX1: 52;

        then |.((x1 - x2) - r).| < g by A33, XXREAL_0: 2;

        then (((a `1 ) `1 ) - ((a `2 ) `1 )) in ].(r - g), (r + g).[ by A21, A29, RCOMP_1: 1;

        hence thesis by A6, A15, A17;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:61

    

     Th61: diffX1_X2_2 is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

      reconsider Dy = diffX1_X2_2 as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (Dy . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (Dy .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (Dy . p) in V and

         A2: V is open;

        

         A3: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A4: ( diffX1_X2_2 . p) = (((p `1 ) `2 ) - ((p `2 ) `2 )) by Def4;

        set r = (((p `1 ) `2 ) - ((p `2 ) `2 ));

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A5: 0 < g and

         A6: ].(r - g), (r + g).[ c= V1 by A1, A4, RCOMP_1: 19;

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

        set W1 = { |[x, y]| where x,y be Real : (((p `1 ) `2 ) - (g / 2)) < y & y < (((p `1 ) `2 ) + (g / 2)) };

        set W2 = { |[x, y]| where x,y be Real : (((p `2 ) `2 ) - (g / 2)) < y & y < (((p `2 ) `2 ) + (g / 2)) };

        W1 c= the carrier of T2

        proof

          let a be object;

          assume a in W1;

          then ex x,y be Real st a = |[x, y]| & (((p `1 ) `2 ) - (g / 2)) < y & y < (((p `1 ) `2 ) + (g / 2));

          hence thesis;

        end;

        then

        reconsider W1 as Subset of T2;

        W2 c= the carrier of T2

        proof

          let a be object;

          assume a in W2;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `2 ) - (g / 2)) < y & y < (((p `2 ) `2 ) + (g / 2));

          hence thesis;

        end;

        then

        reconsider W2 as Subset of T2;

        take [:W1, W2:];

        

         A7: (p `1 ) = |[((p `1 ) `1 ), ((p `1 ) `2 )]| by EUCLID: 53;

        

         A8: ( 0 / 2) < (g / 2) by A5, XREAL_1: 74;

        then

         A9: (((p `1 ) `2 ) - (g / 2)) < (((p `1 ) `2 ) - 0 ) by XREAL_1: 15;

        (((p `1 ) `2 ) + 0 ) < (((p `1 ) `2 ) + (g / 2)) by A8, XREAL_1: 6;

        then

         A10: (p `1 ) in W1 by A7, A9;

        

         A11: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A12: (((p `2 ) `2 ) - (g / 2)) < (((p `2 ) `2 ) - 0 ) by A8, XREAL_1: 15;

        (((p `2 ) `2 ) + 0 ) < (((p `2 ) `2 ) + (g / 2)) by A8, XREAL_1: 6;

        then (p `2 ) in W2 by A11, A12;

        hence p in [:W1, W2:] by A3, A10, ZFMISC_1:def 2;

        

         A13: W1 is open by PSCOMP_1: 21;

        W2 is open by PSCOMP_1: 21;

        hence [:W1, W2:] is open by A13, BORSUK_1: 6;

        let b be object;

        assume b in (Dy .: [:W1, W2:]);

        then

        consider a be Point of [:T2, T2:] such that

         A14: a in [:W1, W2:] and

         A15: (Dy . a) = b by FUNCT_2: 65;

        

         A16: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A17: ( diffX1_X2_2 . a) = (((a `1 ) `2 ) - ((a `2 ) `2 )) by Def4;

        (a `1 ) in W1 by A14, A16, ZFMISC_1: 87;

        then

        consider x1,y1 be Real such that

         A18: (a `1 ) = |[x1, y1]| and

         A19: (((p `1 ) `2 ) - (g / 2)) < y1 and

         A20: y1 < (((p `1 ) `2 ) + (g / 2));

        

         A21: ((a `1 ) `2 ) = y1 by A18, EUCLID: 52;

        

         A22: ((((p `1 ) `2 ) - (g / 2)) + (g / 2)) < (y1 + (g / 2)) by A19, XREAL_1: 6;

        

         A23: (((p `1 ) `2 ) - y1) > (((p `1 ) `2 ) - (((p `1 ) `2 ) + (g / 2))) by A20, XREAL_1: 15;

        

         A24: (((p `1 ) `2 ) - y1) < ((y1 + (g / 2)) - y1) by A22, XREAL_1: 9;

        (((p `1 ) `2 ) - y1) > ( - (g / 2)) by A23;

        then

         A25: |.(((p `1 ) `2 ) - y1).| < (g / 2) by A24, SEQ_2: 1;

        (a `2 ) in W2 by A14, A16, ZFMISC_1: 87;

        then

        consider x2,y2 be Real such that

         A26: (a `2 ) = |[x2, y2]| and

         A27: (((p `2 ) `2 ) - (g / 2)) < y2 and

         A28: y2 < (((p `2 ) `2 ) + (g / 2));

        

         A29: ((a `2 ) `2 ) = y2 by A26, EUCLID: 52;

        

         A30: ((((p `2 ) `2 ) - (g / 2)) + (g / 2)) < (y2 + (g / 2)) by A27, XREAL_1: 6;

        

         A31: (((p `2 ) `2 ) - y2) > (((p `2 ) `2 ) - (((p `2 ) `2 ) + (g / 2))) by A28, XREAL_1: 15;

        

         A32: (((p `2 ) `2 ) - y2) < ((y2 + (g / 2)) - y2) by A30, XREAL_1: 9;

        (((p `2 ) `2 ) - y2) > ( - (g / 2)) by A31;

        then |.(((p `2 ) `2 ) - y2).| < (g / 2) by A32, SEQ_2: 1;

        then

         A33: ( |.(((p `1 ) `2 ) - y1).| + |.(((p `2 ) `2 ) - y2).|) < ((g / 2) + (g / 2)) by A25, XREAL_1: 8;

         |.((((p `1 ) `2 ) - y1) - (((p `2 ) `2 ) - y2)).| <= ( |.(((p `1 ) `2 ) - y1).| + |.(((p `2 ) `2 ) - y2).|) by COMPLEX1: 57;

        then |.( - ((((p `1 ) `2 ) - y1) - (((p `2 ) `2 ) - y2))).| <= ( |.(((p `1 ) `2 ) - y1).| + |.(((p `2 ) `2 ) - y2).|) by COMPLEX1: 52;

        then |.((y1 - y2) - r).| < g by A33, XXREAL_0: 2;

        then (((a `1 ) `2 ) - ((a `2 ) `2 )) in ].(r - g), (r + g).[ by A21, A29, RCOMP_1: 1;

        hence thesis by A6, A15, A17;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:62

    

     Th62: Proj2_1 is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

      reconsider fX2 = Proj2_1 as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (fX2 . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (fX2 .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (fX2 . p) in V and

         A2: V is open;

        

         A3: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A4: (fX2 . p) = ((p `2 ) `1 ) by Def5;

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A5: 0 < g and

         A6: ].(((p `2 ) `1 ) - g), (((p `2 ) `1 ) + g).[ c= V1 by A1, A4, RCOMP_1: 19;

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

        set W1 = { |[x, y]| where x,y be Real : (((p `2 ) `1 ) - g) < x & x < (((p `2 ) `1 ) + g) };

        W1 c= the carrier of T2

        proof

          let a be object;

          assume a in W1;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `1 ) - g) < x & x < (((p `2 ) `1 ) + g);

          hence thesis;

        end;

        then

        reconsider W1 as Subset of T2;

        take [:( [#] T2), W1:];

        

         A7: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A8: (((p `2 ) `1 ) - g) < (((p `2 ) `1 ) - 0 ) by A5, XREAL_1: 15;

        (((p `2 ) `1 ) + 0 ) < (((p `2 ) `1 ) + g) by A5, XREAL_1: 6;

        then (p `2 ) in W1 by A7, A8;

        hence p in [:( [#] T2), W1:] by A3, ZFMISC_1:def 2;

        W1 is open by PSCOMP_1: 19;

        hence [:( [#] T2), W1:] is open by BORSUK_1: 6;

        let b be object;

        assume b in (fX2 .: [:( [#] T2), W1:]);

        then

        consider a be Point of [:T2, T2:] such that

         A9: a in [:( [#] T2), W1:] and

         A10: (fX2 . a) = b by FUNCT_2: 65;

        

         A11: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A12: (fX2 . a) = ((a `2 ) `1 ) by Def5;

        (a `2 ) in W1 by A9, A11, ZFMISC_1: 87;

        then

        consider x1,y1 be Real such that

         A13: (a `2 ) = |[x1, y1]| and

         A14: (((p `2 ) `1 ) - g) < x1 and

         A15: x1 < (((p `2 ) `1 ) + g);

        

         A16: ((a `2 ) `1 ) = x1 by A13, EUCLID: 52;

        

         A17: ((((p `2 ) `1 ) - g) + g) < (x1 + g) by A14, XREAL_1: 6;

        

         A18: (((p `2 ) `1 ) - x1) > (((p `2 ) `1 ) - (((p `2 ) `1 ) + g)) by A15, XREAL_1: 15;

        

         A19: (((p `2 ) `1 ) - x1) < ((x1 + g) - x1) by A17, XREAL_1: 9;

        (((p `2 ) `1 ) - x1) > ( - g) by A18;

        then |.(((p `2 ) `1 ) - x1).| < g by A19, SEQ_2: 1;

        then |.( - (((p `2 ) `1 ) - x1)).| < g by COMPLEX1: 52;

        then |.(x1 - ((p `2 ) `1 )).| < g;

        then ((a `2 ) `1 ) in ].(((p `2 ) `1 ) - g), (((p `2 ) `1 ) + g).[ by A16, RCOMP_1: 1;

        hence thesis by A6, A10, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:63

    

     Th63: Proj2_2 is continuous Function of [:( TOP-REAL 2), ( TOP-REAL 2):], R^1

    proof

      reconsider fY2 = Proj2_2 as Function of [:T2, T2:], R^1 by TOPMETR: 17;

      for p be Point of [:T2, T2:], V be Subset of R^1 st (fY2 . p) in V & V is open holds ex W be Subset of [:T2, T2:] st p in W & W is open & (fY2 .: W) c= V

      proof

        let p be Point of [:T2, T2:], V be Subset of R^1 such that

         A1: (fY2 . p) in V and

         A2: V is open;

        

         A3: p = [(p `1 ), (p `2 )] by Lm5, MCART_1: 21;

        

         A4: (fY2 . p) = ((p `2 ) `2 ) by Def6;

        reconsider V1 = V as open Subset of REAL by A2, BORSUK_5: 39, TOPMETR: 17;

        consider g be Real such that

         A5: 0 < g and

         A6: ].(((p `2 ) `2 ) - g), (((p `2 ) `2 ) + g).[ c= V1 by A1, A4, RCOMP_1: 19;

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

        set W1 = { |[x, y]| where x,y be Real : (((p `2 ) `2 ) - g) < y & y < (((p `2 ) `2 ) + g) };

        W1 c= the carrier of T2

        proof

          let a be object;

          assume a in W1;

          then ex x,y be Real st a = |[x, y]| & (((p `2 ) `2 ) - g) < y & y < (((p `2 ) `2 ) + g);

          hence thesis;

        end;

        then

        reconsider W1 as Subset of T2;

        take [:( [#] T2), W1:];

        

         A7: (p `2 ) = |[((p `2 ) `1 ), ((p `2 ) `2 )]| by EUCLID: 53;

        

         A8: (((p `2 ) `2 ) - g) < (((p `2 ) `2 ) - 0 ) by A5, XREAL_1: 15;

        (((p `2 ) `2 ) + 0 ) < (((p `2 ) `2 ) + g) by A5, XREAL_1: 6;

        then (p `2 ) in W1 by A7, A8;

        hence p in [:( [#] T2), W1:] by A3, ZFMISC_1:def 2;

        W1 is open by PSCOMP_1: 21;

        hence [:( [#] T2), W1:] is open by BORSUK_1: 6;

        let b be object;

        assume b in (fY2 .: [:( [#] T2), W1:]);

        then

        consider a be Point of [:T2, T2:] such that

         A9: a in [:( [#] T2), W1:] and

         A10: (fY2 . a) = b by FUNCT_2: 65;

        

         A11: a = [(a `1 ), (a `2 )] by Lm5, MCART_1: 21;

        

         A12: (fY2 . a) = ((a `2 ) `2 ) by Def6;

        (a `2 ) in W1 by A9, A11, ZFMISC_1: 87;

        then

        consider x1,y1 be Real such that

         A13: (a `2 ) = |[x1, y1]| and

         A14: (((p `2 ) `2 ) - g) < y1 and

         A15: y1 < (((p `2 ) `2 ) + g);

        

         A16: ((a `2 ) `2 ) = y1 by A13, EUCLID: 52;

        

         A17: ((((p `2 ) `2 ) - g) + g) < (y1 + g) by A14, XREAL_1: 6;

        

         A18: (((p `2 ) `2 ) - y1) > (((p `2 ) `2 ) - (((p `2 ) `2 ) + g)) by A15, XREAL_1: 15;

        

         A19: (((p `2 ) `2 ) - y1) < ((y1 + g) - y1) by A17, XREAL_1: 9;

        (((p `2 ) `2 ) - y1) > ( - g) by A18;

        then |.(((p `2 ) `2 ) - y1).| < g by A19, SEQ_2: 1;

        then |.( - (((p `2 ) `2 ) - y1)).| < g by COMPLEX1: 52;

        then |.(y1 - ((p `2 ) `2 )).| < g;

        then ((a `2 ) `2 ) in ].(((p `2 ) `2 ) - g), (((p `2 ) `2 ) + g).[ by A16, RCOMP_1: 1;

        hence thesis by A6, A10, A12;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    registration

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

      cluster ( diffX2_1 o) -> continuous;

      coherence

      proof

        ( diffX2_1 o) is continuous Function of [:T2, T2:], R^1 by Th58;

        hence thesis by JORDAN5A: 27;

      end;

      cluster ( diffX2_2 o) -> continuous;

      coherence

      proof

        ( diffX2_2 o) is continuous Function of [:T2, T2:], R^1 by Th59;

        hence thesis by JORDAN5A: 27;

      end;

    end

    registration

      cluster diffX1_X2_1 -> continuous;

      coherence by Th60, JORDAN5A: 27;

      cluster diffX1_X2_2 -> continuous;

      coherence by Th61, JORDAN5A: 27;

      cluster Proj2_1 -> continuous;

      coherence by Th62, JORDAN5A: 27;

      cluster Proj2_2 -> continuous;

      coherence by Th63, JORDAN5A: 27;

    end

    definition

      let n be non zero Element of NAT , o,p be Point of ( TOP-REAL n), r be positive Real;

      set X = (( TOP-REAL n) | (( cl_Ball (o,r)) \ {p}));

      :: JORDAN:def7

      func DiskProj (o,r,p) -> Function of (( TOP-REAL n) | (( cl_Ball (o,r)) \ {p})), ( Tcircle (o,r)) means

      : Def7: for x be Point of (( TOP-REAL n) | (( cl_Ball (o,r)) \ {p})) holds ex y be Point of ( TOP-REAL n) st x = y & (it . x) = ( HC (p,y,o,r));

      existence

      proof

        

         A2: the carrier of X = (( cl_Ball (o,r)) \ {p}) by PRE_TOPC: 8;

        defpred P[ object, object] means ex z be Point of ( TOP-REAL n) st $1 = z & $2 = ( HC (p,z,o,r));

        

         A3: for x be object st x in the carrier of X holds ex y be object st y in the carrier of ( Tcircle (o,r)) & P[x, y]

        proof

          let x be object such that

           A4: x in the carrier of X;

          reconsider z = x as Point of ( TOP-REAL n) by A4, PRE_TOPC: 25;

          z in ( cl_Ball (o,r)) by A2, A4, XBOOLE_0:def 5;

          then

           A5: z is Point of ( Tdisk (o,r)) by BROUWER: 3;

          p <> z by A2, A4, ZFMISC_1: 56;

          then ( HC (p,z,o,r)) is Point of ( Tcircle (o,r)) by A1, A5, BROUWER: 6;

          hence thesis;

        end;

        consider f be Function of the carrier of X, the carrier of ( Tcircle (o,r)) such that

         A6: for x be object st x in the carrier of X holds P[x, (f . x)] from FUNCT_2:sch 1( A3);

        reconsider f as Function of X, ( Tcircle (o,r));

        take f;

        let x be Point of X;

        thus thesis by A6;

      end;

      uniqueness

      proof

        let f,g be Function of X, ( Tcircle (o,r)) such that

         A7: for x be Point of X holds ex y be Point of ( TOP-REAL n) st x = y & (f . x) = ( HC (p,y,o,r)) and

         A8: for x be Point of X holds ex y be Point of ( TOP-REAL n) st x = y & (g . x) = ( HC (p,y,o,r));

        now

          let x be object such that

           A9: x in the carrier of X;

          

           A10: ex y be Point of ( TOP-REAL n) st x = y & (f . x) = ( HC (p,y,o,r)) by A7, A9;

          ex y be Point of ( TOP-REAL n) st x = y & (g . x) = ( HC (p,y,o,r)) by A8, A9;

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

        end;

        hence thesis by FUNCT_2: 12;

      end;

    end

    theorem :: JORDAN:64

    

     Th64: for o,p be Point of ( TOP-REAL 2), r be positive Real st p is Point of ( Tdisk (o,r)) holds ( DiskProj (o,r,p)) is continuous

    proof

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

      let r be positive Real such that

       A1: p is Point of ( Tdisk (o,r));

      set D = ( Tdisk (o,r));

      set cB = ( cl_Ball (o,r));

      set Bp = (cB \ {p});

      set OK = [:Bp, {p}:];

      set D1 = (T2 | Bp);

      set D2 = (T2 | {p});

      set S1 = ( Tcircle (o,r));

      

       A2: p in {p} by TARSKI:def 1;

      

       A3: the carrier of D = ( cl_Ball (o,r)) by BROUWER: 3;

      

       A4: the carrier of D1 = Bp by PRE_TOPC: 8;

      

       A5: the carrier of D2 = {p} by PRE_TOPC: 8;

      set TD = ( [:T2, T2:] | OK);

      set gg = ( DiskProj (o,r,p));

      set xo = ( diffX2_1 o);

      set yo = ( diffX2_2 o);

      set dx = diffX1_X2_1 ;

      set dy = diffX1_X2_2 ;

      set fx2 = Proj2_1 ;

      set fy2 = Proj2_2 ;

      reconsider rr = (r ^2 ) as Element of REAL by XREAL_0:def 1;

      set f1 = (the carrier of [:T2, T2:] --> rr);

      reconsider f1 as continuous RealMap of [:T2, T2:] by Lm6;

      set Zf1 = (f1 | OK);

      set Zfx2 = (fx2 | OK);

      set Zfy2 = (fy2 | OK);

      set Zdx = (dx | OK);

      set Zdy = (dy | OK);

      set Zxo = (xo | OK);

      set Zyo = (yo | OK);

      set xx = (Zxo (#) Zdx);

      set yy = (Zyo (#) Zdy);

      set m = ((Zdx (#) Zdx) + (Zdy (#) Zdy));

      

       A6: the carrier of TD = OK by PRE_TOPC: 8;

      

       A7: for y be Point of D1, z be Point of D2 holds (Zdx . [y, z]) = (dx . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A8: for y be Point of D1, z be Point of D2 holds (Zdy . [y, z]) = (dy . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A9: for y be Point of D1, z be Point of D2 holds (Zfx2 . [y, z]) = (fx2 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A10: for y be Point of D1, z be Point of D2 holds (Zfy2 . [y, z]) = (fy2 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A11: for y be Point of D1, z be Point of D2 holds (Zf1 . [y, z]) = (f1 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A12: for y be Point of D1, z be Point of D2 holds (Zxo . [y, z]) = (xo . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A13: for y be Point of D1, z be Point of D2 holds (Zyo . [y, z]) = (yo . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A4, A5, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      now

        let b be Real;

        assume b in ( rng m);

        then

        consider a be object such that

         A14: a in ( dom m) and

         A15: (m . a) = b by FUNCT_1:def 3;

        consider y,z be object such that

         A16: y in Bp and

         A17: z in {p} and

         A18: a = [y, z] by A14, ZFMISC_1:def 2;

        

         A19: z = p by A17, TARSKI:def 1;

        reconsider y, z as Point of T2 by A16, A17;

        

         A20: y <> z by A16, A19, ZFMISC_1: 56;

        

         A21: (dx . [y, z]) = ((( [y, z] `1 ) `1 ) - (( [y, z] `2 ) `1 )) by Def3;

        

         A22: (dy . [y, z]) = ((( [y, z] `1 ) `2 ) - (( [y, z] `2 ) `2 )) by Def4;

        set r1 = ((y `1 ) - (z `1 ));

        set r2 = ((y `2 ) - (z `2 ));

        

         A23: (Zdx . [y, z]) = (dx . [y, z]) by A4, A5, A7, A16, A17;

        

         A24: (Zdy . [y, z]) = (dy . [y, z]) by A4, A5, A8, A16, A17;

        ( dom m) c= the carrier of TD by RELAT_1:def 18;

        then a in the carrier of TD by A14;

        

        then

         A25: (m . [y, z]) = (((Zdx (#) Zdx) . [y, z]) + ((Zdy (#) Zdy) . [y, z])) by A18, VALUED_1: 1

        .= (((Zdx . [y, z]) * (Zdx . [y, z])) + ((Zdy (#) Zdy) . [y, z])) by VALUED_1: 5

        .= ((r1 ^2 ) + (r2 ^2 )) by A21, A22, A23, A24, VALUED_1: 5;

        now

          assume

           A26: ((r1 ^2 ) + (r2 ^2 )) = 0 ;

          then

           A27: r1 = 0 by COMPLEX1: 1;

          r2 = 0 by A26, COMPLEX1: 1;

          hence contradiction by A20, A27, TOPREAL3: 6;

        end;

        hence 0 < b by A15, A18, A25;

      end;

      then

      reconsider m as positive-yielding continuous RealMap of TD by PARTFUN3:def 1;

      set p1 = ((xx + yy) (#) (xx + yy));

      set p2 = (((Zxo (#) Zxo) + (Zyo (#) Zyo)) - Zf1);

      

       A28: ( dom p2) = the carrier of TD by FUNCT_2:def 1;

      now

        let b be Real;

        assume b in ( rng p2);

        then

        consider a be object such that

         A29: a in ( dom p2) and

         A30: (p2 . a) = b by FUNCT_1:def 3;

        consider y,z be object such that

         A31: y in Bp and

         A32: z in {p} and

         A33: a = [y, z] by A29, ZFMISC_1:def 2;

        reconsider y, z as Point of T2 by A31, A32;

        set r3 = ((z `1 ) - (o `1 )), r4 = ((z `2 ) - (o `2 ));

        

         A34: (Zf1 . [y, z]) = (f1 . [y, z]) by A4, A5, A11, A31, A32;

        

         A35: (Zxo . [y, z]) = (xo . [y, z]) by A4, A5, A12, A31, A32;

        

         A36: (Zyo . [y, z]) = (yo . [y, z]) by A4, A5, A13, A31, A32;

        

         A37: (xo . [y, z]) = ((( [y, z] `2 ) `1 ) - (o `1 )) by Def1;

        

         A38: (yo . [y, z]) = ((( [y, z] `2 ) `2 ) - (o `2 )) by Def2;

        ( dom p2) c= the carrier of TD by RELAT_1:def 18;

        then

         A39: a in the carrier of TD by A29;

        

         A40: (p2 . [y, z]) = ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, z]) - (Zf1 . [y, z])) by A29, A33, VALUED_1: 13

        .= ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, z]) - (r ^2 )) by A34, FUNCOP_1: 7

        .= ((((Zxo (#) Zxo) . [y, z]) + ((Zyo (#) Zyo) . [y, z])) - (r ^2 )) by A33, A39, VALUED_1: 1

        .= ((((Zxo . [y, z]) * (Zxo . [y, z])) + ((Zyo (#) Zyo) . [y, z])) - (r ^2 )) by VALUED_1: 5

        .= (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) by A35, A36, A37, A38, VALUED_1: 5;

        z = p by A32, TARSKI:def 1;

        then |.(z - o).| <= r by A1, A3, TOPREAL9: 8;

        then

         A41: ( |.(z - o).| ^2 ) <= (r ^2 ) by SQUARE_1: 15;

        ( |.(z - o).| ^2 ) = ((((z - o) `1 ) ^2 ) + (((z - o) `2 ) ^2 )) by JGRAPH_1: 29

        .= ((r3 ^2 ) + (((z - o) `2 ) ^2 )) by TOPREAL3: 3

        .= ((r3 ^2 ) + (r4 ^2 )) by TOPREAL3: 3;

        then (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) <= ((r ^2 ) - (r ^2 )) by A41, XREAL_1: 9;

        hence 0 >= b by A30, A33, A40;

      end;

      then

      reconsider p2 as nonpositive-yielding continuous RealMap of TD by PARTFUN3:def 3;

      set pp = (p1 - (m (#) p2));

      set k = ((( - (xx + yy)) + ( sqrt pp)) / m);

      set x3 = (Zfx2 + (k (#) Zdx));

      set y3 = (Zfy2 + (k (#) Zdy));

      reconsider X3 = x3, Y3 = y3 as Function of TD, R^1 by TOPMETR: 17;

      set F = <:X3, Y3:>;

      set R = R2Homeomorphism ;

      

       A42: for x be Point of D1 holds (gg . x) = ((R * F) . [x, p])

      proof

        let x be Point of D1;

        consider y be Point of T2 such that

         A43: x = y and

         A44: (gg . x) = ( HC (p,y,o,r)) by A1, Def7;

        

         A45: x <> p by A4, ZFMISC_1: 56;

        

         A46: [y, p] in OK by A2, A4, A43, ZFMISC_1:def 2;

        set r1 = ((y `1 ) - (p `1 )), r2 = ((y `2 ) - (p `2 )), r3 = ((p `1 ) - (o `1 )), r4 = ((p `2 ) - (o `2 ));

        set l = ((( - ((r3 * r1) + (r4 * r2))) + ( sqrt ((((r3 * r1) + (r4 * r2)) ^2 ) - (((r1 ^2 ) + (r2 ^2 )) * (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )))))) / ((r1 ^2 ) + (r2 ^2 )));

        

         A47: (fx2 . [y, p]) = (( [y, p] `2 ) `1 ) by Def5;

        

         A48: (fy2 . [y, p]) = (( [y, p] `2 ) `2 ) by Def6;

        

         A49: (dx . [y, p]) = ((( [y, p] `1 ) `1 ) - (( [y, p] `2 ) `1 )) by Def3;

        

         A50: (dy . [y, p]) = ((( [y, p] `1 ) `2 ) - (( [y, p] `2 ) `2 )) by Def4;

        

         A51: (xo . [y, p]) = ((( [y, p] `2 ) `1 ) - (o `1 )) by Def1;

        

         A52: (yo . [y, p]) = ((( [y, p] `2 ) `2 ) - (o `2 )) by Def2;

        

         A53: ( dom X3) = the carrier of TD by FUNCT_2:def 1;

        

         A54: ( dom Y3) = the carrier of TD by FUNCT_2:def 1;

        

         A55: ( dom pp) = the carrier of TD by FUNCT_2:def 1;

        

         A56: p is Point of D2 by A5, TARSKI:def 1;

        then

         A57: (Zdx . [y, p]) = (dx . [y, p]) by A7, A43;

        

         A58: (Zdy . [y, p]) = (dy . [y, p]) by A8, A43, A56;

        

         A59: (Zf1 . [y, p]) = (f1 . [y, p]) by A11, A43, A56;

        

         A60: (Zxo . [y, p]) = (xo . [y, p]) by A12, A43, A56;

        

         A61: (Zyo . [y, p]) = (yo . [y, p]) by A13, A43, A56;

        

         A62: (m . [y, p]) = (((Zdx (#) Zdx) . [y, p]) + ((Zdy (#) Zdy) . [y, p])) by A6, A46, VALUED_1: 1

        .= (((Zdx . [y, p]) * (Zdx . [y, p])) + ((Zdy (#) Zdy) . [y, p])) by VALUED_1: 5

        .= ((r1 ^2 ) + (r2 ^2 )) by A49, A50, A57, A58, VALUED_1: 5;

        

         A63: (xx . [y, p]) = ((Zxo . [y, p]) * (Zdx . [y, p])) by VALUED_1: 5;

        

