jordan2c.miz



    begin

    registration

      let n be Nat;

      cluster ( TOP-REAL n) -> add-continuous Mult-continuous;

      coherence

      proof

        set T = ( TOP-REAL n), E = ( Euclid n), TE = ( TopSpaceMetr E);

        

         A1: the TopStruct of T = TE by EUCLID:def 8;

        thus T is add-continuous

        proof

          let x1,x2 be Point of T, V be Subset of T such that

           A2: V is open and

           A3: (x1 + x2) in V;

          reconsider X1 = x1, X2 = x2, X12 = (x1 + x2) as Point of E by A1, TOPMETR: 12;

          reconsider v = V as Subset of TE by A1;

          V in the topology of T by A2, PRE_TOPC:def 2;

          then v is open by A1, PRE_TOPC:def 2;

          then

          consider r be Real such that

           A4: r > 0 and

           A5: ( Ball (X12,r)) c= v by A3, TOPMETR: 15;

          set r2 = (r / 2);

          reconsider B1 = ( Ball (X1,r2)), B2 = ( Ball (X2,r2)) as Subset of T by A1, TOPMETR: 12;

          take B1, B2;

          thus B1 is open & B2 is open & x1 in B1 & x2 in B2 by A4, GOBOARD6: 1, GOBOARD6: 3, XREAL_1: 215;

          let x be object;

          assume x in (B1 + B2);

          then x in { (b1 + b2) where b1,b2 be Element of T : b1 in B1 & b2 in B2 } by RUSUB_4:def 9;

          then

          consider b1,b2 be Element of T such that

           A6: x = (b1 + b2) and

           A7: b1 in B1 and

           A8: b2 in B2;

          reconsider e1 = b1, e2 = b2, e12 = (b1 + b2) as Point of E by A1, TOPMETR: 12;

          reconsider y1 = x1, y2 = x2, c1 = b1, c2 = b2 as Element of ( REAL n) by EUCLID: 22;

          ( dist (X2,e2)) < r2 by A8, METRIC_1: 11;

          then

           A9: |.(y2 - c2).| < r2 by SPPOL_1: 5;

          ( dist (X1,e1)) < r2 by A7, METRIC_1: 11;

          then |.(y1 - c1).| < r2 by SPPOL_1: 5;

          then

           A10: ( |.(y1 - c1).| + |.(y2 - c2).|) < (r2 + r2) by A9, XREAL_1: 8;

          

           A11: ((y1 + y2) - (c1 + c2)) = ((y1 + y2) + ( - (c2 + c1)))

          .= ((y1 + y2) + (( - c2) + ( - c1))) by RVSUM_1: 26

          .= (((y1 + y2) + ( - c2)) + ( - c1)) by RVSUM_1: 15

          .= (((y2 + ( - c2)) + y1) + ( - c1)) by RVSUM_1: 15

          .= ((y2 + ( - c2)) + (y1 + ( - c1))) by RVSUM_1: 15

          .= ((y2 - c2) + (y1 + ( - c1)))

          .= ((y2 - c2) + (y1 - c1));

          

           A12: ( dist (X12,e12)) = |.((y1 - c1) + (y2 - c2)).| by A11, SPPOL_1: 5;

           |.((y1 - c1) + (y2 - c2)).| <= ( |.(y1 - c1).| + |.(y2 - c2).|) by EUCLID: 12;

          then ( dist (X12,e12)) < r by A10, A12, XXREAL_0: 2;

          then e12 in ( Ball (X12,r)) by METRIC_1: 11;

          hence x in V by A5, A6;

        end;

        let a be Real, x be Point of T, V be Subset of T such that

         A13: V is open and

         A14: (a * x) in V;

        reconsider X = x, AX = (a * x) as Point of E by A1, TOPMETR: 12;

        reconsider v = V as Subset of TE by A1;

        V in the topology of T by A13, PRE_TOPC:def 2;

        then v is open by A1, PRE_TOPC:def 2;

        then

        consider r be Real such that

         A15: r > 0 and

         A16: ( Ball (AX,r)) c= v by A14, TOPMETR: 15;

        set r2 = (r / 2);