         A64: (yy . [y, p]) = ((Zyo . [y, p]) * (Zdy . [y, p])) by VALUED_1: 5;

        

         A65: ((xx + yy) . [y, p]) = ((xx . [y, p]) + (yy . [y, p])) by A6, A46, VALUED_1: 1;

        then

         A66: (p1 . [y, p]) = (((r3 * r1) + (r4 * r2)) ^2 ) by A49, A50, A51, A52, A57, A58, A60, A61, A63, A64, VALUED_1: 5;

        

         A67: (p2 . [y, p]) = ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, p]) - (Zf1 . [y, p])) by A6, A28, A46, VALUED_1: 13

        .= ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, p]) - (r ^2 )) by A59, FUNCOP_1: 7

        .= ((((Zxo (#) Zxo) . [y, p]) + ((Zyo (#) Zyo) . [y, p])) - (r ^2 )) by A6, A46, VALUED_1: 1

        .= ((((Zxo . [y, p]) * (Zxo . [y, p])) + ((Zyo (#) Zyo) . [y, p])) - (r ^2 )) by VALUED_1: 5

        .= (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) by A51, A52, A60, A61, VALUED_1: 5;

        ( dom ( sqrt pp)) = the carrier of TD by FUNCT_2:def 1;

        

        then

         A68: (( sqrt pp) . [y, p]) = ( sqrt (pp . [y, p])) by A6, A46, PARTFUN3:def 5

        .= ( sqrt ((p1 . [y, p]) - ((m (#) p2) . [y, p]))) by A6, A46, A55, VALUED_1: 13

        .= ( sqrt ((((r3 * r1) + (r4 * r2)) ^2 ) - (((r1 ^2 ) + (r2 ^2 )) * (((r3 ^2 ) + (r4 ^2 )) - (r ^2 ))))) by A62, A66, A67, VALUED_1: 5;

        ( dom k) = the carrier of TD by FUNCT_2:def 1;

        

        then

         A69: (k . [y, p]) = (((( - (xx + yy)) + ( sqrt pp)) . [y, p]) * ((m . [y, p]) " )) by A6, A46, RFUNCT_1:def 1

        .= (((( - (xx + yy)) + ( sqrt pp)) . [y, p]) / (m . [y, p])) by XCMPLX_0:def 9

        .= (((( - (xx + yy)) . [y, p]) + (( sqrt pp) . [y, p])) / ((r1 ^2 ) + (r2 ^2 ))) by A6, A46, A62, VALUED_1: 1

        .= l by A49, A50, A51, A52, A57, A58, A60, A61, A63, A64, A65, A68, VALUED_1: 8;

        

         A70: (X3 . [y, p]) = ((Zfx2 . [y, p]) + ((k (#) Zdx) . [y, p])) by A6, A46, VALUED_1: 1

        .= ((p `1 ) + ((k (#) Zdx) . [y, p])) by A9, A43, A47, A56

        .= ((p `1 ) + (l * r1)) by A49, A57, A69, VALUED_1: 5;

        

         A71: (Y3 . [y, p]) = ((Zfy2 . [y, p]) + ((k (#) Zdy) . [y, p])) by A6, A46, VALUED_1: 1

        .= ((p `2 ) + ((k (#) Zdy) . [y, p])) by A10, A43, A48, A56

        .= ((p `2 ) + (l * r2)) by A50, A58, A69, VALUED_1: 5;

        

         A72: y in Bp by A4, A43;

        Bp c= cB by XBOOLE_1: 36;

        

        hence (gg . x) = |[((p `1 ) + (l * r1)), ((p `2 ) + (l * r2))]| by A1, A3, A43, A44, A45, A72, BROUWER: 8

        .= (R . [(X3 . [y, p]), (Y3 . [y, p])]) by A70, A71, TOPREALA:def 2

        .= (R . (F . [y, p])) by A6, A46, A53, A54, FUNCT_3: 49

        .= ((R * F) . [x, p]) by A6, A43, A46, FUNCT_2: 15;

      end;

      

       A73: X3 is continuous by JORDAN5A: 27;

      Y3 is continuous by JORDAN5A: 27;

      then

      reconsider F as continuous Function of TD, [: R^1 , R^1 :] by A73, YELLOW12: 41;

      for pp be Point of D1, V be Subset of S1 st (gg . pp) in V & V is open holds ex W be Subset of D1 st pp in W & W is open & (gg .: W) c= V

      proof

        let pp be Point of D1, V be Subset of S1 such that

         A74: (gg . pp) in V and

         A75: V is open;

        reconsider p1 = pp, fp = p as Point of T2 by PRE_TOPC: 25;

        

         A76: [pp, p] in OK by A2, A4, ZFMISC_1:def 2;

        consider V1 be Subset of T2 such that

         A77: V1 is open and

         A78: (V1 /\ ( [#] S1)) = V by A75, TOPS_2: 24;

        

         A79: (gg . pp) = ((R * F) . [pp, p]) by A42;

        (R " ) is being_homeomorphism by TOPREALA: 34, TOPS_2: 56;

        then

         A80: ((R " ) .: V1) is open by A77, TOPGRP_1: 25;

        

         A81: ( dom F) = the carrier of ( [:T2, T2:] | OK) by FUNCT_2:def 1;

        

         A82: ( dom R) = the carrier of [: R^1 , R^1 :] by FUNCT_2:def 1;

        then

         A83: ( rng F) c= ( dom R);

        then

         A84: ( dom (R * F)) = ( dom F) by RELAT_1: 27;

        

         A85: ( rng R) = ( [#] T2) by TOPREALA: 34, TOPS_2:def 5;

        

         A86: ((R " ) * (R * F)) = (((R " ) * R) * F) by RELAT_1: 36

        .= (( id ( dom R)) * F) by A85, TOPREALA: 34, TOPS_2: 52;

        ( dom ( id ( dom R))) = ( dom R);

        then

         A87: ( dom (( id ( dom R)) * F)) = ( dom F) by A83, RELAT_1: 27;

        for x be object st x in ( dom F) holds ((( id ( dom R)) * F) . x) = (F . x)

        proof

          let x be object such that

           A88: x in ( dom F);

          

           A89: (F . x) in ( rng F) by A88, FUNCT_1:def 3;

          

          thus ((( id ( dom R)) * F) . x) = (( id ( dom R)) . (F . x)) by A88, FUNCT_1: 13

          .= (F . x) by A82, A89, FUNCT_1: 18;

        end;

        then

         A90: (( id ( dom R)) * F) = F by A87, FUNCT_1: 2;

        ((R * F) . [p1, fp]) in V1 by A74, A78, A79, XBOOLE_0:def 4;

        then ((R " ) . ((R * F) . [p1, fp])) in ((R " ) .: V1) by FUNCT_2: 35;

        then (((R " ) * (R * F)) . [p1, fp]) in ((R " ) .: V1) by A6, A76, A81, A84, FUNCT_1: 13;

        then

        consider W be Subset of TD such that

         A91: [p1, fp] in W and

         A92: W is open and

         A93: (F .: W) c= ((R " ) .: V1) by A6, A76, A80, A86, A90, JGRAPH_2: 10;

        consider WW be Subset of [:T2, T2:] such that

         A94: WW is open and

         A95: (WW /\ ( [#] TD)) = W by A92, TOPS_2: 24;

        consider SF be Subset-Family of [:T2, T2:] such that

         A96: WW = ( union SF) and

         A97: for e be set st e in SF holds ex X1 be Subset of T2, Y1 be Subset of T2 st e = [:X1, Y1:] & X1 is open & Y1 is open by A94, BORSUK_1: 5;

         [p1, fp] in WW by A91, A95, XBOOLE_0:def 4;

        then

        consider Z be set such that

         A98: [p1, fp] in Z and

         A99: Z in SF by A96, TARSKI:def 4;

        consider X1,Y1 be Subset of T2 such that

         A100: Z = [:X1, Y1:] and

         A101: X1 is open and Y1 is open by A97, A99;

        set ZZ = (Z /\ ( [#] TD));

        reconsider XX = (X1 /\ ( [#] D1)) as open Subset of D1 by A101, TOPS_2: 24;

        take XX;

        pp in X1 by A98, A100, ZFMISC_1: 87;

        hence pp in XX by XBOOLE_0:def 4;

        thus XX is open;

        let b be object;

        assume b in (gg .: XX);

        then

        consider a be Point of D1 such that

         A102: a in XX and

         A103: b = (gg . a) by FUNCT_2: 65;

        reconsider a1 = a, fa = fp as Point of T2 by PRE_TOPC: 25;

        

         A104: a in X1 by A102, XBOOLE_0:def 4;

        

         A105: [a, p] in OK by A2, A4, ZFMISC_1:def 2;

        fa in Y1 by A98, A100, ZFMISC_1: 87;

        then [a, fa] in Z by A100, A104, ZFMISC_1:def 2;

        then [a, fa] in ZZ by A6, A105, XBOOLE_0:def 4;

        then

         A106: (F . [a1, fa]) in (F .: ZZ) by FUNCT_2: 35;

        

         A107: (R qua Function " ) = (R " ) by TOPREALA: 34, TOPS_2:def 4;

        

         A108: ( dom (R " )) = ( [#] T2) by A85, TOPREALA: 34, TOPS_2: 49;

        Z c= WW by A96, A99, ZFMISC_1: 74;

        then ZZ c= (WW /\ ( [#] TD)) by XBOOLE_1: 27;

        then (F .: ZZ) c= (F .: W) by A95, RELAT_1: 123;

        then (F . [a1, fa]) in (F .: W) by A106;

        then (R . (F . [a1, fa])) in (R .: ((R " ) .: V1)) by A93, FUNCT_2: 35;

        then ((R * F) . [a1, fa]) in (R .: ((R " ) .: V1)) by A6, A105, FUNCT_2: 15;

        then ((R * F) . [a1, fa]) in V1 by A107, A108, PARTFUN3: 1, TOPREALA: 34;

        then (gg . a) in V1 by A42;

        hence thesis by A78, A103, XBOOLE_0:def 4;

      end;

      hence thesis by JGRAPH_2: 10;

    end;

    theorem :: JORDAN:65

    

     Th65: for n be non zero Element of NAT , o,p be Point of ( TOP-REAL n), r be positive Real st p in ( Ball (o,r)) holds (( DiskProj (o,r,p)) | ( Sphere (o,r))) = ( id ( Sphere (o,r)))

    proof

      let n be non zero Element of NAT ;

      let o,p be Point of ( TOP-REAL n);

      let r be positive Real;

      assume

       A1: p in ( Ball (o,r));

      

       A2: the carrier of ( Tdisk (o,r)) = ( cl_Ball (o,r)) by BROUWER: 3;

      

       A3: the carrier of (( TOP-REAL n) | (( cl_Ball (o,r)) \ {p})) = (( cl_Ball (o,r)) \ {p}) by PRE_TOPC: 8;

      

       A4: ( dom ( DiskProj (o,r,p))) = the carrier of (( TOP-REAL n) | (( cl_Ball (o,r)) \ {p})) by FUNCT_2:def 1;

      

       A5: ( Sphere (o,r)) misses ( Ball (o,r)) by TOPREAL9: 19;

      

       A6: ( Sphere (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 17;

      

       A7: ( Ball (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 16;

      

       A8: ( Sphere (o,r)) c= (( cl_Ball (o,r)) \ {p})

      proof

        let a be object;

        assume

         A9: a in ( Sphere (o,r));

        then a <> p by A1, A5, XBOOLE_0: 3;

        hence thesis by A6, A9, ZFMISC_1: 56;

      end;

      then

       A10: ( dom (( DiskProj (o,r,p)) | ( Sphere (o,r)))) = ( Sphere (o,r)) by A3, A4, RELAT_1: 62;

      

       A11: ( dom ( id ( Sphere (o,r)))) = ( Sphere (o,r));

      now

        let x be object;

        assume

         A12: x in ( dom (( DiskProj (o,r,p)) | ( Sphere (o,r))));

        then x in ( dom ( DiskProj (o,r,p))) by RELAT_1: 57;

        then

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

         A13: x = y and

         A14: (( DiskProj (o,r,p)) . x) = ( HC (p,y,o,r)) by A1, A2, A7, Def7;

        y in ( halfline (p,y)) by TOPREAL9: 28;

        then

         A15: x in (( halfline (p,y)) /\ ( Sphere (o,r))) by A12, A13, XBOOLE_0:def 4;

        

         A16: x <> p by A1, A5, A12, XBOOLE_0: 3;

        

        thus ((( DiskProj (o,r,p)) | ( Sphere (o,r))) . x) = (( DiskProj (o,r,p)) . x) by A12, FUNCT_1: 47

        .= x by A1, A2, A6, A7, A10, A12, A13, A14, A15, A16, BROUWER:def 3

        .= (( id ( Sphere (o,r))) . x) by A12, FUNCT_1: 18;

      end;

      hence thesis by A3, A4, A8, A11, FUNCT_1: 2, RELAT_1: 62;

    end;

    definition

      let n be non zero Element of NAT , o,p be Point of ( TOP-REAL n), r be positive Real;

      set X = ( Tcircle (o,r));

      :: JORDAN:def8

      func RotateCircle (o,r,p) -> Function of ( Tcircle (o,r)), ( Tcircle (o,r)) means

      : Def8: for x be Point of ( Tcircle (o,r)) holds ex y be Point of ( TOP-REAL n) st x = y & (it . x) = ( HC (y,p,o,r));

      existence

      proof

        

         A2: the carrier of X = ( Sphere (o,r)) by TOPREALB: 9;

        defpred P[ object, object] means ex z be Point of ( TOP-REAL n) st $1 = z & $2 = ( HC (z,p,o,r));

        

         A3: for x be object st x in the carrier of X holds ex y be object st y in the carrier of X & P[x, y]

        proof

          let x be object such that

           A4: x in the carrier of X;

          reconsider z = x as Point of ( TOP-REAL n) by A4, PRE_TOPC: 25;

          ( Sphere (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 17;

          then

           A5: z is Point of ( Tdisk (o,r)) by A2, A4, BROUWER: 3;

          ( Ball (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 16;

          then

           A6: p is Point of ( Tdisk (o,r)) by A1, BROUWER: 3;

          ( Ball (o,r)) misses ( Sphere (o,r)) by TOPREAL9: 19;

          then p <> z by A1, A2, A4, XBOOLE_0: 3;

          then ( HC (z,p,o,r)) is Point of X by A5, A6, BROUWER: 6;

          hence thesis;

        end;

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

         A7: for x be object st x in the carrier of X holds P[x, (f . x)] from FUNCT_2:sch 1( A3);

        reconsider f as Function of X, X;

        take f;

        let x be Point of X;

        thus thesis by A7;

      end;

      uniqueness

      proof

        let f,g be Function of X, X such that

         A8: for x be Point of X holds ex y be Point of ( TOP-REAL n) st x = y & (f . x) = ( HC (y,p,o,r)) and

         A9: for x be Point of X holds ex y be Point of ( TOP-REAL n) st x = y & (g . x) = ( HC (y,p,o,r));

        now

          let x be object such that

           A10: x in the carrier of X;

          

           A11: ex y be Point of ( TOP-REAL n) st x = y & (f . x) = ( HC (y,p,o,r)) by A8, A10;

          ex y be Point of ( TOP-REAL n) st x = y & (g . x) = ( HC (y,p,o,r)) by A9, A10;

          hence (f . x) = (g . x) by A11;

        end;

        hence thesis by FUNCT_2: 12;

      end;

    end

    theorem :: JORDAN:66

    

     Th66: for o,p be Point of ( TOP-REAL 2), r be positive Real st p in ( Ball (o,r)) holds ( RotateCircle (o,r,p)) is continuous

    proof

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

      let r be positive Real such that

       A1: p in ( Ball (o,r));

      set D = ( Tdisk (o,r));

      set cB = ( cl_Ball (o,r));

      set Bp = ( Sphere (o,r));

      set OK = [: {p}, Bp:];

      set D1 = (T2 | {p});

      set D2 = (T2 | Bp);

      set S1 = ( Tcircle (o,r));

      

       A2: D2 = S1 by TOPREALB:def 6;

      

       A3: ( Ball (o,r)) misses ( Sphere (o,r)) by TOPREAL9: 19;

      

       A4: p in {p} by TARSKI:def 1;

      

       A5: Bp c= cB by TOPREAL9: 17;

      

       A6: ( Ball (o,r)) c= cB by TOPREAL9: 16;

      

       A7: the carrier of D = cB by BROUWER: 3;

      

       A8: the carrier of D1 = {p} by PRE_TOPC: 8;

      

       A9: the carrier of D2 = Bp by PRE_TOPC: 8;

      set TD = ( [:T2, T2:] | OK);

      set gg = ( RotateCircle (o,r,p));

      set xo = ( diffX2_1 o);

      set yo = ( diffX2_2 o);

      set dx = diffX1_X2_1 ;

      set dy = diffX1_X2_2 ;

      set fx2 = Proj2_1 ;

      set fy2 = Proj2_2 ;

      reconsider rr = (r ^2 ) as Element of REAL by XREAL_0:def 1;

      set f1 = (the carrier of [:T2, T2:] --> rr);

      reconsider f1 as continuous RealMap of [:T2, T2:] by Lm6;

      set Zf1 = (f1 | OK);

      set Zfx2 = (fx2 | OK);

      set Zfy2 = (fy2 | OK);

      set Zdx = (dx | OK);

      set Zdy = (dy | OK);

      set Zxo = (xo | OK);

      set Zyo = (yo | OK);

      set xx = (Zxo (#) Zdx);

      set yy = (Zyo (#) Zdy);

      set m = ((Zdx (#) Zdx) + (Zdy (#) Zdy));

      

       A10: the carrier of TD = OK by PRE_TOPC: 8;

      

       A11: for y be Point of D1, z be Point of D2 holds (Zdx . [y, z]) = (dx . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A12: for y be Point of D1, z be Point of D2 holds (Zdy . [y, z]) = (dy . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A13: for y be Point of D1, z be Point of D2 holds (Zfx2 . [y, z]) = (fx2 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A14: for y be Point of D1, z be Point of D2 holds (Zfy2 . [y, z]) = (fy2 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A15: for y be Point of D1, z be Point of D2 holds (Zf1 . [y, z]) = (f1 . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A16: for y be Point of D1, z be Point of D2 holds (Zxo . [y, z]) = (xo . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      

       A17: for y be Point of D1, z be Point of D2 holds (Zyo . [y, z]) = (yo . [y, z])

      proof

        let y be Point of D1;

        let z be Point of D2;

         [y, z] in OK by A8, A9, ZFMISC_1:def 2;

        hence thesis by FUNCT_1: 49;

      end;

      now

        let b be Real;

        assume b in ( rng m);

        then

        consider a be object such that

         A18: a in ( dom m) and

         A19: (m . a) = b by FUNCT_1:def 3;

        consider y,z be object such that

         A20: y in {p} and

         A21: z in Bp and

         A22: a = [y, z] by A18, ZFMISC_1:def 2;

        

         A23: y = p by A20, TARSKI:def 1;

        reconsider y, z as Point of T2 by A20, A21;

        

         A24: y <> z by A1, A3, A21, A23, XBOOLE_0: 3;

        

         A25: (dx . [y, z]) = ((( [y, z] `1 ) `1 ) - (( [y, z] `2 ) `1 )) by Def3;

        

         A26: (dy . [y, z]) = ((( [y, z] `1 ) `2 ) - (( [y, z] `2 ) `2 )) by Def4;

        set r1 = ((y `1 ) - (z `1 ));

        set r2 = ((y `2 ) - (z `2 ));

        

         A27: (Zdx . [y, z]) = (dx . [y, z]) by A8, A9, A11, A20, A21;

        

         A28: (Zdy . [y, z]) = (dy . [y, z]) by A8, A9, A12, A20, A21;

        ( dom m) c= the carrier of TD by RELAT_1:def 18;

        then a in the carrier of TD by A18;

        

        then

         A29: (m . [y, z]) = (((Zdx (#) Zdx) . [y, z]) + ((Zdy (#) Zdy) . [y, z])) by A22, VALUED_1: 1

        .= (((Zdx . [y, z]) * (Zdx . [y, z])) + ((Zdy (#) Zdy) . [y, z])) by VALUED_1: 5

        .= ((r1 ^2 ) + (r2 ^2 )) by A25, A26, A27, A28, VALUED_1: 5;

        now

          assume

           A30: ((r1 ^2 ) + (r2 ^2 )) = 0 ;

          then

           A31: r1 = 0 by COMPLEX1: 1;

          r2 = 0 by A30, COMPLEX1: 1;

          hence contradiction by A24, A31, TOPREAL3: 6;

        end;

        hence 0 < b by A19, A22, A29;

      end;

      then

      reconsider m as positive-yielding continuous RealMap of TD by PARTFUN3:def 1;

      set p1 = ((xx + yy) (#) (xx + yy));

      set p2 = (((Zxo (#) Zxo) + (Zyo (#) Zyo)) - Zf1);

      

       A32: ( dom p2) = the carrier of TD by FUNCT_2:def 1;

      now

        let b be Real;

        assume b in ( rng p2);

        then

        consider a be object such that

         A33: a in ( dom p2) and

         A34: (p2 . a) = b by FUNCT_1:def 3;

        consider y,z be object such that

         A35: y in {p} and

         A36: z in Bp and

         A37: a = [y, z] by A33, ZFMISC_1:def 2;

        reconsider y, z as Point of T2 by A35, A36;

        set r3 = ((z `1 ) - (o `1 )), r4 = ((z `2 ) - (o `2 ));

        

         A38: (Zf1 . [y, z]) = (f1 . [y, z]) by A8, A9, A15, A35, A36;

        

         A39: (Zxo . [y, z]) = (xo . [y, z]) by A8, A9, A16, A35, A36;

        

         A40: (Zyo . [y, z]) = (yo . [y, z]) by A8, A9, A17, A35, A36;

        

         A41: (xo . [y, z]) = ((( [y, z] `2 ) `1 ) - (o `1 )) by Def1;

        

         A42: (yo . [y, z]) = ((( [y, z] `2 ) `2 ) - (o `2 )) by Def2;

        ( dom p2) c= the carrier of TD by RELAT_1:def 18;

        then

         A43: a in the carrier of TD by A33;

        

         A44: (p2 . [y, z]) = ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, z]) - (Zf1 . [y, z])) by A33, A37, VALUED_1: 13

        .= ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [y, z]) - (r ^2 )) by A38, FUNCOP_1: 7

        .= ((((Zxo (#) Zxo) . [y, z]) + ((Zyo (#) Zyo) . [y, z])) - (r ^2 )) by A37, A43, VALUED_1: 1

        .= ((((Zxo . [y, z]) * (Zxo . [y, z])) + ((Zyo (#) Zyo) . [y, z])) - (r ^2 )) by VALUED_1: 5

        .= (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) by A39, A40, A41, A42, VALUED_1: 5;

         |.(z - o).| <= r by A5, A36, TOPREAL9: 8;

        then

         A45: ( |.(z - o).| ^2 ) <= (r ^2 ) by SQUARE_1: 15;

        ( |.(z - o).| ^2 ) = ((((z - o) `1 ) ^2 ) + (((z - o) `2 ) ^2 )) by JGRAPH_1: 29

        .= ((r3 ^2 ) + (((z - o) `2 ) ^2 )) by TOPREAL3: 3

        .= ((r3 ^2 ) + (r4 ^2 )) by TOPREAL3: 3;

        then (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) <= ((r ^2 ) - (r ^2 )) by A45, XREAL_1: 9;

        hence 0 >= b by A34, A37, A44;

      end;

      then

      reconsider p2 as nonpositive-yielding continuous RealMap of TD by PARTFUN3:def 3;

      set pp = (p1 - (m (#) p2));

      set k = ((( - (xx + yy)) + ( sqrt pp)) / m);

      set x3 = (Zfx2 + (k (#) Zdx));

      set y3 = (Zfy2 + (k (#) Zdy));

      reconsider X3 = x3, Y3 = y3 as Function of TD, R^1 by TOPMETR: 17;

      set F = <:X3, Y3:>;

      set R = R2Homeomorphism ;

      

       A46: for x be Point of D2 holds (gg . x) = ((R * F) . [p, x])

      proof

        let x be Point of D2;

        consider y be Point of T2 such that

         A47: x = y and

         A48: (gg . x) = ( HC (y,p,o,r)) by A1, A2, Def8;

        

         A49: x <> p by A1, A3, A9, XBOOLE_0: 3;

        

         A50: [p, y] in OK by A4, A9, A47, ZFMISC_1:def 2;

        set r1 = ((p `1 ) - (y `1 )), r2 = ((p `2 ) - (y `2 )), r3 = ((y `1 ) - (o `1 )), r4 = ((y `2 ) - (o `2 ));

        set l = ((( - ((r3 * r1) + (r4 * r2))) + ( sqrt ((((r3 * r1) + (r4 * r2)) ^2 ) - (((r1 ^2 ) + (r2 ^2 )) * (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )))))) / ((r1 ^2 ) + (r2 ^2 )));

        

         A51: (fx2 . [p, y]) = (( [p, y] `2 ) `1 ) by Def5;

        

         A52: (fy2 . [p, y]) = (( [p, y] `2 ) `2 ) by Def6;

        

         A53: (dx . [p, y]) = ((( [p, y] `1 ) `1 ) - (( [p, y] `2 ) `1 )) by Def3;

        

         A54: (dy . [p, y]) = ((( [p, y] `1 ) `2 ) - (( [p, y] `2 ) `2 )) by Def4;

        

         A55: (xo . [p, y]) = ((( [p, y] `2 ) `1 ) - (o `1 )) by Def1;

        

         A56: (yo . [p, y]) = ((( [p, y] `2 ) `2 ) - (o `2 )) by Def2;

        

         A57: ( dom X3) = the carrier of TD by FUNCT_2:def 1;

        

         A58: ( dom Y3) = the carrier of TD by FUNCT_2:def 1;

        

         A59: ( dom pp) = the carrier of TD by FUNCT_2:def 1;

        

         A60: p is Point of D1 by A8, TARSKI:def 1;

        then

         A61: (Zdx . [p, y]) = (dx . [p, y]) by A11, A47;

        

         A62: (Zdy . [p, y]) = (dy . [p, y]) by A12, A47, A60;

        

         A63: (Zf1 . [p, y]) = (f1 . [p, y]) by A15, A47, A60;

        

         A64: (Zxo . [p, y]) = (xo . [p, y]) by A16, A47, A60;

        

         A65: (Zyo . [p, y]) = (yo . [p, y]) by A17, A47, A60;

        

         A66: (m . [p, y]) = (((Zdx (#) Zdx) . [p, y]) + ((Zdy (#) Zdy) . [p, y])) by A10, A50, VALUED_1: 1

        .= (((Zdx . [p, y]) * (Zdx . [p, y])) + ((Zdy (#) Zdy) . [p, y])) by VALUED_1: 5

        .= ((r1 ^2 ) + (r2 ^2 )) by A53, A54, A61, A62, VALUED_1: 5;

        

         A67: (xx . [p, y]) = ((Zxo . [p, y]) * (Zdx . [p, y])) by VALUED_1: 5;

        

         A68: (yy . [p, y]) = ((Zyo . [p, y]) * (Zdy . [p, y])) by VALUED_1: 5;

        

         A69: ((xx + yy) . [p, y]) = ((xx . [p, y]) + (yy . [p, y])) by A10, A50, VALUED_1: 1;

        then

         A70: (p1 . [p, y]) = (((r3 * r1) + (r4 * r2)) ^2 ) by A53, A54, A55, A56, A61, A62, A64, A65, A67, A68, VALUED_1: 5;

        

         A71: (p2 . [p, y]) = ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [p, y]) - (Zf1 . [p, y])) by A10, A32, A50, VALUED_1: 13

        .= ((((Zxo (#) Zxo) + (Zyo (#) Zyo)) . [p, y]) - (r ^2 )) by A63, FUNCOP_1: 7

        .= ((((Zxo (#) Zxo) . [p, y]) + ((Zyo (#) Zyo) . [p, y])) - (r ^2 )) by A10, A50, VALUED_1: 1

        .= ((((Zxo . [p, y]) * (Zxo . [p, y])) + ((Zyo (#) Zyo) . [p, y])) - (r ^2 )) by VALUED_1: 5

        .= (((r3 ^2 ) + (r4 ^2 )) - (r ^2 )) by A55, A56, A64, A65, VALUED_1: 5;

        ( dom ( sqrt pp)) = the carrier of TD by FUNCT_2:def 1;

        

        then

         A72: (( sqrt pp) . [p, y]) = ( sqrt (pp . [p, y])) by A10, A50, PARTFUN3:def 5

        .= ( sqrt ((p1 . [p, y]) - ((m (#) p2) . [p, y]))) by A10, A50, A59, VALUED_1: 13

        .= ( sqrt ((((r3 * r1) + (r4 * r2)) ^2 ) - (((r1 ^2 ) + (r2 ^2 )) * (((r3 ^2 ) + (r4 ^2 )) - (r ^2 ))))) by A66, A70, A71, VALUED_1: 5;

        ( dom k) = the carrier of TD by FUNCT_2:def 1;

        

        then

         A73: (k . [p, y]) = (((( - (xx + yy)) + ( sqrt pp)) . [p, y]) * ((m . [p, y]) " )) by A10, A50, RFUNCT_1:def 1

        .= (((( - (xx + yy)) + ( sqrt pp)) . [p, y]) / (m . [p, y])) by XCMPLX_0:def 9

        .= (((( - (xx + yy)) . [p, y]) + (( sqrt pp) . [p, y])) / ((r1 ^2 ) + (r2 ^2 ))) by A10, A50, A66, VALUED_1: 1

        .= l by A53, A54, A55, A56, A61, A62, A64, A65, A67, A68, A69, A72, VALUED_1: 8;

        

         A74: (X3 . [p, y]) = ((Zfx2 . [p, y]) + ((k (#) Zdx) . [p, y])) by A10, A50, VALUED_1: 1

        .= ((y `1 ) + ((k (#) Zdx) . [p, y])) by A13, A47, A51, A60

        .= ((y `1 ) + (l * r1)) by A53, A61, A73, VALUED_1: 5;

        

         A75: (Y3 . [p, y]) = ((Zfy2 . [p, y]) + ((k (#) Zdy) . [p, y])) by A10, A50, VALUED_1: 1

        .= ((y `2 ) + ((k (#) Zdy) . [p, y])) by A14, A47, A52, A60

        .= ((y `2 ) + (l * r2)) by A54, A62, A73, VALUED_1: 5;

        y in the carrier of D2 by A47;

        

        hence (gg . x) = |[((y `1 ) + (l * r1)), ((y `2 ) + (l * r2))]| by A1, A5, A6, A7, A9, A47, A48, A49, BROUWER: 8

        .= (R . [(X3 . [p, y]), (Y3 . [p, y])]) by A74, A75, TOPREALA:def 2

        .= (R . (F . [p, y])) by A10, A50, A57, A58, FUNCT_3: 49

        .= ((R * F) . [p, x]) by A10, A47, A50, FUNCT_2: 15;

      end;

      

       A76: X3 is continuous by JORDAN5A: 27;

      Y3 is continuous by JORDAN5A: 27;

      then

      reconsider F as continuous Function of TD, [: R^1 , R^1 :] by A76, YELLOW12: 41;

      for pp be Point of D2, V be Subset of S1 st (gg . pp) in V & V is open holds ex W be Subset of D2 st pp in W & W is open & (gg .: W) c= V

      proof

        let pp be Point of D2, V be Subset of S1 such that

         A77: (gg . pp) in V and

         A78: V is open;

        reconsider p1 = pp, fp = p as Point of T2 by PRE_TOPC: 25;

        

         A79: [p, pp] in OK by A4, A9, ZFMISC_1:def 2;

        consider V1 be Subset of T2 such that

         A80: V1 is open and

         A81: (V1 /\ ( [#] S1)) = V by A78, TOPS_2: 24;

        

         A82: (gg . pp) = ((R * F) . [p, pp]) by A46;

        (R " ) is being_homeomorphism by TOPREALA: 34, TOPS_2: 56;

        then

         A83: ((R " ) .: V1) is open by A80, TOPGRP_1: 25;

        

         A84: ( dom F) = the carrier of ( [:T2, T2:] | OK) by FUNCT_2:def 1;

        

         A85: ( dom R) = the carrier of [: R^1 , R^1 :] by FUNCT_2:def 1;

        then

         A86: ( rng F) c= ( dom R);

        then

         A87: ( dom (R * F)) = ( dom F) by RELAT_1: 27;

        

         A88: ( rng R) = ( [#] T2) by TOPREALA: 34, TOPS_2:def 5;

        

         A89: ((R " ) * (R * F)) = (((R " ) * R) * F) by RELAT_1: 36

        .= (( id ( dom R)) * F) by A88, TOPREALA: 34, TOPS_2: 52;

        ( dom ( id ( dom R))) = ( dom R);

        then

         A90: ( dom (( id ( dom R)) * F)) = ( dom F) by A86, RELAT_1: 27;

        for x be object st x in ( dom F) holds ((( id ( dom R)) * F) . x) = (F . x)

        proof

          let x be object such that

           A91: x in ( dom F);

          

           A92: (F . x) in ( rng F) by A91, FUNCT_1:def 3;

          

          thus ((( id ( dom R)) * F) . x) = (( id ( dom R)) . (F . x)) by A91, FUNCT_1: 13

          .= (F . x) by A85, A92, FUNCT_1: 18;

        end;

        then

         A93: (( id ( dom R)) * F) = F by A90, FUNCT_1: 2;

        ((R * F) . [fp, p1]) in V1 by A77, A81, A82, XBOOLE_0:def 4;

        then ((R " ) . ((R * F) . [fp, p1])) in ((R " ) .: V1) by FUNCT_2: 35;

        then (((R " ) * (R * F)) . [fp, p1]) in ((R " ) .: V1) by A10, A79, A84, A87, FUNCT_1: 13;

        then

        consider W be Subset of TD such that

         A94: [fp, p1] in W and

         A95: W is open and

         A96: (F .: W) c= ((R " ) .: V1) by A10, A79, A83, A89, A93, JGRAPH_2: 10;

        consider WW be Subset of [:T2, T2:] such that

         A97: WW is open and

         A98: (WW /\ ( [#] TD)) = W by A95, TOPS_2: 24;

        consider SF be Subset-Family of [:T2, T2:] such that

         A99: WW = ( union SF) and

         A100: for e be set st e in SF holds ex X1 be Subset of T2, Y1 be Subset of T2 st e = [:X1, Y1:] & X1 is open & Y1 is open by A97, BORSUK_1: 5;

         [fp, p1] in WW by A94, A98, XBOOLE_0:def 4;

        then

        consider Z be set such that

         A101: [fp, p1] in Z and

         A102: Z in SF by A99, TARSKI:def 4;

        consider X1,Y1 be Subset of T2 such that

         A103: Z = [:X1, Y1:] and X1 is open and

         A104: Y1 is open by A100, A102;

        set ZZ = (Z /\ ( [#] TD));

        reconsider XX = (Y1 /\ ( [#] D2)) as open Subset of D2 by A104, TOPS_2: 24;

        take XX;

        pp in Y1 by A101, A103, ZFMISC_1: 87;

        hence pp in XX by XBOOLE_0:def 4;

        thus XX is open;

        let b be object;

        assume b in (gg .: XX);

        then

        consider a be Point of D2 such that

         A105: a in XX and

         A106: b = (gg . a) by A2, FUNCT_2: 65;

        reconsider a1 = a, fa = fp as Point of T2 by PRE_TOPC: 25;

        

         A107: a in Y1 by A105, XBOOLE_0:def 4;

        

         A108: [p, a] in OK by A4, A9, ZFMISC_1:def 2;

        fa in X1 by A101, A103, ZFMISC_1: 87;

        then [fa, a] in Z by A103, A107, ZFMISC_1:def 2;

        then [fa, a] in ZZ by A10, A108, XBOOLE_0:def 4;

        then

         A109: (F . [fa, a1]) in (F .: ZZ) by FUNCT_2: 35;

        

         A110: (R qua Function " ) = (R " ) by TOPREALA: 34, TOPS_2:def 4;

        

         A111: ( dom (R " )) = ( [#] T2) by A88, TOPREALA: 34, TOPS_2: 49;

        

         A112: (gg . a1) in the carrier of S1 by A2, FUNCT_2: 5;

        Z c= WW by A99, A102, ZFMISC_1: 74;

        then ZZ c= (WW /\ ( [#] TD)) by XBOOLE_1: 27;

        then (F .: ZZ) c= (F .: W) by A98, RELAT_1: 123;

        then (F . [fa, a1]) in (F .: W) by A109;

        then (R . (F . [fa, a1])) in (R .: ((R " ) .: V1)) by A96, FUNCT_2: 35;

        then ((R * F) . [fa, a1]) in (R .: ((R " ) .: V1)) by A10, A108, FUNCT_2: 15;

        then ((R * F) . [fa, a1]) in V1 by A110, A111, PARTFUN3: 1, TOPREALA: 34;

        then (gg . a) in V1 by A46;

        hence thesis by A81, A106, A112, XBOOLE_0:def 4;

      end;

      hence thesis by A2, JGRAPH_2: 10;

    end;

    theorem :: JORDAN:67

    

     Th67: for n be non zero Element of NAT holds for o,p be Point of ( TOP-REAL n), r be positive Real st p in ( Ball (o,r)) holds ( RotateCircle (o,r,p)) is without_fixpoints

    proof

      let n be non zero Element of NAT ;

      let o,p be Point of ( TOP-REAL n);

      let r be positive Real;

      assume

       A1: p in ( Ball (o,r));

      set f = ( RotateCircle (o,r,p));

      let x be object;

      assume

       A2: x in ( dom f);

      set S = ( Tcircle (o,r));

      

       A3: ( dom f) = the carrier of S by FUNCT_2:def 1;

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

       A4: x = y and

       A5: (f . x) = ( HC (y,p,o,r)) by A1, A2, Def8;

      

       A6: the carrier of S = ( Sphere (o,r)) by TOPREALB: 9;

      ( Sphere (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 17;

      then

       A7: y is Point of ( Tdisk (o,r)) by A2, A3, A4, A6, BROUWER: 3;

      ( Ball (o,r)) c= ( cl_Ball (o,r)) by TOPREAL9: 16;

      then

       A8: p is Point of ( Tdisk (o,r)) by A1, BROUWER: 3;

      ( Ball (o,r)) misses ( Sphere (o,r)) by TOPREAL9: 19;

      then y <> p by A1, A2, A4, A6, XBOOLE_0: 3;

      hence thesis by A4, A5, A7, A8, BROUWER:def 3;

    end;

    begin

    theorem :: JORDAN:68

    

     Th68: U = P & U is a_component & V is a_component & U <> V implies ( Cl P) misses V

    proof

      assume that

       A1: U = P and

       A2: U is a_component and

       A3: V is a_component and

       A4: U <> V;

      assume ( Cl P) meets V;

      then

       A5: ex x be object st x in ( Cl P) & x in V by XBOOLE_0: 3;

      the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      then

      reconsider V1 = V as Subset of T2 by XBOOLE_1: 1;

      reconsider T2C = (T2 | (C ` )) as non empty SubSpace of T2;

      T2C is locally_connected by JORDAN2C: 81;

      then V is open by A3, CONNSP_2: 15;

      then V1 is open by TSEP_1: 17;

      then P meets V1 by A5, PRE_TOPC:def 7;

      hence thesis by A1, A2, A3, A4, CONNSP_1: 35;

    end;

    theorem :: JORDAN:69

    

     Th69: U is a_component implies ((( TOP-REAL 2) | (C ` )) | U) is pathwise_connected

    proof

      set T = (T2 | (C ` ));

      assume

       A1: U is a_component;

      let a,b be Point of (T | U);

      

       A2: the carrier of (T | U) = U by PRE_TOPC: 8;

      

       A3: U <> ( {} T) by A1, CONNSP_1: 32;

      per cases ;

        suppose

         A4: a = b;

        reconsider TU = (T | U) as non empty TopSpace by A3;

        reconsider a as Point of TU;

        reconsider f = ( I[01] --> a) as Function of I[01] , (T | U);

        take f;

        thus thesis by A4, BORSUK_1:def 14, BORSUK_1:def 15, TOPALG_3: 4;

      end;

        suppose

         A5: a <> b;

        

         A6: (T | U) is SubSpace of T2 by TSEP_1: 7;

        then

        reconsider a1 = a, b1 = b as Point of T2 by A3, PRE_TOPC: 25;

        reconsider V = U as Subset of T2 by PRE_TOPC: 11;

        V is_a_component_of (C ` ) by A1;

        then

         A7: V is open by SPRECT_3: 8;

        U is connected by A1;

        then V is connected by CONNSP_1: 23;

        then

        consider P be Subset of T2 such that

         A8: P is_S-P_arc_joining (a1,b1) and

         A9: P c= V by A2, A3, A5, A7, TOPREAL4: 29;

        

         A10: a1 in P by A8, TOPREAL4: 3;

        P is_an_arc_of (a1,b1) by A8, TOPREAL4: 2;

        then

        consider g be Function of I[01] , (T2 | P) such that

         A11: g is being_homeomorphism and

         A12: (g . 0 ) = a and

         A13: (g . 1) = b by TOPREAL1:def 1;

        

         A14: the carrier of (T2 | P) = P by PRE_TOPC: 8;

        then

        reconsider f = g as Function of I[01] , (T | U) by A2, A9, A10, FUNCT_2: 7;

        take f;

        (T2 | P) is SubSpace of (T | U) by A2, A6, A9, A14, TSEP_1: 4;

        hence f is continuous by A11, PRE_TOPC: 26;

        thus thesis by A12, A13;

      end;

    end;

    

     Lm12: for r be non negative Real st A is_an_arc_of (p1,p2) & A is Subset of ( Tdisk (p,r)) holds ex f be Function of ( Tdisk (p,r)), (( TOP-REAL 2) | A) st f is continuous & (f | A) = ( id A)

    proof

      let r be non negative Real;

      set D = ( Tdisk (p,r));

      assume that

       A1: A is_an_arc_of (p1,p2) and

       A2: A is Subset of D;

      reconsider A1 = A as non empty Subset of D by A1, A2, TOPREAL1: 1;

      reconsider A2 = A as non empty Subset of T2 by A1, TOPREAL1: 1;

      set TA = (T2 | A2);

      consider h be Function of I[01] , TA such that

       A3: h is being_homeomorphism and (h . 0 ) = p1 and (h . 1) = p2 by A1, TOPREAL1:def 1;

      

       A4: h1 is being_homeomorphism by TREAL_1: 17;

      reconsider hh = h as Function of C0, TA by TOPMETR: 20;

      

       A5: TA = (D | A1) by TOPALG_5: 4;

      then

      reconsider f = (h1 * (hh " )) as Function of (D | A1), C1;

      A is closed by A1, JORDAN6: 11;

      then

       A6: A1 is closed by TSEP_1: 12;

      (hh " ) is continuous by A3, TOPMETR: 20, TOPS_2:def 5;

      then

      consider g be continuous Function of D, C1 such that

       A7: (g | A1) = f by A4, A5, A6, TIETZE: 23;

      reconsider R = ((hh * (h1 " )) * g) as Function of D, (( TOP-REAL 2) | A);

      take R;

      (h1 " ) is continuous by A4, TOPS_2:def 5;

      hence R is continuous by A3, TOPMETR: 20;

      

       A8: the carrier of TA = A1 by PRE_TOPC: 8;

      

       A9: ( dom R) = the carrier of D by FUNCT_2:def 1;

      

       A10: ( dom ( id A)) = A;

      now

        let a be object;

        assume

         A11: a in ( dom (R | A));

        then

         A12: a in ( dom R) by RELAT_1: 57;

        

         A13: ( dom g) = the carrier of D by FUNCT_2:def 1;

        

         A14: ( dom (h1 * (hh " ))) = the carrier of TA by FUNCT_2:def 1;

        

         A15: ((hh * (h1 " )) * (h1 * (hh " ))) = (((hh * (h1 " )) * h1) * (hh " )) by RELAT_1: 36

        .= ((hh * ((h1 " ) * h1)) * (hh " )) by RELAT_1: 36

        .= ((hh * ( id C0)) * (hh " )) by A4, GRCAT_1: 41

        .= (hh * (hh " )) by FUNCT_2: 17

        .= ( id TA) by A3, GRCAT_1: 41;

        

        thus ((R | A) . a) = (R . a) by A11, FUNCT_1: 49

        .= ((hh * (h1 " )) . (g . a)) by A13, A12, FUNCT_1: 13

        .= ((hh * (h1 " )) . ((h1 * (hh " )) . a)) by A7, A11, FUNCT_1: 49

        .= (( id A) . a) by A8, A11, A14, A15, FUNCT_1: 13;

      end;

      hence thesis by A2, A9, A10, FUNCT_1: 2, RELAT_1: 62;

    end;

    

     Lm13: for r be positive Real st A is_an_arc_of (p1,p2) & A c= C & C c= ( Ball (p,r)) & p in U & (( Cl P) /\ (P ` )) c= A & P c= ( Ball (p,r)) holds for f be Function of ( Tdisk (p,r)), (( TOP-REAL 2) | A) st f is continuous & (f | A) = ( id A) & U = P & U is a_component & B = (( cl_Ball (p,r)) \ {p}) holds ex g be Function of ( Tdisk (p,r)), (( TOP-REAL 2) | B) st g is continuous & for x be Point of ( Tdisk (p,r)) holds (x in ( Cl P) implies (g . x) = (f . x)) & (x in (P ` ) implies (g . x) = x)

    proof

      let r be positive Real;

      set D = ( Tdisk (p,r));

      assume that

       A1: A is_an_arc_of (p1,p2) and

       A2: A c= C and

       A3: C c= ( Ball (p,r)) and

       A4: p in U and

       A5: (( Cl P) /\ (P ` )) c= A and

       A6: P c= ( Ball (p,r));

      let f be Function of D, (T2 | A);

      assume that

       A7: f is continuous and

       A8: (f | A) = ( id A) and

       A9: U = P and

       A10: U is a_component and

       A11: B = (( cl_Ball (p,r)) \ {p});

      reconsider B1 = B as non empty Subset of T2 by A11;

      reconsider T2B1 = (T2 | B1) as non empty SubSpace of T2;

      

       A12: the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      

       A13: the carrier of (T2 | A) = A by PRE_TOPC: 8;

      

       A14: the carrier of D = ( cl_Ball (p,r)) by BROUWER: 3;

      

       A15: ( Ball (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 16;

      

       A16: A <> {} by A1, TOPREAL1: 1;

      reconsider A1 = A as non empty Subset of T2 by A1, TOPREAL1: 1;

      

       A17: not p in C by A4, A12, XBOOLE_0:def 5;

       |.(p - p).| = 0 by TOPRNS_1: 28;

      then

       A18: p in ( [#] D) by A14, TOPREAL9: 8;

      

       A19: P c= ( Cl P) by PRE_TOPC: 18;

      then

      reconsider F1 = (( Cl P) /\ ( [#] D)) as non empty Subset of D by A4, A9, A18, XBOOLE_0:def 4;

      

       A20: ( Sphere (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 17;

      

       A21: ( Ball (p,r)) misses ( Sphere (p,r)) by TOPREAL9: 19;

      consider e be Point of T2 such that

       A22: e in ( Sphere (p,r)) by SUBSET_1: 4;

       not e in P by A6, A21, A22, XBOOLE_0: 3;

      then e in (P ` ) by SUBSET_1: 29;

      then

      reconsider F3 = ((P ` ) /\ ( [#] D)) as non empty Subset of D by A14, A20, A22, XBOOLE_0:def 4;

      reconsider T1 = (D | F1) as non empty SubSpace of D;

      reconsider T3 = (D | F3) as non empty SubSpace of D;

      

       A23: the carrier of T1 = F1 by PRE_TOPC: 8;

      

       A24: the carrier of T3 = F3 by PRE_TOPC: 8;

      

       A25: the carrier of T2B1 = B1 by PRE_TOPC: 8;

      

       A26: A c= B

      proof

        let a be object;

        assume a in A;

        then

         A27: a in C by A2;

        then a in ( Ball (p,r)) by A3;

        hence thesis by A11, A15, A17, A27, ZFMISC_1: 56;

      end;

      

       A28: F3 c= B

      proof

        let a be object;

        assume

         A29: a in F3;

        then a in (P ` ) by XBOOLE_0:def 4;

        then not a in P by XBOOLE_0:def 5;

        hence thesis by A4, A9, A11, A14, A29, ZFMISC_1: 56;

      end;

      (f | F1) is Function of F1, A by A13, A16, FUNCT_2: 32;

      then

      reconsider f1 = (f | F1) as Function of T1, T2B1 by A16, A23, A25, A26, FUNCT_2: 7;

      reconsider g1 = ( id F3) as Function of T3, T2B1 by A24, A25, A28, FUNCT_2: 7;

      

       A30: F1 = ( [#] T1) by PRE_TOPC: 8;

      

       A31: F3 = ( [#] T3) by PRE_TOPC: 8;

      

       A32: (( [#] T1) \/ ( [#] T3)) = ( [#] D)

      proof

        thus (( [#] T1) \/ ( [#] T3)) c= ( [#] D) by A30, A31, XBOOLE_1: 8;

        let p be object;

        assume

         A33: p in ( [#] D);

        per cases ;

          suppose p in P;

          then p in F1 by A19, A33, XBOOLE_0:def 4;

          hence thesis by A30, XBOOLE_0:def 3;

        end;

          suppose not p in P;

          then p in (P ` ) by A14, A33, SUBSET_1: 29;

          then p in F3 by A33, XBOOLE_0:def 4;

          hence thesis by A31, XBOOLE_0:def 3;

        end;

      end;

      reconsider DT = ( [#] D) as closed Subset of T2 by BORSUK_1:def 11, TSEP_1: 1;

      (DT /\ ( Cl P)) is closed;

      then

       A34: F1 is closed by TSEP_1: 8;

      P is_a_component_of (C ` ) by A9, A10;

      then P is open by SPRECT_3: 8;

      then (DT /\ (P ` )) is closed;

      then

       A35: F3 is closed by TSEP_1: 8;

      reconsider f2 = (f | F1) as Function of T1, (T2 | A1) by A23, FUNCT_2: 32;

      

       A36: (T2 | A1) is SubSpace of T2B1 by A13, A25, A26, TSEP_1: 4;

      T3 is SubSpace of T2 by TSEP_1: 7;

      then

       A37: T3 is SubSpace of T2B1 by A24, A25, A28, TSEP_1: 4;

      f2 is continuous by A7, TOPMETR: 7;

      then

       A38: f1 is continuous by A36, PRE_TOPC: 26;

      reconsider g2 = ( id F3) as Function of T3, T3 by A24;

      g2 = ( id T3) by PRE_TOPC: 8;

      then

       A39: g1 is continuous by A37, PRE_TOPC: 26;

      

       A40: for x be set st x in ( Cl P) & x in (P ` ) holds (f . x) = x

      proof

        let x be set;

        assume that

         A41: x in ( Cl P) and

         A42: x in (P ` );

        

         A43: x in (( Cl P) /\ (P ` )) by A41, A42, XBOOLE_0:def 4;

        then (( id A) . x) = x by A5, FUNCT_1: 18;

        hence thesis by A5, A8, A43, FUNCT_1: 49;

      end;

      for x be object st x in (( [#] T1) /\ ( [#] T3)) holds (f1 . x) = (g1 . x)

      proof

        let x be object;

        assume

         A44: x in (( [#] T1) /\ ( [#] T3));

        then

         A45: x in ( [#] T1) by XBOOLE_0:def 4;

        then

         A46: x in ( Cl P) by A30, XBOOLE_0:def 4;

        x in (P ` ) by A31, A44, XBOOLE_0:def 4;

        then

         A47: (f . x) = x by A40, A46;

        

        thus (f1 . x) = (f . x) by A30, A45, FUNCT_1: 49

        .= (g1 . x) by A31, A44, A47, FUNCT_1: 18;

      end;

      then

      consider g be Function of D, (T2 | B) such that

       A48: g = (f1 +* g1) and

       A49: g is continuous by A30, A31, A32, A34, A35, A38, A39, JGRAPH_2: 1;

      take g;

      thus g is continuous by A49;

      let x be Point of D;

      

       A50: ( dom g1) = the carrier of T3 by FUNCT_2:def 1;

      hereby

        assume

         A51: x in ( Cl P);

        then

         A52: x in F1 by XBOOLE_0:def 4;

        per cases ;

          suppose not x in ( dom g1);

          

          hence (g . x) = (f1 . x) by A48, FUNCT_4: 11

          .= (f . x) by A52, FUNCT_1: 49;

        end;

          suppose

           A53: x in ( dom g1);

          then

           A54: x in (P ` ) by XBOOLE_0:def 4;

          

          thus (g . x) = (g1 . x) by A48, A53, FUNCT_4: 13

          .= x by A53, FUNCT_1: 18

          .= (f . x) by A40, A51, A54;

        end;

      end;

      assume x in (P ` );

      then

       A55: x in F3 by XBOOLE_0:def 4;

      

      hence (g . x) = (g1 . x) by A48, A50, FUNCT_4: 13

      .= x by A55, FUNCT_1: 18;

    end;

    

     Lm14: for A be non empty Subset of T2 st U <> V holds for r be positive Real st A c= C & C c= ( Ball (p,r)) & p in V & (( Cl P) /\ (P ` )) c= A & ( Ball (p,r)) meets P holds for f be Function of ( Tdisk (p,r)), (( TOP-REAL 2) | A) st f is continuous & (f | A) = ( id A) & U = P & U is a_component & V is a_component & B = (( cl_Ball (p,r)) \ {p}) holds ex g be Function of ( Tdisk (p,r)), (( TOP-REAL 2) | B) st g is continuous & for x be Point of ( Tdisk (p,r)) holds (x in ( Cl P) implies (g . x) = x) & (x in (P ` ) implies (g . x) = (f . x))

    proof

      let A be non empty Subset of T2 such that

       A1: U <> V;

      let r be positive Real;

      set D = ( Tdisk (p,r));

      assume that

       A2: A c= C and

       A3: C c= ( Ball (p,r)) and

       A4: p in V and

       A5: (( Cl P) /\ (P ` )) c= A and

       A6: ( Ball (p,r)) meets P;

      let f be Function of D, (T2 | A);

      assume that

       A7: f is continuous and

       A8: (f | A) = ( id A) and

       A9: U = P and

       A10: U is a_component and

       A11: V is a_component and

       A12: B = (( cl_Ball (p,r)) \ {p});

      reconsider B1 = B as non empty Subset of T2 by A12;

      reconsider T2B1 = (T2 | B1) as non empty SubSpace of T2;

      

       A13: the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      

       A14: the carrier of (T2 | A) = A by PRE_TOPC: 8;

      

       A15: the carrier of D = ( cl_Ball (p,r)) by BROUWER: 3;

      

       A16: ( Ball (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 16;

      

       A17: not p in C by A4, A13, XBOOLE_0:def 5;

       |.(p - p).| = 0 by TOPRNS_1: 28;

      then

       A18: p in ( [#] D) by A15, TOPREAL9: 8;

      

       A19: P c= ( Cl P) by PRE_TOPC: 18;

      ex j be object st j in ( Ball (p,r)) & j in P by A6, XBOOLE_0: 3;

      then

      reconsider F1 = (( Cl P) /\ ( [#] D)) as non empty Subset of D by A15, A16, A19, XBOOLE_0:def 4;

       not p in P by A1, A10, A11, CONNSP_1: 35, A4, A9, XBOOLE_0: 3;

      then p in (P ` ) by SUBSET_1: 29;

      then

      reconsider F3 = ((P ` ) /\ ( [#] D)) as non empty Subset of D by A18, XBOOLE_0:def 4;

      set T1 = (D | F1);

      set T3 = (D | F3);

      

       A20: the carrier of T1 = F1 by PRE_TOPC: 8;

      

       A21: the carrier of T3 = F3 by PRE_TOPC: 8;

      

       A22: the carrier of (T2 | B1) = B1 by PRE_TOPC: 8;

      

       A23: A c= B

      proof

        let a be object;

        assume a in A;

        then

         A24: a in C by A2;

        then a in ( Ball (p,r)) by A3;

        hence thesis by A12, A16, A17, A24, ZFMISC_1: 56;

      end;

      

       A25: F1 c= B

      proof

        let a be object;

        assume

         A26: a in F1;

        then

         A27: a in ( Cl P) by XBOOLE_0:def 4;

         not p in ( Cl P) by A4, XBOOLE_0: 3, A1, A9, A10, A11, Th68;

        hence thesis by A12, A15, A26, A27, ZFMISC_1: 56;

      end;

      then

      reconsider f1 = ( id F1) as Function of T1, T2B1 by A20, A22, FUNCT_2: 7;