        

         A17: r2 > 0 by A15, XREAL_1: 215;

        then

         A18: (r2 / 2) > 0 by XREAL_1: 215;

        ex m be positive Real st ( |.a.| * m) < r2

        proof

          per cases by COMPLEX1: 46;

            suppose |.a.| = 0 ;

            then ( |.a.| * 1) < r2 by A15, XREAL_1: 215;

            hence thesis;

          end;

            suppose

             A19: |.a.| > 0 ;

            then

            reconsider m = ((r2 / 2) / |.a.|) as positive Real by A18, XREAL_1: 139;

            take m;

            (r2 / 2) < r2 by A15, XREAL_1: 215, XREAL_1: 216;

            hence thesis by A19, XCMPLX_1: 87;

          end;

        end;

        then

        consider m be positive Real such that

         A20: ( |.a.| * m) < r2;

        reconsider B = ( Ball (X,m)) as Subset of T by A1, TOPMETR: 12;

        reconsider nr = (r2 / ( |.x.| + m)) as positive Real by A17, XREAL_1: 139;

        take nr, B;

        thus B is open & x in B by GOBOARD6: 1, GOBOARD6: 3;

        let s be Real;

        assume

         A21: |.(s - a).| < nr;

        let z be object;

        assume z in (s * B);

        then

        consider b be Element of T such that

         A22: z = (s * b) and

         A23: b in B;

        reconsider e = b, se = (s * b) as Point of E by A1, TOPMETR: 12;

        reconsider y = x, c = b as Element of ( REAL n) by EUCLID: 22;

        reconsider Y = y, C = c as Element of (n -tuples_on REAL );

        c = (C - (n |-> 0 )) by RVSUM_1: 32

        .= (C - (Y - Y)) by RVSUM_1: 37

        .= ((C - Y) + Y) by RVSUM_1: 41;

        then

         A24: |.c.| <= ( |.(c - y).| + |.y.|) by EUCLID: 12;

        

         A25: ( dist (X,e)) < m by A23, METRIC_1: 11;

        then |.(c - y).| < m by SPPOL_1: 5;

        then ( |.(c - y).| + |.y.|) <= (m + |.x.|) by XREAL_1: 6;

        then |.c.| <= (m + |.x.|) by A24, XXREAL_0: 2;

        then

         A26: (nr * |.c.|) <= (nr * (m + |.x.|)) by XREAL_1: 64;

        ((a * y) + ( - (a * c))) = ((a * y) + (( - 1) * (a * c)))

        .= ((a * y) + ((( - 1) * a) * c)) by RVSUM_1: 49

        .= ((a * y) + (a * (( - 1) * c))) by RVSUM_1: 49

        .= (a * (y + (( - 1) * c))) by RVSUM_1: 51

        .= (a * (y + ( - c)))

        .= (a * (y - c));

        then

         A27: |.((a * y) + ( - (a * c))).| = ( |.a.| * |.(y - c).|) by EUCLID: 11;

         |.a.| >= 0 & |.(y - c).| = ( dist (X,e)) by COMPLEX1: 46, SPPOL_1: 5;

        then |.((a * y) + ( - (a * c))).| <= ( |.a.| * m) by A25, A27, XREAL_1: 64;

        then

         A28: |.((a * y) + ( - (a * c))).| < r2 by A20, XXREAL_0: 2;

        ((a * c) + ( - (s * c))) = ((a * c) + (( - 1) * (s * c)))

        .= ((a * c) + ((( - 1) * s) * c)) by RVSUM_1: 49

        .= ((a + (( - 1) * s)) * c) by RVSUM_1: 50;

        

        then |.((a * c) + ( - (s * c))).| = ( |.(a - s).| * |.c.|) by EUCLID: 11

        .= ( |.( - (a - s)).| * |.c.|) by COMPLEX1: 52;

        then (nr * ( |.x.| + m)) = r2 & |.((a * c) + ( - (s * c))).| <= (nr * |.c.|) by A21, XCMPLX_1: 87, XREAL_1: 64;

        then |.((a * c) + ( - (s * c))).| <= r2 by A26, XXREAL_0: 2;

        then

         A29: |.(((a * y) + ( - (a * c))) + ((a * c) + ( - (s * c)))).| <= ( |.((a * y) + ( - (a * c))).| + |.((a * c) + ( - (s * c))).|) & ( |.((a * y) + ( - (a * c))).| + |.((a * c) + ( - (s * c))).|) < (r2 + r2) by A28, EUCLID: 12, XREAL_1: 8;

        ((a * y) - (s * c)) = (((a * Y) - (n |-> 0 )) - (s * C)) by RVSUM_1: 32

        .= (((a * y) - ((a * C) - (a * C))) - (s * c)) by RVSUM_1: 37

        .= ((((a * y) - (a * C)) + (a * C)) - (s * c)) by RVSUM_1: 41

        .= ((((a * y) - (a * C)) + (a * C)) + ( - (s * c)))

        .= (((a * y) - (a * C)) + ((a * c) + ( - (s * c)))) by RVSUM_1: 15

        .= (((a * y) + ( - (a * c))) + ((a * c) + ( - (s * c))));

        then ( dist (AX,se)) = |.(((a * y) + ( - (a * c))) + ((a * c) + ( - (s * c)))).| by SPPOL_1: 5;

        then ( dist (AX,se)) < r by A29, XXREAL_0: 2;

        then se in ( Ball (AX,r)) by METRIC_1: 11;

        hence z in V by A16, A22;

      end;

    end

    begin

    reserve m,n,i,i2,j for Nat,

r,r1,r2,s,t for Real,

x,y,z for object;

    ::$Canceled

    theorem :: JORDAN2C:6

    

     Th1: for f be increasing FinSequence of REAL st ( rng f) = {r, s} & ( len f) = 2 & r <= s holds (f . 1) = r & (f . 2) = s

    proof

      let f be increasing FinSequence of REAL ;

      assume that

       A1: ( rng f) = {r, s} and

       A2: ( len f) = 2 and

       A3: r <= s;

      now

        

         A4: 2 in ( dom f) by A2, FINSEQ_3: 25;

        

         A5: 1 in ( dom f) by A2, FINSEQ_3: 25;

        assume (f . 1) = s & (f . 2) = r;

        hence thesis by A3, A5, A4, SEQM_3:def 1;

      end;

      hence thesis by A1, A2, FINSEQ_3: 151;

    end;

    reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of ( TOP-REAL n);

    ::$Canceled

    theorem :: JORDAN2C:8

     |. |.q.|.| = |.q.| by ABSVALUE:def 1;

    theorem :: JORDAN2C:9

    

     Th3: |.( |.q1.| - |.q2.|).| <= |.(q1 - q2).|

    proof

      per cases ;

        suppose |.q1.| >= |.q2.|;

        then ( |.q1.| - |.q2.|) >= 0 by XREAL_1: 48;

        then ( |.q1.| - |.q2.|) = |.( |.q1.| - |.q2.|).| by ABSVALUE:def 1;

        hence thesis by TOPRNS_1: 32;

      end;

        suppose

         A1: |.q1.| < |.q2.|;

        

         A2: ( |.q2.| - |.q1.|) <= |.(q2 - q1).| by TOPRNS_1: 32;

        ( |.q2.| - |.q1.|) > 0 by A1, XREAL_1: 50;

        then |.( |.q2.| - |.q1.|).| <= |.(q2 - q1).| by A2, ABSVALUE:def 1;

        then |.( |.q2.| - |.q1.|).| <= |.(q1 - q2).| by TOPRNS_1: 27;

        hence thesis by UNIFORM1: 11;

      end;

    end;

    theorem :: JORDAN2C:10

    

     Th4: |. |[r]|.| = |.r.|

    proof

      set p = |[r]|;

      reconsider w = |[r]| as Element of ( REAL 1) by EUCLID: 22;

      reconsider r2 = (r ^2 ) as Element of REAL by XREAL_0:def 1;

      ( sqr w) = <*r2*> by RVSUM_1: 55;