      (f | F3) is Function of F3, A by A14;

      then

      reconsider g1 = (f | F3) as Function of T3, T2B1 by A21, A22, A23, FUNCT_2: 7;

      

       A28: F1 = ( [#] T1) by PRE_TOPC: 8;

      

       A29: F3 = ( [#] T3) by PRE_TOPC: 8;

      

       A30: (( [#] T1) \/ ( [#] T3)) = ( [#] D)

      proof

        thus (( [#] T1) \/ ( [#] T3)) c= ( [#] D) by A28, A29, XBOOLE_1: 8;

        let p be object;

        assume

         A31: p in ( [#] D);

        per cases ;

          suppose p in P;

          then p in F1 by A19, A31, XBOOLE_0:def 4;

          hence thesis by A28, XBOOLE_0:def 3;

        end;

          suppose not p in P;

          then p in (P ` ) by A15, A31, SUBSET_1: 29;

          then p in F3 by A31, XBOOLE_0:def 4;

          hence thesis by A29, XBOOLE_0:def 3;

        end;

      end;

      reconsider DT = ( [#] D) as closed Subset of T2 by BORSUK_1:def 11, TSEP_1: 1;

      (DT /\ ( Cl P)) is closed;

      then

       A32: F1 is closed by TSEP_1: 8;

      P is_a_component_of (C ` ) by A9, A10;

      then P is open by SPRECT_3: 8;

      then (DT /\ (P ` )) is closed;

      then

       A33: F3 is closed by TSEP_1: 8;

      

       A34: ( id T1) = ( id F1) by PRE_TOPC: 8;

      T1 is SubSpace of T2 by TSEP_1: 7;

      then T1 is SubSpace of T2B1 by A20, A22, A25, TSEP_1: 4;

      then

       A35: f1 is continuous by A34, PRE_TOPC: 26;

      

       A36: (T2 | A) is SubSpace of T2B1 by A14, A22, A23, TSEP_1: 4;

      reconsider g2 = g1 as Function of T3, (T2 | A) by A21;

      g2 is continuous by A7, TOPMETR: 7;

      then

       A37: g1 is continuous by A36, PRE_TOPC: 26;

      

       A38: for x be set st x in ( Cl P) & x in (P ` ) holds (f . x) = x

      proof

        let x be set;

        assume that

         A39: x in ( Cl P) and

         A40: x in (P ` );

        

         A41: x in (( Cl P) /\ (P ` )) by A39, A40, XBOOLE_0:def 4;

        then (( id A) . x) = x by A5, FUNCT_1: 18;

        hence thesis by A5, A8, A41, FUNCT_1: 49;

      end;

      for x be object st x in (( [#] T1) /\ ( [#] T3)) holds (f1 . x) = (g1 . x)

      proof

        let x be object;

        assume

         A42: x in (( [#] T1) /\ ( [#] T3));

        then

         A43: x in ( [#] T1) by XBOOLE_0:def 4;

        then

         A44: x in ( Cl P) by A28, XBOOLE_0:def 4;

        x in (P ` ) by A29, A42, XBOOLE_0:def 4;

        then

         A45: (f . x) = x by A38, A44;

        

        thus (f1 . x) = x by A28, A43, FUNCT_1: 18

        .= (g1 . x) by A29, A42, A45, FUNCT_1: 49;

      end;

      then

      consider g be Function of D, (T2 | B) such that

       A46: g = (f1 +* g1) and

       A47: g is continuous by A28, A29, A30, A32, A33, A35, A37, JGRAPH_2: 1;

      take g;

      thus g is continuous by A47;

      let x be Point of D;

      

       A48: ( dom g1) = the carrier of T3 by FUNCT_2:def 1;

      hereby

        assume

         A49: x in ( Cl P);

        then

         A50: x in F1 by XBOOLE_0:def 4;

        per cases ;

          suppose not x in ( dom g1);

          

          hence (g . x) = (f1 . x) by A46, FUNCT_4: 11

          .= x by A50, FUNCT_1: 18;

        end;

          suppose

           A51: x in ( dom g1);

          then

           A52: x in (P ` ) by A21, XBOOLE_0:def 4;

          

          thus (g . x) = (g1 . x) by A46, A51, FUNCT_4: 13

          .= (f . x) by A21, A51, FUNCT_1: 49

          .= x by A38, A49, A52;

        end;

      end;

      assume x in (P ` );

      then

       A53: x in F3 by XBOOLE_0:def 4;

      

      hence (g . x) = (g1 . x) by A21, A46, A48, FUNCT_4: 13

      .= (f . x) by A53, FUNCT_1: 49;

    end;

    

     Lm15: ( BDD C) is non empty & U = P & U is a_component implies C = ( Fr P)

    proof

      assume that

       A1: ( BDD C) is non empty and

       A2: U = P and

       A3: U is a_component and

       A4: C <> ( Fr P);

      

       A5: the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      reconsider T2C = (T2 | (C ` )) as non empty SubSpace of T2;

      

       A6: T2C is locally_connected by JORDAN2C: 81;

      then U is open by A3, CONNSP_2: 15;

      then

      reconsider P as open Subset of T2 by A2, TSEP_1: 17;

      

       A7: ( Fr P) = (( Cl P) /\ (P ` )) by PRE_TOPC: 22;

      set Z = { X where X be Subset of (T2 | (C ` )) : X is a_component & X <> U };

      set V = ( union Z);

      

       A8: ((V \/ U) \/ C) = the carrier of T2

      proof

        

         A9: V c= the carrier of T2

        proof

          let a be object;

          assume a in V;

          then

          consider A be set such that

           A10: a in A and

           A11: A in Z by TARSKI:def 4;

          ex X be Subset of (T2 | (C ` )) st X = A & X is a_component & X <> U by A11;

          hence thesis by A5, A10, TARSKI:def 3;

        end;

        U c= the carrier of T2 by A5, XBOOLE_1: 1;

        then (V \/ U) c= the carrier of T2 by A9, XBOOLE_1: 8;

        hence ((V \/ U) \/ C) c= the carrier of T2 by XBOOLE_1: 8;

        let a be object;

        assume

         A12: a in the carrier of T2;

        per cases ;

          suppose a in C;

          hence thesis by XBOOLE_0:def 3;

        end;

          suppose not a in C;

          then

          reconsider a as Point of (T2 | (C ` )) by A5, A12, SUBSET_1: 29;

          

           A13: a in ( Component_of a) by CONNSP_1: 38;

          per cases ;

            suppose ( Component_of a) = U;

            then a in (V \/ U) by A13, XBOOLE_0:def 3;

            hence thesis by XBOOLE_0:def 3;

          end;

            suppose

             A14: ( Component_of a) <> U;

            ( Component_of a) is a_component by CONNSP_1: 40;

            then ( Component_of a) in Z by A14;

            then a in V by A13, TARSKI:def 4;

            then a in (V \/ U) by XBOOLE_0:def 3;

            hence thesis by XBOOLE_0:def 3;

          end;

        end;

      end;

      

       A15: P misses (P ` ) by XBOOLE_1: 79;

      ( Fr P) c= C

      proof

        let a be object;

        assume

         A16: a in ( Fr P);

        then

         A17: a in ( Cl P) by XBOOLE_0:def 4;

        

         A18: a in (P ` ) by A7, A16, XBOOLE_0:def 4;

        assume not a in C;

        then a in (V \/ U) by A8, A16, XBOOLE_0:def 3;

        then a in V or a in U by XBOOLE_0:def 3;

        then

        consider O be set such that

         A19: a in O and

         A20: O in Z by A2, A15, A18, TARSKI:def 4, XBOOLE_0: 3;

        consider X be Subset of (T2 | (C ` )) such that

         A21: X = O and

         A22: X is a_component and

         A23: X <> U by A20;

        ( Cl P) misses X by A2, A3, A22, A23, Th68;

        hence thesis by A17, A19, A21, XBOOLE_0: 3;

      end;

      then ( Fr P) c< C by A4;

      then

      consider p1, p2, A such that

       A24: A is_an_arc_of (p1,p2) and

       A25: ( Fr P) c= A and

       A26: A c= C by BORSUK_4: 59;

      

       A27: U <> ( {} (T2 | (C ` ))) by A3, CONNSP_1: 32;

      per cases ;

        suppose P is bounded;

        then

        reconsider P as bounded Subset of T2;

        consider p be object such that

         A28: p in U by A27, XBOOLE_0:def 1;

        reconsider p as Point of T2 by A2, A28;

        

         A29: (P \/ C) is bounded by TOPREAL6: 67;

        then

        reconsider PC = (P \/ C) as bounded Subset of ( Euclid 2) by JORDAN2C: 11;

        consider r be positive Real such that

         A30: PC c= ( Ball (p,r)) by A29, Th26;

        C c= PC by XBOOLE_1: 7;

        then

         A31: C c= ( Ball (p,r)) by A30;

        set D = ( Tdisk (p,r));

        set S = ( Tcircle (p,r));

        set B = (( cl_Ball (p,r)) \ {p});

        

         A32: the carrier of S = ( Sphere (p,r)) by TOPREALB: 9;

        

         A33: the carrier of D = ( cl_Ball (p,r)) by BROUWER: 3;

        

         A34: ( Sphere (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 17;

        

         A35: ( Ball (p,r)) misses ( Sphere (p,r)) by TOPREAL9: 19;

        

         A36: ( Ball (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 16;

        A c= ( Ball (p,r)) by A26, A31;

        then A is Subset of D by A33, A36, XBOOLE_1: 1;

        then

        consider R be Function of D, (T2 | A) such that

         A37: R is continuous and

         A38: (R | A) = ( id A) by A24, Lm12;

        P c= PC by XBOOLE_1: 7;

        then

         A39: P c= ( Ball (p,r)) by A30;

        then

        consider f be Function of D, (T2 | B) such that

         A40: f is continuous and

         A41: for x be Point of D holds (x in ( Cl P) implies (f . x) = (R . x)) & (x in (P ` ) implies (f . x) = x) by A2, A3, A7, A24, A25, A26, A28, A31, A37, A38, Lm13;

        set g = ( DiskProj (p,r,p));

        set h = ( RotateCircle (p,r,p));

        

         A42: S is SubSpace of D by A32, A33, A34, TSEP_1: 4;

        reconsider F = (h * (g * f)) as Function of D, D by A32, A33, A34, FUNCT_2: 7;

        p is Point of D by Th17;

        then

         A43: g is continuous by Th64;

         |.(p - p).| = 0 by TOPRNS_1: 28;

        then

         A44: p in ( Ball (p,r)) by TOPREAL9: 7;

        then h is continuous by Th66;

        then

         A45: F is continuous by A40, A42, A43, PRE_TOPC: 26;

        now

          let x be object;

          per cases ;

            suppose

             A46: x in ( dom F);

            

             A47: (( Ball (p,r)) \/ ( Sphere (p,r))) = ( cl_Ball (p,r)) by TOPREAL9: 18;

            now

              per cases by A33, A46, A47, XBOOLE_0:def 3;

                suppose

                 A48: x in ( Ball (p,r));

                (F . x) in the carrier of S by A46, FUNCT_2: 5;

                hence (F . x) <> x by A32, A35, A48, XBOOLE_0: 3;

              end;

                suppose

                 A49: x in ( Sphere (p,r));

                

                 A50: ( dom f) = the carrier of D by FUNCT_2:def 1;

                 not x in P by A35, A39, A49, XBOOLE_0: 3;

                then

                 A51: x in (P ` ) by A49, SUBSET_1: 29;

                

                 A52: (g | ( Sphere (p,r))) = ( id ( Sphere (p,r))) by A44, Th65;

                h is without_fixpoints by A44, Th67;

                then

                 A53: not x is_a_fixpoint_of h;

                

                 A54: ( dom h) = the carrier of S by FUNCT_2:def 1;

                (F . x) = (h . ((g * f) . x)) by A46, FUNCT_1: 12

                .= (h . (g . (f . x))) by A33, A34, A49, A50, FUNCT_1: 13

                .= (h . (g . x)) by A33, A34, A41, A49, A51

                .= (h . (( id ( Sphere (p,r))) . x)) by A49, A52, FUNCT_1: 49

                .= (h . x) by A49, FUNCT_1: 18;

                hence (F . x) <> x by A32, A49, A53, A54;

              end;

            end;

            hence not x is_a_fixpoint_of F;

          end;

            suppose not x in ( dom F);

            hence not x is_a_fixpoint_of F;

          end;

        end;

        then not F is with_fixpoint;

        hence thesis by A45, BROUWER: 14;

      end;

        suppose

         A55: not P is bounded;

        consider p be object such that

         A56: p in ( BDD C) by A1;

        consider Z be set such that

         A57: p in Z and

         A58: Z in { B where B be Subset of T2 : B is_inside_component_of C } by A56, TARSKI:def 4;

        consider P1 be Subset of T2 such that

         A59: Z = P1 and

         A60: P1 is_inside_component_of C by A58;

        consider U1 be Subset of (T2 | (C ` )) such that

         A61: U1 = P1 and

         A62: U1 is a_component and U1 is bounded Subset of ( Euclid 2) by A60, JORDAN2C: 13;

        U1 is open by A6, A62, CONNSP_2: 15;

        then

        reconsider P1 as non empty open bounded Subset of T2 by A57, A59, A60, A61, TSEP_1: 17;

        reconsider p as Point of T2 by A57, A59;

        

         A63: p in P1 by A57, A59;

        

         A64: (P1 \/ C) is bounded by TOPREAL6: 67;

        then

        reconsider PC = (P1 \/ C) as bounded Subset of ( Euclid 2) by JORDAN2C: 11;

        consider rv be positive Real such that

         A65: PC c= ( Ball (p,rv)) by A64, Th26;

         not P c= ( Ball (p,rv)) by A55, RLTOPSP1: 42;

        then

        consider u be object such that

         A66: u in P and

         A67: not u in ( Ball (p,rv));

        reconsider u as Point of T2 by A66;

        set r = |.(u - p).|;

        P misses P1 by A2, A3, A55, A61, A62, CONNSP_1: 35;

        then p <> u by A57, A59, A66, XBOOLE_0: 3;

        then

        reconsider r as non zero non negative Real by TOPRNS_1: 28;

        

         A68: r >= rv by A67, TOPREAL9: 7;

        then ( Ball (p,rv)) c= ( Ball (p,r)) by Th18;

        then

         A69: PC c= ( Ball (p,r)) by A65;

        

         A70: ( Fr ( Ball (p,r))) = ( Sphere (p,r)) by Th24;

        u in ( Sphere (p,r)) by TOPREAL9: 9;

        then

         A71: P meets ( Ball (p,r)) by A66, A70, TOPS_1: 28;

        

         A72: C c= PC by XBOOLE_1: 7;

        then

         A73: C c= ( Ball (p,r)) by A69;

        set D = ( Tdisk (p,r));

        set S = ( Tcircle (p,r));

        set B = (( cl_Ball (p,r)) \ {p});

        

         A74: the carrier of S = ( Sphere (p,r)) by TOPREALB: 9;

        

         A75: the carrier of D = ( cl_Ball (p,r)) by BROUWER: 3;

        

         A76: ( Sphere (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 17;

        

         A77: ( Ball (p,r)) misses ( Sphere (p,r)) by TOPREAL9: 19;

        

         A78: ( Ball (p,r)) c= ( cl_Ball (p,r)) by TOPREAL9: 16;

        A c= ( Ball (p,r)) by A26, A73;

        then A is Subset of D by A75, A78, XBOOLE_1: 1;

        then

        consider R be Function of D, (T2 | A) such that

         A79: R is continuous and

         A80: (R | A) = ( id A) by A24, Lm12;

        p1 in A by A24, TOPREAL1: 1;

        then

        consider f be Function of D, (T2 | B) such that

         A81: f is continuous and

         A82: for x be Point of D holds (x in ( Cl P) implies (f . x) = x) & (x in (P ` ) implies (f . x) = (R . x)) by A2, A3, A7, A25, A26, A55, A61, A62, A63, A71, A73, A79, A80, Lm14;

        set g = ( DiskProj (p,r,p));

        set h = ( RotateCircle (p,r,p));

        

         A83: S is SubSpace of D by A74, A75, A76, TSEP_1: 4;

        reconsider F = (h * (g * f)) as Function of D, D by A74, A75, A76, FUNCT_2: 7;

        p is Point of D by Th17;

        then

         A84: g is continuous by Th64;

         |.(p - p).| = 0 by TOPRNS_1: 28;

        then

         A85: p in ( Ball (p,r)) by TOPREAL9: 7;

        then h is continuous by Th66;

        then

         A86: F is continuous by A81, A83, A84, PRE_TOPC: 26;

        now

          let x be object;

          per cases ;

            suppose

             A87: x in ( dom F);

            

             A88: (( Ball (p,r)) \/ ( Sphere (p,r))) = ( cl_Ball (p,r)) by TOPREAL9: 18;

            now

              per cases by A75, A87, A88, XBOOLE_0:def 3;

                suppose

                 A89: x in ( Ball (p,r));

                (F . x) in the carrier of S by A87, FUNCT_2: 5;

                hence (F . x) <> x by A74, A77, A89, XBOOLE_0: 3;

              end;

                suppose

                 A90: x in ( Sphere (p,r));

                

                 A91: ( dom f) = the carrier of D by FUNCT_2:def 1;

                

                 A92: P c= ( Cl P) by PRE_TOPC: 18;

                set SS = ( Sphere (p,r));

                SS c= (C ` )

                proof

                  let a be object;

                  assume

                   A93: a in SS;

                  assume not a in (C ` );

                  then

                   A94: a in C by A93, SUBSET_1: 29;

                  reconsider a as Point of T2 by A93;

                  a in PC by A72, A94;

                  then |.(a - p).| < rv by A65, TOPREAL9: 7;

                  hence contradiction by A68, A93, TOPREAL9: 9;

                end;

                then

                reconsider SS as Subset of (T2 | (C ` )) by PRE_TOPC: 8;

                

                 A95: u in SS by TOPREAL9: 9;

                SS is connected by CONNSP_1: 23;

                then SS misses U or SS c= U by A3, CONNSP_1: 36;

                then

                 A96: x in P by A2, A66, A90, A95, XBOOLE_0: 3;

                

                 A97: (g | ( Sphere (p,r))) = ( id ( Sphere (p,r))) by A85, Th65;

                h is without_fixpoints by A85, Th67;

                then

                 A98: not x is_a_fixpoint_of h;

                

                 A99: ( dom h) = the carrier of S by FUNCT_2:def 1;

                (F . x) = (h . ((g * f) . x)) by A87, FUNCT_1: 12

                .= (h . (g . (f . x))) by A75, A76, A90, A91, FUNCT_1: 13

                .= (h . (g . x)) by A75, A76, A82, A90, A92, A96

                .= (h . (( id ( Sphere (p,r))) . x)) by A90, A97, FUNCT_1: 49

                .= (h . x) by A90, FUNCT_1: 18;

                hence (F . x) <> x by A74, A90, A98, A99;

              end;

            end;

            hence not x is_a_fixpoint_of F;

          end;

            suppose not x in ( dom F);

            hence not x is_a_fixpoint_of F;

          end;

        end;

        then F is without_fixpoints;

        hence thesis by A86, BROUWER: 14;

      end;

    end;

    set rp = 1;

    set rl = ( - rp);

    set rg = 3;

    set rd = ( - rg);

    set a = |[rl, 0 ]|;

    set b = |[rp, 0 ]|;

    set c = |[ 0 , rg]|;

    set d = |[ 0 , rd]|;

    set lg = |[rl, rg]|;

    set pg = |[rp, rg]|;

    set ld = |[rl, rd]|;

    set pd = |[rp, rd]|;

    set R = ( closed_inside_of_rectangle (rl,rp,rd,rg));

    set dR = ( rectangle (rl,rp,rd,rg));

    set TR = ( Trectangle (rl,rp,rd,rg));

    

     Lm16: (a `1 ) = rl by EUCLID: 52;

    

     Lm17: (b `1 ) = rp by EUCLID: 52;

    

     Lm18: (a `2 ) = 0 by EUCLID: 52;

    

     Lm19: (b `2 ) = 0 by EUCLID: 52;

    

     Lm20: (c `1 ) = 0 by EUCLID: 52;

    

     Lm21: (c `2 ) = rg by EUCLID: 52;

    

     Lm22: (d `1 ) = 0 by EUCLID: 52;

    

     Lm23: (d `2 ) = rd by EUCLID: 52;

    

     Lm24: (lg `1 ) = rl by EUCLID: 52;

    

     Lm25: (lg `2 ) = rg by EUCLID: 52;

    

     Lm26: (ld `1 ) = rl by EUCLID: 52;

    

     Lm27: (ld `2 ) = rd by EUCLID: 52;

    

     Lm28: (pg `1 ) = rp by EUCLID: 52;

    

     Lm29: (pg `2 ) = rg by EUCLID: 52;

    

     Lm30: (pd `1 ) = rp by EUCLID: 52;

    

     Lm31: (pd `2 ) = rd by EUCLID: 52;

    

     Lm32: ld = |[(ld `1 ), (ld `2 )]| by EUCLID: 53;

    

     Lm33: lg = |[(lg `1 ), (lg `2 )]| by EUCLID: 53;

    

     Lm34: pd = |[(pd `1 ), (pd `2 )]| by EUCLID: 53;

    

     Lm35: pg = |[(pg `1 ), (pg `2 )]| by EUCLID: 53;

    

     Lm36: dR = ((( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) \/ (( LSeg (pg,pd)) \/ ( LSeg (pd,ld)))) by SPPOL_2:def 3;

    

     Lm37: ( LSeg (ld,lg)) c= (( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) by XBOOLE_1: 7;

    (( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) c= dR by Lm36, XBOOLE_1: 7;

    then

     Lm38: ( LSeg (ld,lg)) c= dR by Lm37;

    

     Lm39: ( LSeg (lg,pg)) c= (( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) by XBOOLE_1: 7;

    (( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) c= dR by Lm36, XBOOLE_1: 7;

    then

     Lm40: ( LSeg (lg,pg)) c= dR by Lm39;

    

     Lm41: ( LSeg (pg,pd)) c= (( LSeg (pg,pd)) \/ ( LSeg (pd,ld))) by XBOOLE_1: 7;

    (( LSeg (pg,pd)) \/ ( LSeg (pd,ld))) c= dR by Lm36, XBOOLE_1: 7;

    then

     Lm42: ( LSeg (pg,pd)) c= dR by Lm41;

    

     Lm43: ( LSeg (pd,ld)) c= (( LSeg (pg,pd)) \/ ( LSeg (pd,ld))) by XBOOLE_1: 7;

    (( LSeg (pg,pd)) \/ ( LSeg (pd,ld))) c= dR by Lm36, XBOOLE_1: 7;

    then

     Lm44: ( LSeg (pd,ld)) c= dR by Lm43;

    

     Lm45: ( LSeg (ld,lg)) is vertical by Lm24, Lm26, SPPOL_1: 16;

    

     Lm46: ( LSeg (pd,pg)) is vertical by Lm28, Lm30, SPPOL_1: 16;

    

     Lm47: ( LSeg (a,lg)) is vertical by Lm16, Lm24, SPPOL_1: 16;

    

     Lm48: ( LSeg (a,ld)) is vertical by Lm16, Lm26, SPPOL_1: 16;

    

     Lm49: ( LSeg (b,pg)) is vertical by Lm17, Lm28, SPPOL_1: 16;

    

     Lm50: ( LSeg (b,pd)) is vertical by Lm17, Lm30, SPPOL_1: 16;

    

     Lm51: ( LSeg (ld,d)) is horizontal by Lm23, Lm27, SPPOL_1: 15;

    

     Lm52: ( LSeg (pd,d)) is horizontal by Lm23, Lm31, SPPOL_1: 15;

    

     Lm53: ( LSeg (lg,c)) is horizontal by Lm21, Lm25, SPPOL_1: 15;

    

     Lm54: ( LSeg (pg,c)) is horizontal by Lm21, Lm29, SPPOL_1: 15;

    

     Lm55: ( LSeg (lg,pg)) is horizontal by Lm25, Lm29, SPPOL_1: 15;

    

     Lm56: ( LSeg (ld,pd)) is horizontal by Lm27, Lm31, SPPOL_1: 15;

    

     Lm57: ( LSeg (a,lg)) c= ( LSeg (ld,lg)) by Lm16, Lm18, Lm25, Lm26, Lm27, Lm45, Lm47, GOBOARD7: 63;

    

     Lm58: ( LSeg (a,ld)) c= ( LSeg (ld,lg)) by Lm18, Lm25, Lm26, Lm27, Lm45, Lm48, GOBOARD7: 63;

    

     Lm59: ( LSeg (b,pg)) c= ( LSeg (pd,pg)) by Lm17, Lm19, Lm29, Lm30, Lm31, Lm46, Lm49, GOBOARD7: 63;

    

     Lm60: ( LSeg (b,pd)) c= ( LSeg (pd,pg)) by Lm19, Lm29, Lm30, Lm31, Lm46, Lm50, GOBOARD7: 63;

    

     Lm61: dR = { p where p be Point of T2 : (p `1 ) = rl & (p `2 ) <= rg & (p `2 ) >= rd or (p `1 ) <= rp & (p `1 ) >= rl & (p `2 ) = rg or (p `1 ) <= rp & (p `1 ) >= rl & (p `2 ) = rd or (p `1 ) = rp & (p `2 ) <= rg & (p `2 ) >= rd } by SPPOL_2: 54;

    then

     Lm62: c in dR by Lm20, Lm21;

    

     Lm63: d in dR by Lm22, Lm23, Lm61;

    

     Lm64: ((2 + 1) ^2 ) = ((4 + 4) + 1);

    then

     Lm65: ( sqrt 9) = 3 by SQUARE_1:def 2;

    

     Lm66: ( dist (a,b)) = ( sqrt ((((a `1 ) - (b `1 )) ^2 ) + (((a `2 ) - (b `2 )) ^2 ))) by TOPREAL6: 92

    .= ( - ( - 2)) by Lm16, Lm17, Lm18, Lm19, SQUARE_1: 23;

    theorem :: JORDAN:70

    

     Th70: for h be Homeomorphism of ( TOP-REAL 2) holds (h .: C) is being_simple_closed_curve

    proof

      let h be Homeomorphism of T2;

      consider f be Function of (T2 | R^2-unit_square ), (T2 | C) such that

       A1: f is being_homeomorphism by TOPREAL2:def 1;

      reconsider g = (h | C) as Function of (T2 | C), (T2 | (h .: C)) by JORDAN24: 12;

      take (g * f);

      g is being_homeomorphism by JORDAN24: 14;

      hence thesis by A1, TOPS_2: 57;

    end;

    theorem :: JORDAN:71

    

     Th71: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies P c= ( closed_inside_of_rectangle (( - 1),1,( - 3),3))

    proof

      assume that

       A1: a in P and

       A2: b in P and

       A3: for x,y be Point of ( TOP-REAL 2) st x in P & y in P holds ( dist (a,b)) >= ( dist (x,y));

      let p be object;

      assume

       A4: p in P;

      then

      reconsider p as Point of ( TOP-REAL 2);

      

       A5: ( dist (a,p)) = ( sqrt (((rl - (p `1 )) ^2 ) + (( 0 - (p `2 )) ^2 ))) by Lm16, Lm18, TOPREAL6: 92

      .= ( sqrt (((rl - (p `1 )) ^2 ) + ((p `2 ) ^2 )));

       A6:

      now

        assume 9 < ((p `2 ) ^2 );

        then ( 0 + 9) < (((rl - (p `1 )) ^2 ) + ((p `2 ) ^2 )) by XREAL_1: 8;

        then 3 < ( sqrt (((rl - (p `1 )) ^2 ) + ((p `2 ) ^2 ))) by Lm65, SQUARE_1: 27;

        then 2 < ( sqrt (((rl - (p `1 )) ^2 ) + ((p `2 ) ^2 ))) by XXREAL_0: 2;

        hence contradiction by A1, A3, A4, A5, Lm66;

      end;

       A7:

      now

        assume

         A8: rl > (p `1 );

        then ( LSeg (p,b)) meets ( Vertical_Line rl) by Lm17, Th8;

        then

        consider x be object such that

         A9: x in ( LSeg (p,b)) and

         A10: x in ( Vertical_Line rl) by XBOOLE_0: 3;

        reconsider x as Point of T2 by A9;

        

         A11: (x `1 ) = rl by A10, JORDAN6: 31;

        

         A12: ( dist (p,b)) = (( dist (p,x)) + ( dist (x,b))) by A9, JORDAN1K: 29;

        

         A13: ( dist (x,b)) = ( sqrt ((((x `1 ) - (b `1 )) ^2 ) + (((x `2 ) - (b `2 )) ^2 ))) by TOPREAL6: 92

        .= ( sqrt ((( - 2) ^2 ) + (((x `2 ) - 0 ) ^2 ))) by A11, Lm17, EUCLID: 52

        .= ( sqrt (4 + ((x `2 ) ^2 )));

        now

          assume ( dist (x,b)) < ( dist (a,b));

          then (4 + ((x `2 ) ^2 )) < (4 + 0 ) by A13, Lm66, SQUARE_1: 20, SQUARE_1: 26;

          hence contradiction by XREAL_1: 6;

        end;

        then (( dist (p,b)) + 0 ) > (( dist (a,b)) + 0 ) by A8, A11, A12, JORDAN1K: 22, XREAL_1: 8;

        hence contradiction by A2, A3, A4;

      end;

       A14:

      now

        assume

         A15: (p `1 ) > rp;

        then ( LSeg (p,a)) meets ( Vertical_Line rp) by Lm16, Th8;

        then

        consider x be object such that

         A16: x in ( LSeg (p,a)) and

         A17: x in ( Vertical_Line rp) by XBOOLE_0: 3;

        reconsider x as Point of T2 by A16;

        

         A18: (x `1 ) = rp by A17, JORDAN6: 31;

        

         A19: ( dist (p,a)) = (( dist (p,x)) + ( dist (x,a))) by A16, JORDAN1K: 29;

        

         A20: ( dist (x,a)) = ( sqrt ((((x `1 ) - (a `1 )) ^2 ) + (((x `2 ) - (a `2 )) ^2 ))) by TOPREAL6: 92

        .= ( sqrt (4 + ((x `2 ) ^2 ))) by A18, Lm16, Lm18;

        now

          assume ( dist (x,a)) < ( dist (a,b));

          then (4 + ((x `2 ) ^2 )) < (4 + 0 ) by A20, Lm66, SQUARE_1: 20, SQUARE_1: 26;

          hence contradiction by XREAL_1: 6;

        end;

        then (( dist (p,a)) + 0 ) > (( dist (a,b)) + 0 ) by A15, A18, A19, JORDAN1K: 22, XREAL_1: 8;

        hence contradiction by A1, A3, A4;

      end;

       A21:

      now

        assume rd > (p `2 );

        then ((p `2 ) ^2 ) > (rd ^2 ) by SQUARE_1: 44;

        hence contradiction by A6;

      end;

      rg >= (p `2 ) by A6, Lm64, SQUARE_1: 16;

      hence thesis by A7, A14, A21;

    end;

    

     Lm67: dR c= R by Th45;

    

     Lm68: (lg `2 ) = (lg `2 );

    

     Lm69: (lg `1 ) <= (c `1 ) by Lm24, EUCLID: 52;

    (c `1 ) <= (pg `1 ) by Lm28, EUCLID: 52;

    then ( LSeg (lg,c)) c= ( LSeg (lg,pg)) by Lm53, Lm55, Lm68, Lm69, GOBOARD7: 64;

    then

     Lm70: ( LSeg (lg,c)) c= dR by Lm40;

    ( LSeg (pg,c)) c= ( LSeg (lg,pg)) by Lm20, Lm21, Lm24, Lm25, Lm28, Lm54, Lm55, GOBOARD7: 64;

    then

     Lm71: ( LSeg (pg,c)) c= dR by Lm40;

    

     Lm72: (ld `2 ) = (ld `2 );

    

     Lm73: (ld `1 ) <= (d `1 ) by Lm26, EUCLID: 52;

    (d `1 ) <= (pd `1 ) by Lm30, EUCLID: 52;

    then ( LSeg (ld,d)) c= ( LSeg (ld,pd)) by Lm51, Lm56, Lm72, Lm73, GOBOARD7: 64;

    then

     Lm74: ( LSeg (ld,d)) c= dR by Lm44;

    ( LSeg (pd,d)) c= ( LSeg (ld,pd)) by Lm22, Lm23, Lm26, Lm27, Lm30, Lm52, Lm56, GOBOARD7: 64;

    then

     Lm75: ( LSeg (pd,d)) c= dR by Lm44;

    

     Lm76: 0 <= (p `2 ) & p in dR implies p in ( LSeg (a,lg)) or p in ( LSeg (lg,c)) or p in ( LSeg (c,pg)) or p in ( LSeg (pg,b))

    proof

      assume

       A1: 0 <= (p `2 );

      assume p in dR;

      then

      consider p1 such that

       A2: p1 = p and

       A3: (p1 `1 ) = rl & (p1 `2 ) <= rg & (p1 `2 ) >= rd or (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rg or (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rd or (p1 `1 ) = rp & (p1 `2 ) <= rg & (p1 `2 ) >= rd by Lm61;

      per cases by A3;

        suppose (p1 `1 ) = rl & (p1 `2 ) <= rg & (p1 `2 ) >= rd;

        hence thesis by A1, A2, Lm16, Lm18, Lm24, Lm25, GOBOARD7: 7;

      end;

        suppose

         A4: (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rg;

        per cases ;

          suppose (p1 `1 ) <= (c `1 );

          hence thesis by A2, A4, Lm21, Lm24, Lm25, GOBOARD7: 8;

        end;

          suppose (c `1 ) <= (p1 `1 );

          hence thesis by A2, A4, Lm21, Lm28, Lm29, GOBOARD7: 8;

        end;

      end;

        suppose (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rd;

        hence thesis by A1, A2;

      end;

        suppose (p1 `1 ) = rp & (p1 `2 ) <= rg & (p1 `2 ) >= rd;

        hence thesis by A1, A2, Lm17, Lm19, Lm28, Lm29, GOBOARD7: 7;

      end;

    end;

    

     Lm77: (p `2 ) <= 0 & p in dR implies p in ( LSeg (a,ld)) or p in ( LSeg (ld,d)) or p in ( LSeg (d,pd)) or p in ( LSeg (pd,b))

    proof

      assume

       A1: (p `2 ) <= 0 ;

      assume p in dR;

      then

      consider p1 such that

       A2: p1 = p and

       A3: (p1 `1 ) = rl & (p1 `2 ) <= rg & (p1 `2 ) >= rd or (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rg or (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rd or (p1 `1 ) = rp & (p1 `2 ) <= rg & (p1 `2 ) >= rd by Lm61;

      per cases by A3;

        suppose (p1 `1 ) = rl & (p1 `2 ) <= rg & (p1 `2 ) >= rd;

        hence thesis by A1, A2, Lm16, Lm18, Lm26, Lm27, GOBOARD7: 7;

      end;

        suppose (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rg;

        hence thesis by A1, A2;

      end;

        suppose

         A4: (p1 `1 ) <= rp & (p1 `1 ) >= rl & (p1 `2 ) = rd;

        per cases ;

          suppose (p1 `1 ) <= (d `1 );

          hence thesis by A2, A4, Lm23, Lm26, Lm27, GOBOARD7: 8;

        end;

          suppose (d `1 ) <= (p1 `1 );

          hence thesis by A2, A4, Lm23, Lm30, Lm31, GOBOARD7: 8;

        end;

      end;

        suppose (p1 `1 ) = rp & (p1 `2 ) <= rg & (p1 `2 ) >= rd;

        hence thesis by A1, A2, Lm17, Lm19, Lm30, Lm31, GOBOARD7: 7;

      end;

    end;

    theorem :: JORDAN:72

    

     Th72: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies P misses ( LSeg ( |[( - 1), 3]|, |[1, 3]|))

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      assume P meets ( LSeg (lg,pg));

      then

      consider x be object such that

       A2: x in P and

       A3: x in ( LSeg (lg,pg)) by XBOOLE_0: 3;

      reconsider x as Point of T2 by A2;

      lg in ( LSeg (lg,pg)) by RLTOPSP1: 68;

      then

       A4: (x `2 ) = rg by A3, Lm25, Lm55;

      

       A5: ( dist (a,x)) = ( sqrt ((((a `1 ) - (x `1 )) ^2 ) + (((a `2 ) - (x `2 )) ^2 ))) by TOPREAL6: 92

      .= ( sqrt (((rl - (x `1 )) ^2 ) + (rg ^2 ))) by A4, Lm18, EUCLID: 52;

      ( 0 + 4) < (((rl - (x `1 )) ^2 ) + 9) by XREAL_1: 8;

      then 2 < ( dist (a,x)) by A5, SQUARE_1: 20, SQUARE_1: 27;

      hence thesis by A1, A2, Lm66;

    end;

    theorem :: JORDAN:73

    

     Th73: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies P misses ( LSeg ( |[( - 1), ( - 3)]|, |[1, ( - 3)]|))

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      assume P meets ( LSeg (ld,pd));

      then

      consider x be object such that

       A2: x in P and

       A3: x in ( LSeg (ld,pd)) by XBOOLE_0: 3;

      reconsider x as Point of T2 by A2;

      ld in ( LSeg (ld,pd)) by RLTOPSP1: 68;

      then

       A4: (x `2 ) = rd by A3, Lm27, Lm56;

      

       A5: ( dist (a,x)) = ( sqrt ((((a `1 ) - (x `1 )) ^2 ) + (((a `2 ) - (x `2 )) ^2 ))) by TOPREAL6: 92

      .= ( sqrt (((rl - (x `1 )) ^2 ) + (( - rd) ^2 ))) by A4, Lm18, EUCLID: 52;

      ( 0 + 4) < (((rl - (x `1 )) ^2 ) + 9) by XREAL_1: 8;

      then 2 < ( dist (a,x)) by A5, SQUARE_1: 20, SQUARE_1: 27;

      hence thesis by A1, A2, Lm66;

    end;

    theorem :: JORDAN:74

    

     Th74: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies (P /\ ( rectangle (( - 1),1,( - 3),3))) = { |[( - 1), 0 ]|, |[1, 0 ]|}

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: a in P;

      

       A3: b in P by A1;

      thus (P /\ dR) c= {a, b}

      proof

        let x be object;

        assume

         A4: x in (P /\ dR);

        then

         A5: x in P by XBOOLE_0:def 4;

        x in dR by A4, XBOOLE_0:def 4;

        then

         A6: x in (( LSeg (ld,lg)) \/ ( LSeg (lg,pg))) or x in (( LSeg (pg,pd)) \/ ( LSeg (pd,ld))) by Lm36, XBOOLE_0:def 3;

        reconsider x as Point of T2 by A4;

        per cases by A6, XBOOLE_0:def 3;

          suppose

           A7: x in ( LSeg (ld,lg));

          ld in ( LSeg (ld,lg)) by RLTOPSP1: 68;

          then

           A8: (x `1 ) = rl by A7, Lm26, Lm45;

          per cases ;

            suppose (x `2 ) = 0 ;

            then x = a by A8, Lm16, Lm18, TOPREAL3: 6;

            hence thesis by TARSKI:def 2;

          end;

            suppose (x `2 ) <> 0 ;

            then

             A9: ((x `2 ) ^2 ) > 0 by SQUARE_1: 12;

            

             A10: ( dist (b,x)) = ( sqrt (((rp - rl) ^2 ) + (( 0 - (x `2 )) ^2 ))) by A8, Lm17, Lm19, TOPREAL6: 92

            .= ( sqrt (4 + ((x `2 ) ^2 )));

            ( 0 + 4) < (((x `2 ) ^2 ) + 4) by A9, XREAL_1: 6;

            then 2 < ( sqrt (((x `2 ) ^2 ) + 4)) by SQUARE_1: 20, SQUARE_1: 27;

            hence thesis by A1, A5, A10, Lm66;

          end;

        end;

          suppose x in ( LSeg (lg,pg));

          then ( LSeg (lg,pg)) meets P by A5, XBOOLE_0: 3;

          hence thesis by A1, Th72;

        end;

          suppose

           A11: x in ( LSeg (pg,pd));

          pd in ( LSeg (pd,pg)) by RLTOPSP1: 68;

          then

           A12: (x `1 ) = rp by A11, Lm30, Lm46;

          per cases ;

            suppose (x `2 ) = 0 ;

            then x = b by A12, Lm17, Lm19, TOPREAL3: 6;

            hence thesis by TARSKI:def 2;

          end;

            suppose (x `2 ) <> 0 ;

            then

             A13: ((x `2 ) ^2 ) > 0 by SQUARE_1: 12;

            

             A14: ( dist (x,a)) = ( sqrt ((((x `1 ) - (a `1 )) ^2 ) + (((x `2 ) - (a `2 )) ^2 ))) by TOPREAL6: 92

            .= ( sqrt (4 + ((x `2 ) ^2 ))) by A12, Lm16, Lm18;

            ( 0 + 4) < (((x `2 ) ^2 ) + 4) by A13, XREAL_1: 6;

            then 2 < ( sqrt (((x `2 ) ^2 ) + 4)) by SQUARE_1: 20, SQUARE_1: 27;

            hence thesis by A1, A5, A14, Lm66;

          end;

        end;

          suppose x in ( LSeg (pd,ld));

          then ( LSeg (pd,ld)) meets P by A5, XBOOLE_0: 3;

          hence thesis by A1, Th73;

        end;

      end;

      let x be object;

      assume x in {a, b};

      then

       A15: x = a or x = b by TARSKI:def 2;

      

       A16: a in dR by Lm16, Lm18, Lm61;

      b in dR by Lm17, Lm19, Lm61;

      hence thesis by A2, A3, A15, A16, XBOOLE_0:def 4;

    end;

    

     Lm78: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ( LSeg (lg,c)) misses C

    proof

      assume (a,b) realize-max-dist-in C;

      then

       A1: (C /\ dR) = {a, b} by Th74;

      assume ( LSeg (lg,c)) meets C;

      then

      consider q be object such that

       A2: q in ( LSeg (lg,c)) and

       A3: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A3;

      q in (dR /\ C) by A2, A3, Lm70, XBOOLE_0:def 4;

      then q = a or q = b by A1, TARSKI:def 2;

      hence contradiction by A2, Lm18, Lm19, TOPREAL3: 12;

    end;

    

     Lm79: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ( LSeg (pg,c)) misses C

    proof

      assume (a,b) realize-max-dist-in C;

      then

       A1: (C /\ dR) = {a, b} by Th74;

      assume ( LSeg (pg,c)) meets C;

      then

      consider q be object such that

       A2: q in ( LSeg (pg,c)) and

       A3: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A3;

      q in (dR /\ C) by A2, A3, Lm71, XBOOLE_0:def 4;

      then q = a or q = b by A1, TARSKI:def 2;

      hence contradiction by A2, Lm18, Lm19, TOPREAL3: 12;

    end;

    

     Lm80: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ( LSeg (ld,d)) misses C

    proof

      assume (a,b) realize-max-dist-in C;

      then

       A1: (C /\ dR) = {a, b} by Th74;

      assume ( LSeg (ld,d)) meets C;

      then

      consider q be object such that

       A2: q in ( LSeg (ld,d)) and

       A3: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A3;

      q in (dR /\ C) by A2, A3, Lm74, XBOOLE_0:def 4;

      then q = a or q = b by A1, TARSKI:def 2;

      hence contradiction by A2, Lm18, Lm19, TOPREAL3: 12;

    end;

    

     Lm81: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ( LSeg (pd,d)) misses C

    proof

      assume (a,b) realize-max-dist-in C;

      then

       A1: (C /\ dR) = {a, b} by Th74;

      assume ( LSeg (pd,d)) meets C;

      then

      consider q be object such that

       A2: q in ( LSeg (pd,d)) and

       A3: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A3;

      q in (dR /\ C) by A2, A3, Lm75, XBOOLE_0:def 4;

      then q = a or q = b by A1, TARSKI:def 2;

      hence contradiction by A2, Lm18, Lm19, TOPREAL3: 12;

    end;

    

     Lm82: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & p in (C ` ) & p in ( LSeg (a,lg)) implies ( LSeg (p,lg)) misses C

    proof

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: p in (C ` ) and

       A3: p in ( LSeg (a,lg));

      

       A4: (C /\ dR) = {a, b} by A1, Th74;

      assume ( LSeg (p,lg)) meets C;

      then

      consider q be object such that

       A5: q in ( LSeg (p,lg)) and

       A6: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A6;

      lg in ( LSeg (a,lg)) by RLTOPSP1: 68;

      then

       A7: (p `1 ) = (lg `1 ) by A3, Lm47;

      

       A8: (p `2 ) <= (lg `2 ) by A3, Lm25, JGRAPH_6: 1;

      

       A9: ( LSeg (p,lg)) is vertical by A7, SPPOL_1: 16;

      (a `2 ) <= (p `2 ) by A3, Lm18, JGRAPH_6: 1;

      then ( LSeg (p,lg)) c= ( LSeg (ld,lg)) by A7, A8, A9, Lm18, Lm24, Lm26, Lm27, Lm45, GOBOARD7: 63;

      then ( LSeg (p,lg)) c= dR by Lm38;

      then q in (dR /\ C) by A5, A6, XBOOLE_0:def 4;

      then

       A10: q = a or q = b by A4, TARSKI:def 2;

      a in ( LSeg (a,lg)) by RLTOPSP1: 68;

      then

       A11: (a `1 ) = (p `1 ) by A3, Lm47;

      

       A12: a in C by A1;

       not p in C by A2, XBOOLE_0:def 5;

      then (a `2 ) <> (p `2 ) by A11, A12, TOPREAL3: 6;

      then

       A13: (a `2 ) < (p `2 ) by A3, Lm18, JGRAPH_6: 1;

      p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      hence contradiction by A5, A7, A8, A10, A13, Lm17, Lm24, Lm33, JGRAPH_6: 1;

    end;

    

     Lm83: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & p in (C ` ) & p in ( LSeg (b,pg)) implies ( LSeg (p,pg)) misses C

    proof

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: p in (C ` ) and

       A3: p in ( LSeg (b,pg));

      

       A4: (C /\ dR) = {a, b} by A1, Th74;

      assume ( LSeg (p,pg)) meets C;

      then

      consider q be object such that

       A5: q in ( LSeg (p,pg)) and

       A6: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A6;

      pg in ( LSeg (b,pg)) by RLTOPSP1: 68;

      then

       A7: (p `1 ) = (pg `1 ) by A3, Lm49;

      

       A8: (p `2 ) <= (pg `2 ) by A3, Lm29, JGRAPH_6: 1;

      

       A9: ( LSeg (p,pg)) is vertical by A7, SPPOL_1: 16;

      (b `2 ) <= (p `2 ) by A3, Lm19, JGRAPH_6: 1;

      then ( LSeg (p,pg)) c= ( LSeg (pd,pg)) by A7, A8, A9, Lm19, Lm28, Lm30, Lm31, Lm46, GOBOARD7: 63;

      then ( LSeg (p,pg)) c= dR by Lm42;

      then q in (dR /\ C) by A5, A6, XBOOLE_0:def 4;

      then

       A10: q = a or q = b by A4, TARSKI:def 2;

      b in ( LSeg (b,pg)) by RLTOPSP1: 68;

      then

       A11: (b `1 ) = (p `1 ) by A3, Lm49;

      

       A12: b in C by A1;

       not p in C by A2, XBOOLE_0:def 5;

      then (b `2 ) <> (p `2 ) by A11, A12, TOPREAL3: 6;

      then

       A13: (b `2 ) < (p `2 ) by A3, Lm19, JGRAPH_6: 1;

      p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      hence contradiction by A5, A7, A8, A10, A13, Lm16, Lm28, Lm35, JGRAPH_6: 1;

    end;

    

     Lm84: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & p in (C ` ) & p in ( LSeg (a,ld)) implies ( LSeg (p,ld)) misses C

    proof

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: p in (C ` ) and

       A3: p in ( LSeg (a,ld));

      

       A4: (C /\ dR) = {a, b} by A1, Th74;

      assume ( LSeg (p,ld)) meets C;

      then

      consider q be object such that

       A5: q in ( LSeg (p,ld)) and

       A6: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A6;

      ld in ( LSeg (a,ld)) by RLTOPSP1: 68;

      then

       A7: (p `1 ) = (ld `1 ) by A3, Lm48;

      

       A8: (ld `2 ) <= (p `2 ) by A3, Lm27, JGRAPH_6: 1;

      

       A9: ( LSeg (p,ld)) is vertical by A7, SPPOL_1: 16;

      (p `2 ) <= (a `2 ) by A3, Lm18, JGRAPH_6: 1;

      then ( LSeg (p,ld)) c= ( LSeg (ld,lg)) by A7, A8, A9, Lm18, Lm25, Lm45, GOBOARD7: 63;

      then ( LSeg (p,ld)) c= dR by Lm38;

      then q in (dR /\ C) by A5, A6, XBOOLE_0:def 4;

      then

       A10: q = a or q = b by A4, TARSKI:def 2;

      a in ( LSeg (a,ld)) by RLTOPSP1: 68;

      then

       A11: (a `1 ) = (p `1 ) by A3, Lm48;

      

       A12: a in C by A1;

       not p in C by A2, XBOOLE_0:def 5;

      then (a `2 ) <> (p `2 ) by A11, A12, TOPREAL3: 6;

      then

       A13: (p `2 ) < (a `2 ) by A3, Lm18, JGRAPH_6: 1;

      p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      hence contradiction by A5, A7, A8, A10, A13, Lm17, Lm26, Lm32, JGRAPH_6: 1;

    end;

    

     Lm85: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & p in (C ` ) & p in ( LSeg (b,pd)) implies ( LSeg (p,pd)) misses C

    proof

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: p in (C ` ) and

       A3: p in ( LSeg (b,pd));

      

       A4: (C /\ dR) = {a, b} by A1, Th74;

      assume ( LSeg (p,pd)) meets C;

      then

      consider q be object such that

       A5: q in ( LSeg (p,pd)) and

       A6: q in C by XBOOLE_0: 3;

      reconsider q as Point of T2 by A6;

      pd in ( LSeg (b,pd)) by RLTOPSP1: 68;

      then

       A7: (p `1 ) = (pd `1 ) by A3, Lm50;

      

       A8: (pd `2 ) <= (p `2 ) by A3, Lm31, JGRAPH_6: 1;

      

       A9: ( LSeg (p,pd)) is vertical by A7, SPPOL_1: 16;

      (p `2 ) <= (b `2 ) by A3, Lm19, JGRAPH_6: 1;

      then ( LSeg (p,pd)) c= ( LSeg (pd,pg)) by A7, A8, A9, Lm19, Lm29, Lm46, GOBOARD7: 63;

      then ( LSeg (p,pd)) c= dR by Lm42;

      then q in (dR /\ C) by A5, A6, XBOOLE_0:def 4;

      then

       A10: q = a or q = b by A4, TARSKI:def 2;

      b in ( LSeg (b,pd)) by RLTOPSP1: 68;

      then

       A11: (b `1 ) = (p `1 ) by A3, Lm50;

      

       A12: b in C by A1;

       not p in C by A2, XBOOLE_0:def 5;

      then (b `2 ) <> (p `2 ) by A11, A12, TOPREAL3: 6;

      then

       A13: (p `2 ) < (b `2 ) by A3, Lm19, JGRAPH_6: 1;

      p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      hence contradiction by A5, A7, A8, A10, A13, Lm16, Lm30, Lm34, JGRAPH_6: 1;

    end;

    

     Lm86: |[ 0 , r]| in ( rectangle (rl,rp,rd,rg)) implies r = rd or r = rg

    proof

      assume |[ 0 , r]| in dR;

      then ex p st p = |[ 0 , r]| & ((p `1 ) = rl & (p `2 ) <= rg & (p `2 ) >= rd or (p `1 ) <= rp & (p `1 ) >= rl & (p `2 ) = rg or (p `1 ) <= rp & (p `1 ) >= rl & (p `2 ) = rd or (p `1 ) = rp & (p `2 ) <= rg & (p `2 ) >= rd) by Lm61;

      hence thesis by EUCLID: 52;

    end;

    theorem :: JORDAN:75

    

     Th75: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( W-bound P) = ( - 1)

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: P c= R by Th71;

      

       A3: P = the carrier of (T2 | P) by PRE_TOPC: 8;

      

       A4: a in P by A1;

      reconsider P as non empty Subset of T2 by A1;

      reconsider Z = (( proj1 | P) .: the carrier of (T2 | P)) as Subset of REAL ;

      

       A5: for p be Real st p in Z holds p >= rl

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A6: p0 in the carrier of (T2 | P) and p0 in the carrier of (T2 | P) and

         A7: p = (( proj1 | P) . p0) by FUNCT_2: 64;

        p0 in R by A2, A3, A6;

        then ex p1 st p0 = p1 & rl <= (p1 `1 ) & (p1 `1 ) <= rp & rd <= (p1 `2 ) & (p1 `2 ) <= rg;

        hence thesis by A3, A6, A7, PSCOMP_1: 22;

      end;

      for q be Real st for p be Real st p in Z holds p >= q holds rl >= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p >= q;

        (( proj1 | P) . a) = (a `1 ) by A4, PSCOMP_1: 22;

        hence thesis by A3, A4, A8, Lm16, FUNCT_2: 35;

      end;

      hence thesis by A5, SEQ_4: 44;

    end;

    theorem :: JORDAN:76

    

     Th76: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( E-bound P) = 1

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: P c= R by Th71;

      

       A3: b in P by A1;

      reconsider P as non empty Subset of T2 by A1;

      reconsider Z = (( proj1 | P) .: the carrier of (T2 | P)) as Subset of REAL ;

      

       A4: P = the carrier of (T2 | P) by PRE_TOPC: 8;

      

       A5: for p be Real st p in Z holds p <= rp

      proof

        let p be Real;

        assume p in Z;

        then

        consider p0 be object such that

         A6: p0 in the carrier of (T2 | P) and p0 in the carrier of (T2 | P) and

         A7: p = (( proj1 | P) . p0) by FUNCT_2: 64;

        p0 in R by A2, A4, A6;

        then ex p1 st p0 = p1 & rl <= (p1 `1 ) & (p1 `1 ) <= rp & rd <= (p1 `2 ) & (p1 `2 ) <= rg;

        hence thesis by A4, A6, A7, PSCOMP_1: 22;

      end;

      for q be Real st for p be Real st p in Z holds p <= q holds rp <= q

      proof

        let q be Real such that

         A8: for p be Real st p in Z holds p <= q;

        (( proj1 | P) . b) = (b `1 ) by A3, PSCOMP_1: 22;

        hence thesis by A3, A4, A8, Lm17, FUNCT_2: 35;

      end;

      hence thesis by A5, SEQ_4: 46;

    end;

    theorem :: JORDAN:77

    

     Th77: for P be compact Subset of ( TOP-REAL 2) holds ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( W-most P) = { |[( - 1), 0 ]|}

    proof

      let P be compact Subset of T2;

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: P c= R by Th71;

      set L = ( LSeg (( SW-corner P),( NW-corner P)));

      

       A3: a in P by A1;

      

       A4: (( SW-corner P) `1 ) = ( |[rl, ( S-bound P)]| `1 ) by A1, Th75

      .= rl by EUCLID: 52;

      

       A5: (( NW-corner P) `1 ) = ( |[rl, ( N-bound P)]| `1 ) by A1, Th75

      .= rl by EUCLID: 52;

      thus ( W-most P) c= {a}

      proof

        let x be object;

        assume

         A6: x in ( W-most P);

        then

         A7: x in P by XBOOLE_0:def 4;

        reconsider x as Point of T2 by A6;

        

         A8: x in L by A6, XBOOLE_0:def 4;

        ( SW-corner P) in L by RLTOPSP1: 68;

        then

         A9: (x `1 ) = rl by A4, A8, SPPOL_1:def 3;

        x in R by A2, A7;

        then ex p st x = p & rl <= (p `1 ) & (p `1 ) <= rp & rd <= (p `2 ) & (p `2 ) <= rg;

        then x in dR by A9, Lm61;

        then x in (P /\ dR) by A7, XBOOLE_0:def 4;

        then x in {a, b} by A1, Th74;

        then x = a or x = b by TARSKI:def 2;

        hence thesis by A9, EUCLID: 52, TARSKI:def 1;

      end;

      let x be object;

      assume x in {a};

      then

       A10: x = a by TARSKI:def 1;

      

       A11: (( SW-corner P) `2 ) = ( S-bound P) by EUCLID: 52;

      

       A12: (( NW-corner P) `2 ) = ( N-bound P) by EUCLID: 52;

      

       A13: (( SW-corner P) `2 ) <= (a `2 ) by A3, A11, PSCOMP_1: 24;

      (a `2 ) <= (( NW-corner P) `2 ) by A3, A12, PSCOMP_1: 24;

      then a in L by A4, A5, A13, Lm16, GOBOARD7: 7;

      hence thesis by A3, A10, XBOOLE_0:def 4;

    end;

    theorem :: JORDAN:78

    

     Th78: for P be compact Subset of ( TOP-REAL 2) holds ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( E-most P) = { |[1, 0 ]|}

    proof

      let P be compact Subset of T2;

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: P c= R by Th71;

      set L = ( LSeg (( SE-corner P),( NE-corner P)));

      

       A3: b in P by A1;

      

       A4: (( SE-corner P) `1 ) = ( |[rp, ( S-bound P)]| `1 ) by A1, Th76

      .= rp by EUCLID: 52;

      

       A5: (( NE-corner P) `1 ) = ( |[rp, ( N-bound P)]| `1 ) by A1, Th76

      .= rp by EUCLID: 52;

      thus ( E-most P) c= {b}

      proof

        let x be object;

        assume

         A6: x in ( E-most P);

        then

         A7: x in P by XBOOLE_0:def 4;

        reconsider x as Point of T2 by A6;

        

         A8: x in L by A6, XBOOLE_0:def 4;

        ( SE-corner P) in L by RLTOPSP1: 68;

        then

         A9: (x `1 ) = rp by A4, A8, SPPOL_1:def 3;

        x in R by A2, A7;

        then ex p st x = p & rl <= (p `1 ) & (p `1 ) <= rp & rd <= (p `2 ) & (p `2 ) <= rg;

        then x in dR by A9, Lm61;

        then x in (P /\ dR) by A7, XBOOLE_0:def 4;

        then x in {a, b} by A1, Th74;

        then x = a or x = b by TARSKI:def 2;

        hence thesis by A9, EUCLID: 52, TARSKI:def 1;

      end;

      let x be object;

      assume x in {b};

      then

       A10: x = b by TARSKI:def 1;

      

       A11: (( SE-corner P) `2 ) = ( S-bound P) by EUCLID: 52;

      

       A12: (( NE-corner P) `2 ) = ( N-bound P) by EUCLID: 52;

      

       A13: (( SE-corner P) `2 ) <= (b `2 ) by A3, A11, PSCOMP_1: 24;

      (b `2 ) <= (( NE-corner P) `2 ) by A3, A12, PSCOMP_1: 24;

      then b in L by A4, A5, A13, Lm17, GOBOARD7: 7;

      hence thesis by A3, A10, XBOOLE_0:def 4;

    end;

    theorem :: JORDAN:79

    

     Th79: for P be compact Subset of ( TOP-REAL 2) holds ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( W-min P) = |[( - 1), 0 ]| & ( W-max P) = |[( - 1), 0 ]|

    proof

      let P be compact Subset of T2;

      set M = ( W-most P);

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: M = {a} by Th77;

      set f = ( proj2 | M);

      

       A3: ( dom f) = the carrier of (T2 | M) by FUNCT_2:def 1;

      

       A4: the carrier of (T2 | M) = M by PRE_TOPC: 8;

      

       A5: a in {a} by TARSKI:def 1;

      

       A6: (f .: the carrier of (T2 | M)) = ( Im (f,a)) by A1, A4, Th77

      .= {(f . a)} by A2, A3, A4, A5, FUNCT_1: 59

      .= {( proj2 . a)} by A2, A5, FUNCT_1: 49

      .= {(a `2 )} by PSCOMP_1:def 6;

      then

       A7: ( lower_bound ( proj2 | M)) = (a `2 ) by SEQ_4: 9;

      

       A8: ( upper_bound ( proj2 | M)) = (a `2 ) by A6, SEQ_4: 9;

      a = |[(a `1 ), (a `2 )]| by EUCLID: 53;

      hence thesis by A1, A7, A8, Lm16, Th75;

    end;

    theorem :: JORDAN:80

    

     Th80: for P be compact Subset of ( TOP-REAL 2) holds ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( E-min P) = |[1, 0 ]| & ( E-max P) = |[1, 0 ]|

    proof

      let P be compact Subset of T2;

      set M = ( E-most P);

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: M = {b} by Th78;

      set f = ( proj2 | M);

      

       A3: ( dom f) = the carrier of (T2 | M) by FUNCT_2:def 1;

      

       A4: the carrier of (T2 | M) = M by PRE_TOPC: 8;

      

       A5: b in {b} by TARSKI:def 1;

      

       A6: (f .: the carrier of (T2 | M)) = ( Im (f,b)) by A1, A4, Th78

      .= {(f . b)} by A2, A3, A4, A5, FUNCT_1: 59

      .= {( proj2 . b)} by A2, A5, FUNCT_1: 49

      .= {(b `2 )} by PSCOMP_1:def 6;

      then

       A7: ( lower_bound ( proj2 | M)) = (b `2 ) by SEQ_4: 9;

      

       A8: ( upper_bound ( proj2 | M)) = (b `2 ) by A6, SEQ_4: 9;

      b = |[(b `1 ), (b `2 )]| by EUCLID: 53;

      hence thesis by A1, A7, A8, Lm17, Th76;

    end;

    

     Lm87: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies (c `1 ) = ((( W-bound P) + ( E-bound P)) / 2)

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: ( W-bound P) = rl by Th75;

      ( E-bound P) = rp by A1, Th76;

      hence thesis by A2, EUCLID: 52;

    end;

    

     Lm88: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies (d `1 ) = ((( W-bound P) + ( E-bound P)) / 2)

    proof

      assume

       A1: (a,b) realize-max-dist-in P;

      then

       A2: ( W-bound P) = rl by Th75;

      ( E-bound P) = rp by A1, Th76;

      hence thesis by A2, EUCLID: 52;

    end;

    theorem :: JORDAN:81

    

     Th81: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( LSeg ( |[ 0 , 3]|,( UMP P))) is vertical

    proof

      assume (a,b) realize-max-dist-in P;

      

      then (c `1 ) = ((( W-bound P) + ( E-bound P)) / 2) by Lm87

      .= (( UMP P) `1 ) by EUCLID: 52;

      hence thesis by SPPOL_1: 16;

    end;

    theorem :: JORDAN:82

    

     Th82: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P implies ( LSeg (( LMP P), |[ 0 , ( - 3)]|)) is vertical

    proof

      assume (a,b) realize-max-dist-in P;

      

      then (d `1 ) = ((( W-bound P) + ( E-bound P)) / 2) by Lm88

      .= (( LMP P) `1 ) by EUCLID: 52;

      hence thesis by SPPOL_1: 16;

    end;

    theorem :: JORDAN:83

    

     Th83: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P & p in P implies (p `2 ) < 3

    proof

      assume that

       A1: (a,b) realize-max-dist-in P and

       A2: p in P;

      

       A3: (P /\ dR) = {a, b} by A1, Th74;

      P c= R by A1, Th71;

      then p in R by A2;

      then

       A4: ex p1 st p1 = p & rl <= (p1 `1 ) & (p1 `1 ) <= rp & rd <= (p1 `2 ) & (p1 `2 ) <= rg;

      now

        assume

         A5: (p `2 ) = (c `2 );

        then p in ( LSeg (lg,pg)) by A4, Lm21, Lm24, Lm25, Lm28, Lm29, GOBOARD7: 8;

        then p in (P /\ dR) by A2, Lm40, XBOOLE_0:def 4;

        hence contradiction by A3, A5, Lm18, Lm19, Lm21, TARSKI:def 2;

      end;

      hence thesis by A4, Lm21, XXREAL_0: 1;

    end;

    theorem :: JORDAN:84

    

     Th84: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P & p in P implies ( - 3) < (p `2 )

    proof

      assume that

       A1: (a,b) realize-max-dist-in P and

       A2: p in P;

      

       A3: (P /\ dR) = {a, b} by A1, Th74;

      P c= R by A1, Th71;

      then p in R by A2;

      then

       A4: ex p1 st p1 = p & rl <= (p1 `1 ) & (p1 `1 ) <= rp & rd <= (p1 `2 ) & (p1 `2 ) <= rg;

      now

        assume

         A5: (p `2 ) = (d `2 );

        then p in ( LSeg (ld,pd)) by A4, Lm23, Lm26, Lm27, Lm30, Lm31, GOBOARD7: 8;

        then p in (P /\ dR) by A2, Lm44, XBOOLE_0:def 4;

        then p = a or p = b by A3, TARSKI:def 2;

        hence contradiction by A5, Lm23, EUCLID: 52;

      end;

      hence thesis by A4, Lm23, XXREAL_0: 1;

    end;

    theorem :: JORDAN:85

    

     Th85: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D & p in ( LSeg ( |[ 0 , 3]|,( UMP D))) implies (( UMP D) `2 ) <= (p `2 )

    proof

      set x = ( UMP D);

      assume that

       A1: (a,b) realize-max-dist-in D and

       A2: p in ( LSeg (c,x));

      

       A3: x in ( LSeg (c,x)) by RLTOPSP1: 68;

      

       A4: ( LSeg (c,x)) is vertical by A1, Th81;

      

       A5: c = |[(c `1 ), (c `2 )]| by EUCLID: 53;

      

       A6: x = |[(x `1 ), (x `2 )]| by EUCLID: 53;

      c in ( LSeg (c,x)) by RLTOPSP1: 68;

      then

       A7: (c `1 ) = (x `1 ) by A3, A4;

      (x `2 ) <= (c `2 ) by A1, Lm21, Th83, JORDAN21: 30;

      hence thesis by A2, A5, A6, A7, JGRAPH_6: 1;

    end;

    theorem :: JORDAN:86

    

     Th86: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D & p in ( LSeg (( LMP D), |[ 0 , ( - 3)]|)) implies (p `2 ) <= (( LMP D) `2 )

    proof

      set x = ( LMP D);

      assume that

       A1: (a,b) realize-max-dist-in D and

       A2: p in ( LSeg (x,d));

      

       A3: x in ( LSeg (x,d)) by RLTOPSP1: 68;

      

       A4: ( LSeg (x,d)) is vertical by A1, Th82;

      

       A5: d = |[(d `1 ), (d `2 )]| by EUCLID: 53;

      

       A6: x = |[(x `1 ), (x `2 )]| by EUCLID: 53;

      d in ( LSeg (x,d)) by RLTOPSP1: 68;

      then

       A7: (d `1 ) = (x `1 ) by A3, A4;

      (d `2 ) <= (x `2 ) by A1, Lm23, Th84, JORDAN21: 31;

      hence thesis by A2, A5, A6, A7, JGRAPH_6: 1;

    end;

    theorem :: JORDAN:87

    

     Th87: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D implies ( LSeg ( |[ 0 , 3]|,( UMP D))) c= ( north_halfline ( UMP D))

    proof

      set p = ( UMP D);

      assume

       A1: (a,b) realize-max-dist-in D;

      let x be object;

      assume

       A2: x in ( LSeg (c,p));

      then

      reconsider x as Point of T2;

      

       A3: p in ( LSeg (c,p)) by RLTOPSP1: 68;

      ( LSeg (c,p)) is vertical by A1, Th81;

      then

       A4: (x `1 ) = (p `1 ) by A2, A3;

      (p `2 ) <= (x `2 ) by A1, A2, Th85;

      hence thesis by A4, TOPREAL1:def 10;

    end;

    theorem :: JORDAN:88

    

     Th88: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D implies ( LSeg (( LMP D), |[ 0 , ( - 3)]|)) c= ( south_halfline ( LMP D))

    proof

      set p = ( LMP D);

      assume

       A1: (a,b) realize-max-dist-in D;

      let x be object;

      assume

       A2: x in ( LSeg (p,d));

      then

      reconsider x as Point of T2;

      

       A3: p in ( LSeg (p,d)) by RLTOPSP1: 68;

      

       A4: ( LSeg (p,d)) is vertical by A1, Th82;

      then

       A5: (x `1 ) = (p `1 ) by A2, A3;

      

       A6: d = |[(d `1 ), (d `2 )]| by EUCLID: 53;

      

       A7: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      d in ( LSeg (p,d)) by RLTOPSP1: 68;

      then

       A8: (d `1 ) = (p `1 ) by A3, A4;

      (d `2 ) <= (p `2 ) by A1, Lm23, Th84, JORDAN21: 31;

      then (x `2 ) <= (p `2 ) by A2, A6, A7, A8, JGRAPH_6: 1;

      hence thesis by A5, TOPREAL1:def 12;

    end;

    theorem :: JORDAN:89

    

     Th89: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & P is_inside_component_of C implies ( LSeg ( |[ 0 , 3]|,( UMP C))) misses P

    proof

      set m = ( UMP C);

      set L = ( LSeg (c,m));

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: P is_inside_component_of C;

      

       A3: ex VP be Subset of (T2 | (C ` )) st (VP = P) & (VP is a_component) & (VP is bounded Subset of ( Euclid 2)) by A2, JORDAN2C: 13;

      m in L by RLTOPSP1: 68;

      then {m} c= L by ZFMISC_1: 31;

      then

       A4: L = ((L \ {m}) \/ {m}) by XBOOLE_1: 45;

      

       A5: (L \ {m}) c= (( north_halfline m) \ {m}) by A1, Th87, XBOOLE_1: 33;

      (( north_halfline m) \ {m}) c= ( UBD C) by Th12;

      then (L \ {m}) c= ( UBD C) by A5;

      then

       A6: (L \ {m}) misses P by A2, Th14, XBOOLE_1: 63;

       {m} misses P by A3, Lm4, JORDAN21: 30;

      hence thesis by A4, A6, XBOOLE_1: 70;

    end;

    theorem :: JORDAN:90

    

     Th90: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C & P is_inside_component_of C implies ( LSeg (( LMP C), |[ 0 , ( - 3)]|)) misses P

    proof

      set m = ( LMP C);

      set L = ( LSeg (m,d));

      assume that

       A1: (a,b) realize-max-dist-in C and

       A2: P is_inside_component_of C;

      

       A3: ex VP be Subset of (T2 | (C ` )) st (VP = P) & (VP is a_component) & (VP is bounded Subset of ( Euclid 2)) by A2, JORDAN2C: 13;

      m in L by RLTOPSP1: 68;

      then {m} c= L by ZFMISC_1: 31;

      then

       A4: L = ((L \ {m}) \/ {m}) by XBOOLE_1: 45;

      

       A5: (L \ {m}) c= (( south_halfline m) \ {m}) by A1, Th88, XBOOLE_1: 33;

      (( south_halfline m) \ {m}) c= ( UBD C) by Th13;

      then (L \ {m}) c= ( UBD C) by A5;

      then

       A6: (L \ {m}) misses P by A2, Th14, XBOOLE_1: 63;

       {m} misses P by A3, Lm4, JORDAN21: 31;

      hence thesis by A4, A6, XBOOLE_1: 70;

    end;

    theorem :: JORDAN:91

    

     Th91: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D implies (( LSeg ( |[ 0 , 3]|,( UMP D))) /\ D) = {( UMP D)}

    proof

      assume

       A1: (a,b) realize-max-dist-in D;

      set m = ( UMP D);

      set w = ((( W-bound D) + ( E-bound D)) / 2);

      

       A2: (c `1 ) = w by A1, Lm87;

      

       A3: (m `1 ) = w by EUCLID: 52;

      

       A4: m in ( LSeg (c,m)) by RLTOPSP1: 68;

      

       A5: m in D by JORDAN21: 30;

      thus (( LSeg (c,m)) /\ D) c= {m}

      proof

        let x be object;

        assume

         A6: x in (( LSeg (c,m)) /\ D);

        then

         A7: x in ( LSeg (c,m)) by XBOOLE_0:def 4;

        

         A8: x in D by A6, XBOOLE_0:def 4;

        reconsider x as Point of T2 by A6;

        ( LSeg (c,m)) is vertical by A2, A3, SPPOL_1: 16;

        then

         A9: (x `1 ) = (m `1 ) by A4, A7;

        then x in ( Vertical_Line w) by A3, JORDAN6: 31;

        then x in (D /\ ( Vertical_Line w)) by A8, XBOOLE_0:def 4;

        then

         A10: (x `2 ) <= (m `2 ) by JORDAN21: 28;

        (m `2 ) <= (x `2 ) by A1, A7, Th85;

        then (x `2 ) = (m `2 ) by A10, XXREAL_0: 1;

        then x = m by A9, TOPREAL3: 6;

        hence thesis by TARSKI:def 1;

      end;

      let x be object;

      assume x in {m};

      then x = m by TARSKI:def 1;

      hence thesis by A4, A5, XBOOLE_0:def 4;

    end;

    theorem :: JORDAN:92

    ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in D implies (( LSeg ( |[ 0 , ( - 3)]|,( LMP D))) /\ D) = {( LMP D)}

    proof

      assume

       A1: (a,b) realize-max-dist-in D;

      set m = ( LMP D);

      set w = ((( W-bound D) + ( E-bound D)) / 2);

      

       A2: (d `1 ) = w by A1, Lm88;

      

       A3: (m `1 ) = w by EUCLID: 52;

      

       A4: m in ( LSeg (d,m)) by RLTOPSP1: 68;

      

       A5: m in D by JORDAN21: 31;

      thus (( LSeg (d,m)) /\ D) c= {m}

      proof

        let x be object;

        assume

         A6: x in (( LSeg (d,m)) /\ D);

        then

         A7: x in ( LSeg (d,m)) by XBOOLE_0:def 4;

        

         A8: x in D by A6, XBOOLE_0:def 4;

        reconsider x as Point of T2 by A6;

        ( LSeg (d,m)) is vertical by A2, A3, SPPOL_1: 16;

        then

         A9: (x `1 ) = (m `1 ) by A4, A7;

        then x in ( Vertical_Line w) by A3, JORDAN6: 31;

        then x in (D /\ ( Vertical_Line w)) by A8, XBOOLE_0:def 4;

        then

         A10: (m `2 ) <= (x `2 ) by JORDAN21: 29;

        (x `2 ) <= (m `2 ) by A1, A7, Th86;

        then (x `2 ) = (m `2 ) by A10, XXREAL_0: 1;

        then x = m by A9, TOPREAL3: 6;

        hence thesis by TARSKI:def 1;

      end;

      let x be object;

      assume x in {m};

      then x = m by TARSKI:def 1;

      hence thesis by A4, A5, XBOOLE_0:def 4;

    end;

    theorem :: JORDAN:93

    

     Th93: P is compact & ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P & A is_inside_component_of P implies A c= ( closed_inside_of_rectangle (( - 1),1,( - 3),3))

    proof

      assume that

       A1: P is compact and

       A2: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in P and

       A3: A is_inside_component_of P;

      let x be object;

      assume that

       A4: x in A and

       A5: not x in R;

      P c= R by A2, Th71;

      then

       A6: (R ` ) c= (P ` ) by SUBSET_1: 12;

      reconsider x as Point of T2 by A4;

      

       A7: not (rl <= (x `1 ) & (x `1 ) <= rp & rd <= (x `2 ) & (x `2 ) <= rg) by A5;

      per cases ;

        suppose

         A8: 0 <= (x `1 );

        set E = ( east_halfline x);

        E c= (R ` )

        proof

          let e be object;

          assume

           A9: e in E;

          then

          reconsider e as Point of T2;

          

           A10: (e `1 ) >= (x `1 ) by A9, TOPREAL1:def 11;

          now

            assume e in R;

            then ex p st e = p & rl <= (p `1 ) & (p `1 ) <= rp & rd <= (p `2 ) & (p `2 ) <= rg;

            hence contradiction by A7, A8, A9, A10, TOPREAL1:def 11, XXREAL_0: 2;

          end;

          hence thesis by SUBSET_1: 29;

        end;

        then E c= (P ` ) by A6;

        then E misses P by SUBSET_1: 23;

        then

         A11: E c= ( UBD P) by A1, JORDAN2C: 127;

        x in E by TOPREAL1: 38;

        then A meets ( UBD P) by A4, A11, XBOOLE_0: 3;

        hence thesis by A3, Th14;

      end;

        suppose

         A12: (x `1 ) < 0 ;

        set E = ( west_halfline x);

        E c= (R ` )

        proof

          let e be object;

          assume

           A13: e in E;

          then

          reconsider e as Point of T2;

          

           A14: (e `1 ) <= (x `1 ) by A13, TOPREAL1:def 13;

          now

            assume e in R;

            then ex p st e = p & rl <= (p `1 ) & (p `1 ) <= rp & rd <= (p `2 ) & (p `2 ) <= rg;

            hence contradiction by A7, A12, A13, A14, TOPREAL1:def 13, XXREAL_0: 2;

          end;

          hence thesis by SUBSET_1: 29;

        end;

        then E c= (P ` ) by A6;

        then E misses P by SUBSET_1: 23;

        then

         A15: E c= ( UBD P) by A1, JORDAN2C: 126;

        x in E by TOPREAL1: 38;

        then A meets ( UBD P) by A4, A15, XBOOLE_0: 3;

        hence thesis by A3, Th14;

      end;

    end;

    

     Lm89: p in R implies R c= ( Ball (p,10))

    proof

      assume p in R;

      then

      consider p1 such that

       A1: p1 = p and

       A2: rl <= (p1 `1 ) and

       A3: (p1 `1 ) <= rp and

       A4: rd <= (p1 `2 ) and

       A5: (p1 `2 ) <= rg;

      let x be object;

      assume

       A6: x in R;

      then

      reconsider x as Point of T2;

      consider p2 such that

       A7: p2 = x and

       A8: rl <= (p2 `1 ) and

       A9: (p2 `1 ) <= rp and

       A10: rd <= (p2 `2 ) and

       A11: (p2 `2 ) <= rg by A6;

      

       A12: ex s,t be Point of ( Euclid 2) st s = p1 & t = p2 & ( dist (p1,p2)) = ( dist (s,t)) by TOPREAL6:def 1;

      ( dist (p1,p2)) <= ((rp - rl) + (rg - rd)) by A2, A3, A4, A5, A8, A9, A10, A11, TOPREAL6: 95;

      then ( dist (p1,p2)) < 10 by XXREAL_0: 2;

      then |.(x - p).| < 10 by A1, A7, A12, SPPOL_1: 39;

      hence thesis by TOPREAL9: 7;

    end;

    theorem :: JORDAN:94

    ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ( LSeg ( |[ 0 , 3]|, |[ 0 , ( - 3)]|)) meets C

    proof

      assume

       A1: (a,b) realize-max-dist-in C;

      set Jc = ( Upper_Arc C);

      consider Pf be Path of c, d, f be Function of I[01] , (T2 | ( LSeg (c,d))) such that

       A2: ( rng f) = ( LSeg (c,d)) and

       A3: Pf = f by Th43;

      

       A4: a = ( W-min C) by A1, Th79;

      b = ( E-max C) by A1, Th80;

      then Jc is_an_arc_of (a,b) by A4, JORDAN6:def 8;

      then

      consider Pg be Path of a, b, g be Function of I[01] , (T2 | Jc) such that

       A5: ( rng g) = Jc and

       A6: Pg = g by Th42;

      

       A7: Jc c= C by JORDAN6: 61;

      

       A8: C c= R by A1, Th71;

      

       A9: a in C by A1;

      

       A10: b in C by A1;

      

       A11: the carrier of TR = R by PRE_TOPC: 8;

      reconsider AR = a, BR = b, CR = c, DR = d as Point of TR by A8, A9, A10, Lm62, Lm63, Lm67, PRE_TOPC: 8;

      ( rng Pg) c= the carrier of TR by A5, A6, A7, A8, A11;

      then

      reconsider h = Pg as Path of AR, BR by Th30;

      ( LSeg (c,d)) c= R by Lm62, Lm63, Lm67, JORDAN1:def 1;

      then

      reconsider v = Pf as Path of CR, DR by A2, A3, A11, Th30;

      consider s,t be Point of I[01] such that

       A12: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

      

       A13: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

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

      then

       A14: (v . t) in ( rng Pf) by FUNCT_1:def 3;

      (h . s) in ( rng Pg) by A13, FUNCT_1:def 3;

      hence thesis by A2, A3, A5, A6, A7, A12, A14, XBOOLE_0: 3;

    end;

    

     Lm90: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies ex Jc,Jd be compact with_the_max_arc Subset of T2 st Jc is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & Jd is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & C = (Jc \/ Jd) & (Jc /\ Jd) = { |[( - 1), 0 ]|, |[1, 0 ]|} & ( UMP C) in Jc & ( LMP C) in Jd & ( W-bound C) = ( W-bound Jc) & ( E-bound C) = ( E-bound Jc)

    proof

      assume

       A1: (a,b) realize-max-dist-in C;

      set U = ( Upper_Arc C);

      set L = ( Lower_Arc C);

      

       A2: (U \/ L) = C by JORDAN6:def 9;

      

       A3: ( UMP C) in C by JORDAN21: 30;

      ( LMP C) in C by JORDAN21: 31;

      then

       A4: ( LMP C) in U or ( LMP C) in L by A2, XBOOLE_0:def 3;

      

       A5: ( W-min C) = a by A1, Th79;

      

       A6: ( E-max C) = b by A1, Th80;

      per cases by A2, A3, XBOOLE_0:def 3;

        suppose

         A7: ( UMP C) in U;

        take U, L;

        thus thesis by A4, A5, A6, A7, JORDAN21: 17, JORDAN21: 18, JORDAN21: 50, JORDAN6: 50;

      end;

        suppose

         A8: ( UMP C) in L;

        take L, U;

        thus thesis by A4, A5, A6, A8, JORDAN21: 19, JORDAN21: 20, JORDAN21: 49, JORDAN6: 50;

      end;

    end;

    theorem :: JORDAN:95

    

     Th95: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies for Jc,Jd be compact with_the_max_arc Subset of ( TOP-REAL 2) st Jc is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & Jd is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & C = (Jc \/ Jd) & (Jc /\ Jd) = { |[( - 1), 0 ]|, |[1, 0 ]|} & ( UMP C) in Jc & ( LMP C) in Jd & ( W-bound C) = ( W-bound Jc) & ( E-bound C) = ( E-bound Jc) holds for Ux be Subset of ( TOP-REAL 2) st Ux = ( Component_of ( Down (((1 / 2) * (( UMP (( LSeg (( LMP Jc), |[ 0 , ( - 3)]|)) /\ Jd)) + ( LMP Jc))),(C ` )))) holds Ux is_inside_component_of C & for V be Subset of ( TOP-REAL 2) st V is_inside_component_of C holds V = Ux

    proof

      set m = ( UMP C);

      set j = ( LMP C);

      assume

       A1: (a,b) realize-max-dist-in C;

      let Jc,Jd be compact with_the_max_arc Subset of T2 such that

       A2: Jc is_an_arc_of (a,b) and

       A3: Jd is_an_arc_of (a,b) and

       A4: C = (Jc \/ Jd) and

       A5: (Jc /\ Jd) = {a, b} and

       A6: ( UMP C) in Jc and

       A7: ( LMP C) in Jd and

       A8: ( W-bound C) = ( W-bound Jc) and

       A9: ( E-bound C) = ( E-bound Jc);

      set l = ( LMP Jc);

      set LJ = (( LSeg (l,d)) /\ Jd);

      set k = ( UMP LJ);

      set x = ((1 / 2) * (k + l));

      set w = ((( W-bound C) + ( E-bound C)) / 2);

      let Ux be Subset of ( TOP-REAL 2) such that

       A10: Ux = ( Component_of ( Down (x,(C ` ))));

      

       A11: C c= R by A1, Th71;

      

       A12: ( W-bound C) = rl by A1, Th75;

      

       A13: ( E-bound C) = rp by A1, Th76;

      

       A14: a in C by A1;

      

       A15: b in C by A1;

      

       A16: m in C by JORDAN21: 30;

      

       A17: l in Jc by JORDAN21: 31;

      

       A18: Jd c= C by A4, XBOOLE_1: 7;

      

       A19: Jc c= C by A4, XBOOLE_1: 7;

      then

       A20: l in C by A17;

      

       A21: (m `2 ) < (c `2 ) by A1, Lm21, Th83, JORDAN21: 30;

      

       A22: (l `1 ) = 0 by A8, A9, A12, A13, EUCLID: 52;

      

       A23: (c `1 ) = w by A1, Lm87;

      

       A24: (m `1 ) = w by EUCLID: 52;

      

       A25: m <> a by A12, A13, Lm16, EUCLID: 52;

      

       A26: m <> b by A12, A13, Lm17, EUCLID: 52;

      

       A27: l <> a by A8, A9, A12, A13, Lm16, EUCLID: 52;

      

       A28: l <> b by A8, A9, A12, A13, Lm17, EUCLID: 52;

      then

      consider Pml be Path of m, l such that

       A29: ( rng Pml) c= Jc and

       A30: ( rng Pml) misses {a, b} by A2, A6, A17, A25, A26, A27, Th44;

      set ml = ( rng Pml);

      

       A31: ml c= C by A19, A29;

      

       A32: j in C by A7, A18;

      

       A33: ( LSeg (l,d)) is vertical by A22, Lm22, SPPOL_1: 16;

      

       A34: (d `2 ) <= (j `2 ) by A1, A7, A18, Lm23, Th84;

      

       A35: (j `1 ) = 0 by A12, A13, EUCLID: 52;

      l in ( Vertical_Line w) by A12, A13, A22, JORDAN6: 31;

      then

       A36: l in (C /\ ( Vertical_Line w)) by A17, A19, XBOOLE_0:def 4;

      then (j `2 ) <= (l `2 ) by JORDAN21: 29;

      then j in ( LSeg (l,d)) by A22, A34, A35, Lm22, GOBOARD7: 7;

      then

       A37: LJ is non empty by A7, XBOOLE_0:def 4;

      

       A38: LJ is vertical by A33, Th4;

      then

       A39: k in LJ by A37, JORDAN21: 30;

      then

       A40: k in ( LSeg (l,d)) by XBOOLE_0:def 4;

      

       A41: k in Jd by A39, XBOOLE_0:def 4;

      then

       A42: k in C by A18;

      

       A43: d in ( LSeg (l,d)) by RLTOPSP1: 68;

      then

       A44: (k `1 ) = 0 by A33, A40, Lm22;

      then

       A45: k <> a by EUCLID: 52;

      

       A46: k <> b by A44, EUCLID: 52;

      

       A47: j <> a by A35, EUCLID: 52;

      j <> b by A35, EUCLID: 52;

      then

      consider Pkj be Path of k, j such that

       A48: ( rng Pkj) c= Jd and

       A49: ( rng Pkj) misses {a, b} by A3, A7, A41, A45, A46, A47, Th44;

      set kj = ( rng Pkj);

      

       A50: kj c= C by A18, A48;

      

       A51: x in ( LSeg (k,l)) by RLTOPSP1: 69;

      

       A52: ( Component_of ( Down (x,(C ` )))) is a_component by CONNSP_1: 40;

      

       A53: the carrier of (T2 | (C ` )) = (C ` ) by PRE_TOPC: 8;

      

       A54: ( LSeg (l,k)) is vertical by A22, A44, SPPOL_1: 16;

      

       A55: k in ( LSeg (l,k)) by RLTOPSP1: 68;

      

       A56: l = |[(l `1 ), (l `2 )]| by EUCLID: 53;

      

       A57: k = |[(k `1 ), (k `2 )]| by EUCLID: 53;

      

       A58: d = |[(d `1 ), (d `2 )]| by EUCLID: 53;

      (d `2 ) <= (l `2 ) by A1, A17, A19, Lm23, Th84;

      then

       A59: (k `2 ) <= (l `2 ) by A22, A40, A56, A58, Lm22, JGRAPH_6: 1;

      

       A60: a <> k by A44, EUCLID: 52;

      b <> k by A44, EUCLID: 52;

      then not k in {a, b} by A60, TARSKI:def 2;

      then

       A61: k <> l by A5, A17, A41, XBOOLE_0:def 4;

      then (k `2 ) <> (l `2 ) by A22, A44, TOPREAL3: 6;

      then

       A62: (k `2 ) < (l `2 ) by A59, XXREAL_0: 1;

      k in ( Vertical_Line w) by A12, A13, A44, JORDAN6: 31;

      then k in (C /\ ( Vertical_Line w)) by A18, A41, XBOOLE_0:def 4;

      then (j `2 ) <= (k `2 ) by JORDAN21: 29;

      then (d `2 ) <= (k `2 ) by A1, A7, A18, Lm23, Th84, XXREAL_0: 2;

      then

       A63: ( LSeg (l,k)) c= ( LSeg (l,d)) by A33, A44, A54, A59, Lm22, GOBOARD7: 63;

      

       A64: (( LSeg (l,k)) \ {l, k}) c= (C ` )

      proof

        let q be object;

        assume that

         A65: q in (( LSeg (l,k)) \ {l, k}) and

         A66: not q in (C ` );

        

         A67: q in ( LSeg (l,k)) by A65, XBOOLE_0:def 5;

        reconsider q as Point of T2 by A65;

        

         A68: q in C by A66, SUBSET_1: 29;

        

         A69: (q `1 ) = w by A12, A13, A44, A54, A55, A67;

        then

         A70: q in ( Vertical_Line w) by JORDAN6: 31;

        per cases by A4, A68, XBOOLE_0:def 3;

          suppose q in Jc;

          then q in (Jc /\ ( Vertical_Line w)) by A70, XBOOLE_0:def 4;

          then

           A71: (l `2 ) <= (q `2 ) by A8, A9, JORDAN21: 29;

          (q `2 ) <= (l `2 ) by A22, A44, A56, A57, A59, A67, JGRAPH_6: 1;

          then (l `2 ) = (q `2 ) by A71, XXREAL_0: 1;

          then l = q by A12, A13, A22, A69, TOPREAL3: 6;

          then q in {l, k} by TARSKI:def 2;

          hence contradiction by A65, XBOOLE_0:def 5;

        end;

          suppose q in Jd;

          then

           A72: q in LJ by A63, A67, XBOOLE_0:def 4;

          

           A73: (q `1 ) = (d `1 ) by A33, A43, A63, A67;

          

           A74: ( W-bound ( LSeg (l,d))) <= ( W-bound LJ) by A72, PSCOMP_1: 69, XBOOLE_1: 17;

          

           A75: ( E-bound LJ) <= ( E-bound ( LSeg (l,d))) by A72, PSCOMP_1: 67, XBOOLE_1: 17;

          

           A76: ( W-bound LJ) = ( E-bound LJ) by A37, A38, SPRECT_1: 15;

          

           A77: ( W-bound ( LSeg (l,d))) = (d `1 ) by A22, Lm22, SPRECT_1: 54;

          then ( W-bound ( LSeg (l,d))) = ( W-bound LJ) by A22, A74, A75, A76, Lm22, SPRECT_1: 57;

          then q in ( Vertical_Line ((( W-bound LJ) + ( E-bound LJ)) / 2)) by A73, A76, A77, JORDAN6: 31;

          then q in (LJ /\ ( Vertical_Line ((( W-bound LJ) + ( E-bound LJ)) / 2))) by A72, XBOOLE_0:def 4;

          then

           A78: (q `2 ) <= (k `2 ) by JORDAN21: 28;

          (k `2 ) <= (q `2 ) by A22, A44, A56, A57, A59, A67, JGRAPH_6: 1;

          then (k `2 ) = (q `2 ) by A78, XXREAL_0: 1;

          then k = q by A12, A13, A44, A69, TOPREAL3: 6;

          then q in {l, k} by TARSKI:def 2;

          hence contradiction by A65, XBOOLE_0:def 5;

        end;

      end;

      then

      reconsider X = (( LSeg (l,k)) \ {l, k}) as Subset of (T2 | (C ` )) by PRE_TOPC: 8;

      now

        assume x in {l, k};

        then x = l or x = k by TARSKI:def 2;

        hence contradiction by A61, Th1;

      end;

      then

       A79: x in (( LSeg (l,k)) \ {l, k}) by A51, XBOOLE_0:def 5;

      then ( Component_of (x,(C ` ))) = ( Component_of ( Down (x,(C ` )))) by A64, CONNSP_3: 27;

      then

       A80: x in ( Component_of ( Down (x,(C ` )))) by A64, A79, CONNSP_3: 26;

      then

       A81: X meets Ux by A10, A79, XBOOLE_0: 3;

      (( LSeg (l,k)) \ {l, k}) is convex by JORDAN1: 46;

      then X is connected by CONNSP_1: 23;

      then

       A82: X c= ( Component_of ( Down (x,(C ` )))) by A10, A52, A81, CONNSP_1: 36;

      

       A83: ( LSeg (l,k)) c= R by A11, A20, A42, JORDAN1:def 1;

      

       A84: the carrier of TR = R by PRE_TOPC: 8;

      reconsider AR = a, BR = b, CR = c, DR = d as Point of TR by A11, A14, A15, Lm62, Lm63, Lm67, PRE_TOPC: 8;

      consider Pcm be Path of c, m, fcm be Function of I[01] , (T2 | ( LSeg (c,m))) such that

       A85: ( rng fcm) = ( LSeg (c,m)) and

       A86: Pcm = fcm by Th43;

      

       A87: ( LSeg (c,m)) c= R by A11, A16, Lm62, Lm67, JORDAN1:def 1;

      

       A88: ml c= R by A11, A31;

      thus Ux is_inside_component_of C

      proof

        thus

         A89: Ux is_a_component_of (C ` ) by A10, A52;

        assume not Ux is bounded;

        then not Ux c= ( Ball (x,10)) by RLTOPSP1: 42;

        then

        consider u be object such that

         A90: u in Ux and

         A91: not u in ( Ball (x,10));

        

         A92: R c= ( Ball (x,10)) by A51, A83, Lm89;

        reconsider u as Point of T2 by A90;

        

         A93: Ux is open by A89, SPRECT_3: 8;

        ( Component_of ( Down (x,(C ` )))) is connected by A52;

        then

         A94: Ux is connected by A10, CONNSP_1: 23;

        x in ( Ball (x,10)) by Th16;

        then

        consider P1 be Subset of T2 such that

         A95: P1 is_S-P_arc_joining (x,u) and

         A96: P1 c= Ux by A10, A80, A90, A91, A93, A94, TOPREAL4: 29;

        

         A97: P1 is_an_arc_of (x,u) by A95, TOPREAL4: 2;

        reconsider P2 = P1 as Subset of (T2 | (C ` )) by A10, A96, XBOOLE_1: 1;

        

         A98: P2 c= ( Component_of ( Down (x,(C ` )))) by A10, A96;

        

         A99: P2 misses C by A53, SUBSET_1: 23;

        then

         A100: P2 misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

        

         A101: P2 misses Jd by A4, A99, XBOOLE_1: 7, XBOOLE_1: 63;

        

         A102: (x `1 ) = ((1 / 2) * ((k + l) `1 )) by TOPREAL3: 4

        .= ((1 / 2) * ((k `1 ) + (l `1 ))) by TOPREAL3: 2

        .= 0 by A22, A44;

        then

         A103: ( LSeg (d,x)) is vertical by Lm22, SPPOL_1: 16;

        

         A104: x = |[(x `1 ), (x `2 )]| by EUCLID: 53;

        

         A105: (x `2 ) < (l `2 ) by A62, Th3;

        

         A106: (k `2 ) < (x `2 ) by A62, Th2;

        then

         A107: (d `2 ) <= (x `2 ) by A1, A18, A41, Lm23, Th84, XXREAL_0: 2;

        (d `1 ) = (d `1 );

        then

         A108: ( LSeg (d,x)) c= ( LSeg (d,l)) by A33, A103, A105, A107, GOBOARD7: 63;

        

         A109: ( LSeg (d,x)) misses Jc

        proof

          assume not thesis;

          then

          consider q be object such that

           A110: q in ( LSeg (d,x)) and

           A111: q in Jc by XBOOLE_0: 3;

          reconsider q as Point of T2 by A110;

          (q `2 ) <= (x `2 ) by A58, A102, A104, A107, A110, Lm22, JGRAPH_6: 1;

          then

           A112: (q `2 ) < (l `2 ) by A105, XXREAL_0: 2;

          (q `1 ) = 0 by A33, A43, A108, A110, Lm22;

          then q in ( Vertical_Line w) by A12, A13, JORDAN6: 31;

          then q in (Jc /\ ( Vertical_Line w)) by A111, XBOOLE_0:def 4;

          hence contradiction by A8, A9, A112, JORDAN21: 29;

        end;

        set n = ( First_Point (P1,x,u,dR));

        

         A113: not u in R by A91, A92;

        

         A114: ( Fr R) = dR by Th52;

        u in P1 by A97, TOPREAL1: 1;

        then

         A115: (P1 \ R) <> ( {} T2) by A113, XBOOLE_0:def 5;

        x in P1 by A97, TOPREAL1: 1;

        then P1 meets R by A51, A83, XBOOLE_0: 3;

        then

         A116: P1 meets dR by A97, A114, A115, CONNSP_1: 22, JORDAN6: 10;

        P1 is closed by A95, JORDAN6: 11, TOPREAL4: 2;

        then

         A117: n in (P1 /\ dR) by A97, A116, JORDAN5C:def 1;

        then

         A118: n in dR by XBOOLE_0:def 4;

        

         A119: n in P1 by A117, XBOOLE_0:def 4;

        set alpha = ( Segment (P1,x,u,x,n));

        

         A120: rd < (k `2 ) by A1, A18, A41, Th84;

        (l `2 ) <= (m `2 ) by A36, JORDAN21: 28;

        then (x `2 ) < (m `2 ) by A105, XXREAL_0: 2;

        then not x in dR by A21, A102, A104, A106, A120, Lm86;

        then

         A121: alpha is_an_arc_of (x,n) by A95, A118, A119, JORDAN16: 24, TOPREAL4: 2;

        

         A122: alpha misses Jc by A100, JORDAN16: 2, XBOOLE_1: 63;

        

         A123: alpha misses Jd by A101, JORDAN16: 2, XBOOLE_1: 63;

        consider Pdx be Path of d, x, fdx be Function of I[01] , (T2 | ( LSeg (d,x))) such that

         A124: ( rng fdx) = ( LSeg (d,x)) and

         A125: Pdx = fdx by Th43;

        consider PJc be Path of a, b, fJc be Function of I[01] , (T2 | Jc) such that

         A126: ( rng fJc) = Jc and

         A127: PJc = fJc by A2, Th42;

        consider PJd be Path of a, b, fJd be Function of I[01] , (T2 | Jd) such that

         A128: ( rng fJd) = Jd and

         A129: PJd = fJd by A3, Th42;

        consider Palpha be Path of x, n, falpha be Function of I[01] , (T2 | alpha) such that

         A130: ( rng falpha) = alpha and

         A131: Palpha = falpha by A121, Th42;

        n in R by A118, Lm67;

        then

         A132: ex p st p = n & rl <= (p `1 ) & (p `1 ) <= rp & rd <= (p `2 ) & (p `2 ) <= rg;

        ( rng PJc) c= the carrier of TR by A11, A19, A84, A126, A127;

        then

        reconsider h = PJc as Path of AR, BR by Th30;

        ( rng PJd) c= the carrier of TR by A11, A18, A84, A128, A129;

        then

        reconsider H = PJd as Path of AR, BR by Th30;

        

         A133: ( LSeg (d,x)) c= R by A51, A83, Lm63, Lm67, JORDAN1:def 1;

        

         A134: alpha c= R by A51, A83, A95, A113, Th57, TOPREAL4: 2;

        

         A135: ld in ( LSeg (ld,lg)) by RLTOPSP1: 68;

        

         A136: pd in ( LSeg (pd,pg)) by RLTOPSP1: 68;

        ( LSeg (lg,c)) misses C by A1, Lm78;

        then

         A137: ( LSeg (lg,c)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

        

         A138: ( LSeg (lg,c)) c= R by Lm67, Lm70;

        

         A139: ( LSeg (pg,c)) c= R by Lm67, Lm71;

        ( LSeg (pg,c)) misses C by A1, Lm79;

        then

         A140: ( LSeg (pg,c)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

        consider Plx be Path of l, x, flx be Function of I[01] , (T2 | ( LSeg (l,x))) such that

         A141: ( rng flx) = ( LSeg (l,x)) and

         A142: Plx = flx by Th43;

        set PCX = ((Pcm + Pml) + Plx);

        

         A143: ( rng PCX) = ((( rng Pcm) \/ ( rng Pml)) \/ ( rng Plx)) by Th40;

        

         A144: ml misses Jd

        proof

          assume ml meets Jd;

          then

          consider q be object such that

           A145: q in ml and

           A146: q in Jd by XBOOLE_0: 3;

          q in {a, b} by A5, A29, A145, A146, XBOOLE_0:def 4;

          hence contradiction by A30, A145, XBOOLE_0: 3;

        end;

        

         A147: (( LSeg (c,m)) /\ C) = {m} by A1, Th91;

        

         A148: ( LSeg (c,m)) misses Jd

        proof

          assume ( LSeg (c,m)) meets Jd;

          then

          consider q be object such that

           A149: q in ( LSeg (c,m)) and

           A150: q in Jd by XBOOLE_0: 3;

          q in {m} by A18, A147, A149, A150, XBOOLE_0:def 4;

          then q = m by TARSKI:def 1;

          then m in {a, b} by A5, A6, A150, XBOOLE_0:def 4;

          hence contradiction by A25, A26, TARSKI:def 2;

        end;

        ( LSeg (l,x)) is vertical by A22, A102, SPPOL_1: 16;

        then

         A151: ( LSeg (l,x)) c= ( LSeg (l,k)) by A44, A54, A102, A105, A106, GOBOARD7: 63;

        l in ( LSeg (l,x)) by RLTOPSP1: 68;

        then {l} c= ( LSeg (l,x)) by ZFMISC_1: 31;

        then

         A152: ( LSeg (l,x)) = ((( LSeg (l,x)) \ {l}) \/ {l}) by XBOOLE_1: 45;

        (( LSeg (l,x)) \ {l}) c= (( LSeg (l,k)) \ {l, k})

        proof

          let q be object;

          assume

           A153: q in (( LSeg (l,x)) \ {l});

          then

           A154: q in ( LSeg (l,x)) by ZFMISC_1: 56;

          

           A155: q <> l by A153, ZFMISC_1: 56;

          q <> k by A22, A56, A102, A104, A105, A106, A154, JGRAPH_6: 1;

          then not q in {l, k} by A155, TARSKI:def 2;

          hence thesis by A151, A154, XBOOLE_0:def 5;

        end;

        then (( LSeg (l,x)) \ {l}) c= (C ` ) by A64;

        then (( LSeg (l,x)) \ {l}) misses C by SUBSET_1: 23;

        then

         A156: (( LSeg (l,x)) \ {l}) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

         {l} misses Jd

        proof

          assume {l} meets Jd;

          then l in Jd by ZFMISC_1: 50;

          then l in {a, b} by A5, A17, XBOOLE_0:def 4;

          hence thesis by A27, A28, TARSKI:def 2;

        end;

        then ( LSeg (l,x)) misses Jd by A152, A156, XBOOLE_1: 70;

        then

         A157: ( rng PCX) misses Jd by A85, A86, A141, A142, A143, A144, A148, XBOOLE_1: 114;

        ( LSeg (l,x)) c= R by A83, A151;

        then

         A158: ( rng PCX) c= R by A85, A86, A87, A88, A141, A142, A143, Lm1;

        ( LSeg (ld,d)) misses C by A1, Lm80;

        then

         A159: ( LSeg (ld,d)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

        ( LSeg (pd,d)) misses C by A1, Lm81;

        then

         A160: ( LSeg (pd,d)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

        per cases ;

          suppose

           A161: (n `2 ) < 0 ;

          per cases by A118, A161, Lm77;

            suppose

             A162: n in ( LSeg (a,ld));

            consider Pnld be Path of n, ld, fnld be Function of I[01] , (T2 | ( LSeg (n,ld))) such that

             A163: ( rng fnld) = ( LSeg (n,ld)) and

             A164: Pnld = fnld by Th43;

            consider Pldd be Path of ld, d, fldd be Function of I[01] , (T2 | ( LSeg (ld,d))) such that

             A165: ( rng fldd) = ( LSeg (ld,d)) and

             A166: Pldd = fldd by Th43;

            

             A167: (ld `1 ) = (n `1 ) by A135, A162, Lm45, Lm58;

            then ( LSeg (n,ld)) is vertical by SPPOL_1: 16;

            then ( LSeg (n,ld)) c= ( LSeg (ld,lg)) by A132, A167, Lm25, Lm27, Lm45, GOBOARD7: 63;

            then

             A168: ( LSeg (n,ld)) c= dR by Lm38;

            set K1 = (((PCX + Palpha) + Pnld) + Pldd);

            ( LSeg (n,ld)) misses C by A1, A53, A98, A119, A162, Lm84;

            then

             A169: ( LSeg (n,ld)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A170: ( rng K1) = (((( rng PCX) \/ ( rng Palpha)) \/ ( rng Pnld)) \/ ( rng Pldd)) by Lm9;

            then

             A171: ( rng PJd) misses ( rng K1) by A123, A128, A129, A130, A131, A157, A159, A163, A164, A165, A166, A169, Lm3;

            

             A172: ( LSeg (ld,d)) c= R by Lm67, Lm74;

            ( LSeg (n,ld)) c= R by A168, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A130, A131, A134, A158, A163, A164, A165, A166, A170, A172, Lm2;

            then

            reconsider v = K1 as Path of CR, DR by Th30;

            consider s,t be Point of I[01] such that

             A173: (H . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A174: ( dom H) = the carrier of I[01] by FUNCT_2:def 1;

            

             A175: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A176: (H . s) in ( rng PJd) by A174, FUNCT_1:def 3;

            (v . t) in ( rng K1) by A175, FUNCT_1:def 3;

            hence contradiction by A171, A173, A176, XBOOLE_0: 3;

          end;

            suppose

             A177: n in ( LSeg (ld,d));

            consider Pnd be Path of n, d, fnd be Function of I[01] , (T2 | ( LSeg (n,d))) such that

             A178: ( rng fnd) = ( LSeg (n,d)) and

             A179: Pnd = fnd by Th43;

            set K1 = ((PCX + Palpha) + Pnd);

            ld in ( LSeg (ld,d)) by RLTOPSP1: 68;

            then

             A180: (ld `2 ) = (n `2 ) by A177, Lm51;

            then

             A181: ( LSeg (n,d)) is horizontal by Lm23, Lm27, SPPOL_1: 15;

            

             A182: (ld `1 ) <= (n `1 ) by A177, Lm26, JGRAPH_6: 3;

            (n `1 ) <= (d `1 ) by A177, Lm22, JGRAPH_6: 3;

            then

             A183: ( LSeg (n,d)) c= ( LSeg (ld,d)) by A180, A181, A182, Lm51, GOBOARD7: 64;

            then

             A184: ( LSeg (n,d)) c= dR by Lm74;

            ( LSeg (n,d)) misses C by A1, A183, Lm80, XBOOLE_1: 63;

            then

             A185: ( LSeg (n,d)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A186: ( rng K1) = ((( rng PCX) \/ ( rng Palpha)) \/ ( rng Pnd)) by Th40;

            then

             A187: ( rng K1) misses Jd by A123, A130, A131, A157, A178, A179, A185, XBOOLE_1: 114;

            ( LSeg (n,d)) c= R by A184, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A130, A131, A134, A158, A178, A179, A186, Lm1;

            then

            reconsider v = K1 as Path of CR, DR by Th30;

            consider s,t be Point of I[01] such that

             A188: (H . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A189: ( dom H) = the carrier of I[01] by FUNCT_2:def 1;

            

             A190: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A191: (H . s) in ( rng PJd) by A189, FUNCT_1:def 3;

            (v . t) in ( rng K1) by A190, FUNCT_1:def 3;

            hence contradiction by A128, A129, A187, A188, A191, XBOOLE_0: 3;

          end;

            suppose

             A192: n in ( LSeg (d,pd));

            consider Pnd be Path of n, d, fnd be Function of I[01] , (T2 | ( LSeg (n,d))) such that

             A193: ( rng fnd) = ( LSeg (n,d)) and

             A194: Pnd = fnd by Th43;

            set K1 = ((PCX + Palpha) + Pnd);

            pd in ( LSeg (pd,d)) by RLTOPSP1: 68;

            then (pd `2 ) = (n `2 ) by A192, Lm52;

            then

             A195: ( LSeg (n,d)) is horizontal by Lm23, Lm31, SPPOL_1: 15;

            

             A196: (d `2 ) = (d `2 );

            

             A197: (d `1 ) <= (n `1 ) by A192, Lm22, JGRAPH_6: 3;

            (n `1 ) <= (pd `1 ) by A192, Lm30, JGRAPH_6: 3;

            then

             A198: ( LSeg (n,d)) c= ( LSeg (pd,d)) by A195, A196, A197, Lm52, GOBOARD7: 64;

            then

             A199: ( LSeg (n,d)) c= dR by Lm75;

            ( LSeg (n,d)) misses C by A1, A198, Lm81, XBOOLE_1: 63;

            then

             A200: ( LSeg (n,d)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A201: ( rng K1) = ((( rng PCX) \/ ( rng Palpha)) \/ ( rng Pnd)) by Th40;

            then

             A202: ( rng K1) misses Jd by A123, A130, A131, A157, A193, A194, A200, XBOOLE_1: 114;

            ( LSeg (n,d)) c= R by A199, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A130, A131, A134, A158, A193, A194, A201, Lm1;

            then

            reconsider v = K1 as Path of CR, DR by Th30;

            consider s,t be Point of I[01] such that

             A203: (H . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A204: ( dom H) = the carrier of I[01] by FUNCT_2:def 1;

            

             A205: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A206: (H . s) in ( rng PJd) by A204, FUNCT_1:def 3;

            (v . t) in ( rng K1) by A205, FUNCT_1:def 3;

            hence contradiction by A128, A129, A202, A203, A206, XBOOLE_0: 3;

          end;

            suppose

             A207: n in ( LSeg (pd,b));

            consider Pnpd be Path of n, pd, fnpd be Function of I[01] , (T2 | ( LSeg (n,pd))) such that

             A208: ( rng fnpd) = ( LSeg (n,pd)) and

             A209: Pnpd = fnpd by Th43;

            consider Ppdd be Path of pd, d, fpdd be Function of I[01] , (T2 | ( LSeg (pd,d))) such that

             A210: ( rng fpdd) = ( LSeg (pd,d)) and

             A211: Ppdd = fpdd by Th43;

            

             A212: (pd `1 ) = (n `1 ) by A136, A207, Lm46, Lm60;

            then ( LSeg (n,pd)) is vertical by SPPOL_1: 16;

            then ( LSeg (n,pd)) c= ( LSeg (pd,pg)) by A132, A212, Lm29, Lm31, Lm46, GOBOARD7: 63;

            then

             A213: ( LSeg (n,pd)) c= dR by Lm42;

            set K1 = (((PCX + Palpha) + Pnpd) + Ppdd);

            ( LSeg (n,pd)) misses C by A1, A53, A98, A119, A207, Lm85;

            then

             A214: ( LSeg (n,pd)) misses Jd by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A215: ( rng K1) = (((( rng PCX) \/ ( rng Palpha)) \/ ( rng Pnpd)) \/ ( rng Ppdd)) by Lm9;

            then

             A216: ( rng PJd) misses ( rng K1) by A123, A128, A129, A130, A131, A157, A160, A208, A209, A210, A211, A214, Lm3;

            

             A217: ( LSeg (pd,d)) c= R by Lm67, Lm75;

            ( LSeg (n,pd)) c= R by A213, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A130, A131, A134, A158, A208, A209, A210, A211, A215, A217, Lm2;

            then

            reconsider v = K1 as Path of CR, DR by Th30;

            consider s,t be Point of I[01] such that

             A218: (H . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A219: ( dom H) = the carrier of I[01] by FUNCT_2:def 1;

            

             A220: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A221: (H . s) in ( rng PJd) by A219, FUNCT_1:def 3;

            (v . t) in ( rng K1) by A220, FUNCT_1:def 3;

            hence contradiction by A216, A218, A221, XBOOLE_0: 3;

          end;

        end;

          suppose

           A222: (n `2 ) >= 0 ;

          per cases by A118, A222, Lm76;

            suppose

             A223: n in ( LSeg (a,lg));

            consider Pnlg be Path of n, lg, fnlg be Function of I[01] , (T2 | ( LSeg (n,lg))) such that

             A224: ( rng fnlg) = ( LSeg (n,lg)) and

             A225: Pnlg = fnlg by Th43;

            consider Plgc be Path of lg, c, flgc be Function of I[01] , (T2 | ( LSeg (lg,c))) such that

             A226: ( rng flgc) = ( LSeg (lg,c)) and

             A227: Plgc = flgc by Th43;

            

             A228: (ld `1 ) = (n `1 ) by A135, A223, Lm45, Lm57;

            then ( LSeg (n,lg)) is vertical by Lm24, Lm26, SPPOL_1: 16;

            then ( LSeg (n,lg)) c= ( LSeg (ld,lg)) by A132, A228, Lm25, Lm27, Lm45, GOBOARD7: 63;

            then

             A229: ( LSeg (n,lg)) c= dR by Lm38;

            set K1 = (((Pdx + Palpha) + Pnlg) + Plgc);

            ( LSeg (n,lg)) misses C by A1, A53, A98, A119, A223, Lm82;

            then

             A230: ( LSeg (n,lg)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A231: ( rng K1) = (((( rng Pdx) \/ ( rng Palpha)) \/ ( rng Pnlg)) \/ ( rng Plgc)) by Lm9;

            then

             A232: ( rng K1) misses Jc by A109, A122, A124, A125, A130, A131, A137, A224, A225, A226, A227, A230, Lm3;

            

             A233: ( rng K1) = ( rng ( - K1)) by Th32;

            ( LSeg (n,lg)) c= R by A229, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A124, A125, A130, A131, A133, A134, A138, A224, A225, A226, A227, A231, Lm2;

            then

            reconsider v = ( - K1) as Path of CR, DR by A233, Th30;

            consider s,t be Point of I[01] such that

             A234: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A235: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

            

             A236: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A237: (h . s) in ( rng PJc) by A235, FUNCT_1:def 3;

            (v . t) in ( rng ( - K1)) by A236, FUNCT_1:def 3;

            hence contradiction by A126, A127, A232, A233, A234, A237, XBOOLE_0: 3;

          end;

            suppose

             A238: n in ( LSeg (lg,c));

            consider Pnc be Path of n, c, fnc be Function of I[01] , (T2 | ( LSeg (n,c))) such that

             A239: ( rng fnc) = ( LSeg (n,c)) and

             A240: Pnc = fnc by Th43;

            set K1 = ((Pdx + Palpha) + Pnc);

            lg in ( LSeg (lg,c)) by RLTOPSP1: 68;

            then

             A241: (lg `2 ) = (n `2 ) by A238, Lm53;

            then

             A242: ( LSeg (n,c)) is horizontal by Lm21, Lm25, SPPOL_1: 15;

            

             A243: (lg `1 ) <= (n `1 ) by A238, Lm24, JGRAPH_6: 3;

            (n `1 ) <= (c `1 ) by A238, Lm20, JGRAPH_6: 3;

            then

             A244: ( LSeg (n,c)) c= ( LSeg (lg,c)) by A241, A242, A243, Lm53, GOBOARD7: 64;

            then

             A245: ( LSeg (n,c)) c= dR by Lm70;

            ( LSeg (n,c)) misses C by A1, A244, Lm78, XBOOLE_1: 63;

            then

             A246: ( LSeg (n,c)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A247: ( rng K1) = ((( rng Pdx) \/ ( rng Palpha)) \/ ( rng Pnc)) by Th40;

            then

             A248: ( rng K1) misses Jc by A109, A122, A124, A125, A130, A131, A239, A240, A246, XBOOLE_1: 114;

            

             A249: ( rng K1) = ( rng ( - K1)) by Th32;

            ( LSeg (n,c)) c= R by A245, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A124, A125, A130, A131, A133, A134, A239, A240, A247, Lm1;

            then

            reconsider v = ( - K1) as Path of CR, DR by A249, Th30;

            consider s,t be Point of I[01] such that

             A250: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A251: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

            

             A252: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A253: (h . s) in ( rng PJc) by A251, FUNCT_1:def 3;

            (v . t) in ( rng ( - K1)) by A252, FUNCT_1:def 3;

            hence contradiction by A126, A127, A248, A249, A250, A253, XBOOLE_0: 3;

          end;

            suppose

             A254: n in ( LSeg (c,pg));

            consider Pnc be Path of n, c, fnc be Function of I[01] , (T2 | ( LSeg (n,c))) such that

             A255: ( rng fnc) = ( LSeg (n,c)) and

             A256: Pnc = fnc by Th43;

            set K1 = ((Pdx + Palpha) + Pnc);

            pg in ( LSeg (pg,c)) by RLTOPSP1: 68;

            then (pg `2 ) = (n `2 ) by A254, Lm54;

            then

             A257: ( LSeg (n,c)) is horizontal by Lm21, Lm29, SPPOL_1: 15;

            

             A258: (c `2 ) = (c `2 );

            

             A259: (c `1 ) <= (n `1 ) by A254, Lm20, JGRAPH_6: 3;

            (n `1 ) <= (pg `1 ) by A254, Lm28, JGRAPH_6: 3;

            then

             A260: ( LSeg (c,n)) c= ( LSeg (c,pg)) by A257, A258, A259, Lm54, GOBOARD7: 64;

            then

             A261: ( LSeg (n,c)) c= dR by Lm71;

            ( LSeg (n,c)) misses C by A1, A260, Lm79, XBOOLE_1: 63;

            then

             A262: ( LSeg (n,c)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A263: ( rng K1) = ((( rng Pdx) \/ ( rng Palpha)) \/ ( rng Pnc)) by Th40;

            then

             A264: ( rng K1) misses Jc by A109, A122, A124, A125, A130, A131, A255, A256, A262, XBOOLE_1: 114;

            

             A265: ( rng K1) = ( rng ( - K1)) by Th32;

            ( LSeg (n,c)) c= R by A261, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A124, A125, A130, A131, A133, A134, A255, A256, A263, Lm1;

            then

            reconsider v = ( - K1) as Path of CR, DR by A265, Th30;

            consider s,t be Point of I[01] such that

             A266: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A267: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

            

             A268: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A269: (h . s) in ( rng PJc) by A267, FUNCT_1:def 3;

            (v . t) in ( rng ( - K1)) by A268, FUNCT_1:def 3;

            hence contradiction by A126, A127, A264, A265, A266, A269, XBOOLE_0: 3;

          end;

            suppose

             A270: n in ( LSeg (pg,b));

            consider Pnpg be Path of n, pg, fnpg be Function of I[01] , (T2 | ( LSeg (n,pg))) such that

             A271: ( rng fnpg) = ( LSeg (n,pg)) and

             A272: Pnpg = fnpg by Th43;

            consider Ppgc be Path of pg, c, fpgc be Function of I[01] , (T2 | ( LSeg (pg,c))) such that

             A273: ( rng fpgc) = ( LSeg (pg,c)) and

             A274: Ppgc = fpgc by Th43;

            

             A275: (pd `1 ) = (n `1 ) by A136, A270, Lm46, Lm59;

            then ( LSeg (n,pg)) is vertical by Lm28, Lm30, SPPOL_1: 16;

            then ( LSeg (n,pg)) c= ( LSeg (pd,pg)) by A132, A275, Lm29, Lm31, Lm46, GOBOARD7: 63;

            then

             A276: ( LSeg (n,pg)) c= dR by Lm42;

            set K1 = (((Pdx + Palpha) + Pnpg) + Ppgc);

            ( LSeg (n,pg)) misses C by A1, A53, A98, A119, A270, Lm83;

            then

             A277: ( LSeg (n,pg)) misses Jc by A4, XBOOLE_1: 7, XBOOLE_1: 63;

            

             A278: ( rng K1) = (((( rng Pdx) \/ ( rng Palpha)) \/ ( rng Pnpg)) \/ ( rng Ppgc)) by Lm9;

            then

             A279: ( rng K1) misses Jc by A109, A122, A124, A125, A130, A131, A140, A271, A272, A273, A274, A277, Lm3;

            

             A280: ( rng K1) = ( rng ( - K1)) by Th32;

            ( LSeg (n,pg)) c= R by A276, Lm67;

            then ( rng K1) c= the carrier of TR by A84, A124, A125, A130, A131, A133, A134, A139, A271, A272, A273, A274, A278, Lm2;

            then

            reconsider v = ( - K1) as Path of CR, DR by A280, Th30;

            consider s,t be Point of I[01] such that

             A281: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

            

             A282: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

            

             A283: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

            

             A284: (h . s) in ( rng PJc) by A282, FUNCT_1:def 3;

            (v . t) in ( rng ( - K1)) by A283, FUNCT_1:def 3;

            hence contradiction by A126, A127, A279, A280, A281, A284, XBOOLE_0: 3;

          end;

        end;

      end;

      let V be Subset of T2;

      assume

       A285: V is_inside_component_of C;

      assume

       A286: V <> Ux;

      consider VP be Subset of (T2 | (C ` )) such that

       A287: VP = V and

       A288: VP is a_component and VP is bounded Subset of ( Euclid 2) by A285, JORDAN2C: 13;

      reconsider T2C = (T2 | (C ` )) as non empty SubSpace of T2;

      VP <> ( {} (T2 | (C ` ))) by A288, CONNSP_1: 32;

      then

      reconsider VP as non empty Subset of T2C;

      

       A289: V misses C by A53, A287, SUBSET_1: 23;

      consider Pjd be Path of j, d, fjd be Function of I[01] , (T2 | ( LSeg (j,d))) such that

       A290: ( rng fjd) = ( LSeg (j,d)) and

       A291: Pjd = fjd by Th43;

      consider Plk be Path of l, k, flk be Function of I[01] , (T2 | ( LSeg (l,k))) such that

       A292: ( rng flk) = ( LSeg (l,k)) and

       A293: Plk = flk by Th43;

      set beta = ((((Pcm + Pml) + Plk) + Pkj) + Pjd);

      

       A294: ( rng beta) = ((((( rng Pcm) \/ ( rng Pml)) \/ ( rng Plk)) \/ ( rng Pkj)) \/ ( rng Pjd)) by Lm11;

      ( dom beta) = ( [#] I[01] ) by FUNCT_2:def 1;

      then (beta .: ( dom beta)) is compact by WEIERSTR: 8;

      then

       A295: ( rng beta) is closed by RELAT_1: 113;

      

       A296: ml misses V by A19, A29, A289, XBOOLE_1: 1, XBOOLE_1: 63;

       {l, k} c= ( LSeg (l,k))

      proof

        let x be object;

        assume x in {l, k};

        then x = l or x = k by TARSKI:def 2;

        hence thesis by RLTOPSP1: 68;

      end;

      then

       A297: ( LSeg (l,k)) = ((( LSeg (l,k)) \ {l, k}) \/ {l, k}) by XBOOLE_1: 45;

      

       A298: (( LSeg (l,k)) \ {l, k}) misses V

      proof

        assume not thesis;

        then ex q be object st (q in (( LSeg (l,k)) \ {l, k})) & (q in V) by XBOOLE_0: 3;

        then V meets Ux by A10, A82, XBOOLE_0: 3;

        hence contradiction by A10, A52, A286, A287, A288, CONNSP_1: 35;

      end;

      

       A299: not l in V by A17, A19, A289, XBOOLE_0: 3;

       not k in V by A18, A41, A289, XBOOLE_0: 3;

      then {l, k} misses V by A299, ZFMISC_1: 51;

      then

       A300: ( LSeg (l,k)) misses V by A297, A298, XBOOLE_1: 70;

      

       A301: kj misses V by A50, A289, XBOOLE_1: 63;

      

       A302: ( LSeg (j,d)) misses V by A1, A285, Th90;

      ( LSeg (c,m)) misses V by A1, A285, Th89;

      then ((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) misses V by A296, A300, XBOOLE_1: 114;

      then

       A303: ( rng beta) misses V by A85, A86, A290, A291, A292, A293, A294, A301, A302, XBOOLE_1: 114;

      

       A304: m = |[(m `1 ), (m `2 )]| by EUCLID: 53;

      

       A305: c = |[(c `1 ), (c `2 )]| by EUCLID: 53;

      

       A306: j = |[(j `1 ), (j `2 )]| by EUCLID: 53;

      

       A307: not a in ( LSeg (c,m)) by A12, A13, A21, A23, A24, A304, A305, Lm16, JGRAPH_6: 1;

       not a in ml by A30, ZFMISC_1: 49;

      then

       A308: not a in (( LSeg (c,m)) \/ ml) by A307, XBOOLE_0:def 3;

       not a in ( LSeg (l,k)) by A22, A44, A56, A57, A59, Lm16, JGRAPH_6: 1;

      then

       A309: not a in ((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) by A308, XBOOLE_0:def 3;

       not a in kj by A49, ZFMISC_1: 49;

      then

       A310: not a in (((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) \/ kj) by A309, XBOOLE_0:def 3;

       not a in ( LSeg (j,d)) by A34, A35, A58, A306, Lm16, Lm22, JGRAPH_6: 1;

      then not a in ( rng beta) by A85, A86, A290, A291, A292, A293, A294, A310, XBOOLE_0:def 3;

      then

      consider ra be positive Real such that

       A311: ( Ball (a,ra)) misses ( rng beta) by A295, Th25;

      

       A312: not b in ( LSeg (c,m)) by A12, A13, A21, A23, A24, A304, A305, Lm17, JGRAPH_6: 1;

       not b in ml by A30, ZFMISC_1: 49;

      then

       A313: not b in (( LSeg (c,m)) \/ ml) by A312, XBOOLE_0:def 3;

       not b in ( LSeg (l,k)) by A22, A44, A56, A57, A59, Lm17, JGRAPH_6: 1;

      then

       A314: not b in ((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) by A313, XBOOLE_0:def 3;

       not b in kj by A49, ZFMISC_1: 49;

      then

       A315: not b in (((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) \/ kj) by A314, XBOOLE_0:def 3;

       not b in ( LSeg (j,d)) by A34, A35, A58, A306, Lm17, Lm22, JGRAPH_6: 1;

      then not b in ( rng beta) by A85, A86, A290, A291, A292, A293, A294, A315, XBOOLE_0:def 3;

      then

      consider rb be positive Real such that

       A316: ( Ball (b,rb)) misses ( rng beta) by A295, Th25;

      set A = ( Ball (a,ra)), B = ( Ball (b,rb));

      

       A317: a in A by Th16;

      

       A318: b in B by Th16;

      VP is non empty;

      then

      consider t be object such that

       A319: t in V by A287;

      V in { W where W be Subset of T2 : W is_inside_component_of C } by A285;

      then t in ( BDD C) by A319, TARSKI:def 4;

      then

       A320: C = ( Fr V) by A287, A288, Lm15;

      then a in ( Cl V) by A14, XBOOLE_0:def 4;

      then A meets V by A317, PRE_TOPC:def 7;

      then

      consider u be object such that

       A321: u in A and

       A322: u in V by XBOOLE_0: 3;

      b in ( Cl V) by A15, A320, XBOOLE_0:def 4;

      then B meets V by A318, PRE_TOPC:def 7;

      then

      consider v be object such that

       A323: v in B and

       A324: v in V by XBOOLE_0: 3;

      reconsider u, v as Point of T2 by A321, A323;

      

       A325: the carrier of (T2C | VP) = VP by PRE_TOPC: 8;

      reconsider u1 = u, v1 = v as Point of (T2C | VP) by A287, A322, A324, PRE_TOPC: 8;

      (T2C | VP) is pathwise_connected by A288, Th69;

      then

       A326: (u1,v1) are_connected ;

      then

      consider fuv be Function of I[01] , (T2C | VP) such that

       A327: fuv is continuous and

       A328: (fuv . 0 ) = u1 and

       A329: (fuv . 1) = v1;

      

       A330: (T2C | VP) = (T2 | V) by A287, GOBOARD9: 2;

      fuv is Path of u1, v1 by A326, A327, A328, A329, BORSUK_2:def 2;

      then

      reconsider uv = fuv as Path of u, v by A326, A330, TOPALG_2: 1;

      

       A331: ( rng fuv) c= the carrier of (T2C | VP);

      then

       A332: ( rng uv) misses ( rng beta) by A287, A303, A325, XBOOLE_1: 63;

      consider au be Path of a, u, fau be Function of I[01] , (T2 | ( LSeg (a,u))) such that

       A333: ( rng fau) = ( LSeg (a,u)) and

       A334: au = fau by Th43;

      consider vb be Path of v, b, fvb be Function of I[01] , (T2 | ( LSeg (v,b))) such that

       A335: ( rng fvb) = ( LSeg (v,b)) and

       A336: vb = fvb by Th43;

      set AB = ((au + uv) + vb);

      

       A337: ( rng AB) = ((( rng au) \/ ( rng uv)) \/ ( rng vb)) by Th40;

      a in A by Th16;

      then ( LSeg (a,u)) c= A by A321, JORDAN1:def 1;

      then

       A338: ( LSeg (a,u)) misses ( rng beta) by A311, XBOOLE_1: 63;

      b in B by Th16;

      then ( LSeg (v,b)) c= B by A323, JORDAN1:def 1;

      then ( LSeg (v,b)) misses ( rng beta) by A316, XBOOLE_1: 63;

      then

       A339: ( rng AB) misses ( rng beta) by A332, A333, A334, A335, A336, A337, A338, XBOOLE_1: 114;

      

       A340: (a,b) are_connected by BORSUK_2:def 3;

      

       A341: V c= R by A1, A285, Th93;

      then

       A342: ( LSeg (a,u)) c= R by A11, A14, A322, JORDAN1:def 1;

      

       A343: ( LSeg (v,b)) c= R by A11, A15, A324, A341, JORDAN1:def 1;

      ( rng uv) c= R by A287, A325, A331, A341;

      then (( LSeg (a,u)) \/ ( rng uv)) c= R by A342, XBOOLE_1: 8;

      then ( rng AB) c= the carrier of TR by A84, A333, A334, A335, A336, A337, A343, XBOOLE_1: 8;

      then

      reconsider h = AB as Path of AR, BR by A340, Th29;

      

       A344: (c,d) are_connected by BORSUK_2:def 3;

      (( LSeg (c,m)) \/ ml) c= R by A87, A88, XBOOLE_1: 8;

      then

       A345: ((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) c= R by A83, XBOOLE_1: 8;

      kj c= R by A11, A50;

      then

       A346: (((( LSeg (c,m)) \/ ml) \/ ( LSeg (l,k))) \/ kj) c= R by A345, XBOOLE_1: 8;

      ( LSeg (j,d)) c= R by A11, A32, Lm63, Lm67, JORDAN1:def 1;

      then ( rng beta) c= the carrier of TR by A84, A85, A86, A290, A291, A292, A293, A294, A346, XBOOLE_1: 8;

      then

      reconsider v = beta as Path of CR, DR by A344, Th29;

      consider s,t be Point of I[01] such that

       A347: (h . s) = (v . t) by Lm16, Lm17, Lm21, Lm23, JGRAPH_8: 6;

      

       A348: ( dom h) = the carrier of I[01] by FUNCT_2:def 1;

      

       A349: ( dom v) = the carrier of I[01] by FUNCT_2:def 1;

      

       A350: (h . s) in ( rng AB) by A348, FUNCT_1:def 3;

      (v . t) in ( rng beta) by A349, FUNCT_1:def 3;

      hence contradiction by A339, A347, A350, XBOOLE_0: 3;

    end;

    theorem :: JORDAN:96

    

     Th96: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies for Jc,Jd be compact with_the_max_arc Subset of ( TOP-REAL 2) st Jc is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & Jd is_an_arc_of ( |[( - 1), 0 ]|, |[1, 0 ]|) & C = (Jc \/ Jd) & (Jc /\ Jd) = { |[( - 1), 0 ]|, |[1, 0 ]|} & ( UMP C) in Jc & ( LMP C) in Jd & ( W-bound C) = ( W-bound Jc) & ( E-bound C) = ( E-bound Jc) holds ( BDD C) = ( Component_of ( Down (((1 / 2) * (( UMP (( LSeg (( LMP Jc), |[ 0 , ( - 3)]|)) /\ Jd)) + ( LMP Jc))),(C ` ))))

    proof

      assume

       A1: (a,b) realize-max-dist-in C;

      let Jc,Jd be compact with_the_max_arc Subset of T2 such that

       A2: Jc is_an_arc_of (a,b) and

       A3: Jd is_an_arc_of (a,b) and

       A4: C = (Jc \/ Jd) and

       A5: (Jc /\ Jd) = {a, b} and

       A6: ( UMP C) in Jc and

       A7: ( LMP C) in Jd and

       A8: ( W-bound C) = ( W-bound Jc) and

       A9: ( E-bound C) = ( E-bound Jc);

      reconsider Ux = ( Component_of ( Down (((1 / 2) * (( UMP (( LSeg (( LMP Jc),d)) /\ Jd)) + ( LMP Jc))),(C ` )))) as Subset of T2 by PRE_TOPC: 11;

      Ux = ( BDD C)

      proof

        Ux is_inside_component_of C by A1, A2, A3, A4, A5, A6, A7, A8, A9, Th95;

        hence Ux c= ( BDD C) by JORDAN2C: 22;

        set F = { B where B be Subset of T2 : B is_inside_component_of C };

        let q be object;

        assume q in ( BDD C);

        then

        consider Z be set such that

         A10: q in Z and

         A11: Z in F by TARSKI:def 4;

        ex B be Subset of T2 st Z = B & B is_inside_component_of C by A11;

        hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, Th95;

      end;

      hence thesis;

    end;

    

     Lm91: ( |[( - 1), 0 ]|, |[1, 0 ]|) realize-max-dist-in C implies C is Jordan

    proof

      assume

       A1: (a,b) realize-max-dist-in C;

      then

      consider Jc,Jd be compact with_the_max_arc Subset of T2 such that

       A2: Jc is_an_arc_of (a,b) and

       A3: Jd is_an_arc_of (a,b) and

       A4: C = (Jc \/ Jd) and

       A5: (Jc /\ Jd) = {a, b} and

       A6: ( UMP C) in Jc and

       A7: ( LMP C) in Jd and

       A8: ( W-bound C) = ( W-bound Jc) and

       A9: ( E-bound C) = ( E-bound Jc) by Lm90;

      set l = ( LMP Jc);

      set LJ = (( LSeg (l,d)) /\ Jd);

      set k = ( UMP LJ);

      set x = ((1 / 2) * (k + l));

      

       A10: ( Component_of ( Down (x,(C ` )))) is a_component by CONNSP_1: 40;

      

       A11: ( Component_of ( Down (x,(C ` )))) = ( BDD C) by A1, A2, A3, A4, A5, A6, A7, A8, A9, Th96;

      thus (C ` ) <> {} ;

      take A1 = ( UBD C), A2 = ( BDD C);

      thus (C ` ) = (A1 \/ A2) by JORDAN2C: 27;

      thus A1 misses A2 by JORDAN2C: 24;

      

       A12: ( Component_of ( Down (x,(C ` )))) <> ( {} (T2 | (C ` ))) by A10, CONNSP_1: 32;

      A1 is_a_component_of (C ` ) by JORDAN2C: 124;

      then

       A13: ex B1 be Subset of (T2 | (C ` )) st B1 = A1 & B1 is a_component;

      

      then

       A14: C = ( Fr A1) by A11, A12, Lm15

      .= (( Cl A1) /\ ( Cl (A1 ` )));

      

       A15: C = ( Fr A2) by A10, A11, A12, Lm15

      .= (( Cl A2) /\ ( Cl (A2 ` )));

      A2 c= (C ` ) by JORDAN2C: 25;

      then C misses A2 by SUBSET_1: 23;

      then

       A16: C c= (( Cl A2) \ A2) by A15, XBOOLE_1: 17, XBOOLE_1: 86;

      

       A17: A1 misses A2 by JORDAN2C: 24;

      then A2 c= (A1 ` ) by SUBSET_1: 23;

      then

       A18: ( Cl A2) c= (A1 ` ) by TOPS_1: 5;

      (A1 \/ A2) = (C ` ) by JORDAN2C: 27;

      then (A1 \/ A2) misses C by SUBSET_1: 23;

      then C misses A1 by XBOOLE_1: 70;

      then

       A19: (A2 \/ C) misses A1 by A17, XBOOLE_1: 70;

      (A2 \/ A1) = (C ` ) by JORDAN2C: 27;

      then ((A2 \/ A1) ` ) misses (C ` ) by SUBSET_1: 23;

      then (((A2 \/ A1) ` ) /\ (C ` )) = {} ;

      then (((A2 \/ A1) \/ C) ` ) = {} by XBOOLE_1: 53;

      then (((A2 \/ C) \/ A1) ` ) = {} by XBOOLE_1: 4;

      then (((A2 \/ C) ` ) /\ (A1 ` )) = {} by XBOOLE_1: 53;

      then ((A2 \/ C) ` ) misses (A1 ` );

      then ( Cl A2) c= (A2 \/ C) by A18, A19, SUBSET_1: 25;

      then

       A20: (( Cl A2) \ A2) c= C by XBOOLE_1: 43;

      A1 c= (C ` ) by JORDAN2C: 26;

      then C misses A1 by SUBSET_1: 23;

      then

       A21: C c= (( Cl A1) \ A1) by A14, XBOOLE_1: 17, XBOOLE_1: 86;

      A1 c= (A2 ` ) by A17, SUBSET_1: 23;

      then

       A22: ( Cl A1) c= (A2 ` ) by TOPS_1: 5;

      (A2 \/ A1) = (C ` ) by JORDAN2C: 27;

      then (A2 \/ A1) misses C by SUBSET_1: 23;

      then C misses A2 by XBOOLE_1: 70;

      then

       A23: (A1 \/ C) misses A2 by A17, XBOOLE_1: 70;

      (A1 \/ A2) = (C ` ) by JORDAN2C: 27;

      then ((A1 \/ A2) ` ) misses (C ` ) by SUBSET_1: 23;

      then (((A1 \/ A2) ` ) /\ (C ` )) = {} ;

      then (((A1 \/ A2) \/ C) ` ) = {} by XBOOLE_1: 53;

      then (((A1 \/ C) \/ A2) ` ) = {} by XBOOLE_1: 4;

      then (((A1 \/ C) ` ) /\ (A2 ` )) = {} by XBOOLE_1: 53;

      then ((A1 \/ C) ` ) misses (A2 ` );

      then ( Cl A1) c= (A1 \/ C) by A22, A23, SUBSET_1: 25;

      then (( Cl A1) \ A1) c= C by XBOOLE_1: 43;

      

      hence (( Cl A1) \ A1) = C by A21

      .= (( Cl A2) \ A2) by A16, A20;

      thus thesis by A11, A13, CONNSP_1: 40;

    end;

    

     Lm92: C is Jordan

    proof

      consider f be Homeomorphism of T2 such that

       A1: (a,b) realize-max-dist-in (f .: C) by JORDAN24: 7;

      

       A2: (f " ) is Homeomorphism of T2 by TOPGRP_1: 30;

      (f .: C) is Simple_closed_curve by Th70;

      then (f .: C) is Jordan by A1, Lm91;

      then

       A3: ((f " ) .: (f .: C)) is Jordan by A2, JORDAN24: 16;

      

       A4: (f " ) = (f qua Function " ) by TOPS_2:def 4;

      ( dom f) = the carrier of T2 by FUNCT_2:def 1;

      hence thesis by A3, A4, FUNCT_1: 107;

    end;

    registration

      let C;

      cluster ( BDD C) -> non empty;

      coherence

      proof

        C is Jordan by Lm92;

        then ( BDD C) is_inside_component_of C by JORDAN2C: 108;

        then ( BDD C) is_a_component_of (C ` );

        then ex B1 be Subset of (T2 | (C ` )) st B1 = ( BDD C) & B1 is a_component;

        then ( BDD C) <> ( {} (T2 | (C ` ))) by CONNSP_1: 32;

        hence thesis;

      end;

    end

    theorem :: JORDAN:97

    U = P & U is a_component implies C = ( Fr P)

    proof

      ( BDD C) is non empty;

      hence thesis by Lm15;

    end;

    theorem :: JORDAN:98

    for C be Simple_closed_curve holds ex A1,A2 be Subset of ( TOP-REAL 2) st (C ` ) = (A1 \/ A2) & A1 misses A2 & (( Cl A1) \ A1) = (( Cl A2) \ A2) & for C1,C2 be Subset of (( TOP-REAL 2) | (C ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component

    proof

      let C;

      C is Jordan by Lm92;

      hence thesis;

    end;

    ::$Notion-Name

    theorem :: JORDAN:99

    for C be Simple_closed_curve holds C is Jordan by Lm92;