      

      then |.p.| = ( sqrt r2) by FINSOP_1: 11

      .= |.r.| by COMPLEX1: 72;

      hence thesis;

    end;

    

     Lm1: for n be Nat, r be Real st r > 0 holds for x,y,z be Element of ( Euclid n) st x = ( 0* n) holds for p be Element of ( TOP-REAL n) st p = y & (r * p) = z holds (r * ( dist (x,y))) = ( dist (x,z))

    proof

      let n be Nat, r be Real such that

       A1: r > 0 ;

      let x,y,z be Element of ( Euclid n) such that

       A2: x = ( 0* n);

      let p be Element of ( TOP-REAL n) such that

       A3: p = y and

       A4: (r * p) = z;

      reconsider x1 = x, y1 = y as Element of ( REAL n);

      

       A5: ( dist (x,z)) = (( Pitag_dist n) . (x,z))

      .= |.(x1 - (r * y1)).| by A3, A4, EUCLID:def 6;

      

       A6: (r * x1) = (n |-> ( 0 * r)) by A2, RVSUM_1: 48

      .= x1 by A2;

      ( dist (x,y)) = (( Pitag_dist n) . (x,y))

      .= |.(x1 - y1).| by EUCLID:def 6;

      

      hence (r * ( dist (x,y))) = ( |.r.| * |.(x1 - y1).|) by A1, ABSVALUE:def 1

      .= |.(r * (x1 - y1)).| by EUCLID: 11

      .= |.((r * x1) + (r * ( - y1))).| by RVSUM_1: 51

      .= |.((r * x1) + ((( - 1) * r) * y1)).| by RVSUM_1: 49

      .= ( dist (x,z)) by A5, A6, RVSUM_1: 49;

    end;

    

     Lm2: for n be Nat, r,s be Real st r > 0 holds for x be Element of ( Euclid n) st x = ( 0* n) holds for A be Subset of ( TOP-REAL n) st A = ( Ball (x,s)) holds (r * A) = ( Ball (x,(r * s)))

    proof

      let n be Nat, r,s be Real such that

       A1: r > 0 ;

      let x be Element of ( Euclid n) such that

       A2: x = ( 0* n);

      let A be Subset of ( TOP-REAL n) such that

       A3: A = ( Ball (x,s));

      thus (r * A) c= ( Ball (x,(r * s)))

      proof

        let y be object;

        assume y in (r * A);

        then

        consider v be Element of ( TOP-REAL n) such that

         A4: y = (r * v) and

         A5: v in A;

        v in { q where q be Element of ( Euclid n) : ( dist (x,q)) < s } by A5, A3, METRIC_1:def 14;

        then

        consider q be Element of ( Euclid n) such that

         A6: v = q and

         A7: ( dist (x,q)) < s;

        reconsider p = y as Element of ( Euclid n) by A4, EUCLID: 67;

        (r * ( dist (x,q))) = ( dist (x,p)) by A1, A2, A6, A4, Lm1;

        then ( dist (x,p)) < (r * s) by A7, A1, XREAL_1: 68;

        then y in { e where e be Element of ( Euclid n) : ( dist (x,e)) < (r * s) };

        hence y in ( Ball (x,(r * s))) by METRIC_1:def 14;

      end;

      let y be object;

      assume y in ( Ball (x,(r * s)));

      then y in { q where q be Element of ( Euclid n) : ( dist (x,q)) < (r * s) } by METRIC_1:def 14;

      then

      consider z be Element of ( Euclid n) such that

       A8: y = z and

       A9: ( dist (x,z)) < (r * s);

      reconsider q = z as Element of ( TOP-REAL n) by EUCLID: 67;

      set p = ((r " ) * q);

      

       A10: y = (1 * q) by A8, RVSUM_1: 52

      .= (((r " ) * r) * q) by A1, XCMPLX_0:def 7

      .= (r * p) by RVSUM_1: 49;

      reconsider f = p as Element of ( Euclid n) by EUCLID: 67;

      

       A11: ( dist (x,f)) = ((r " ) * ( dist (x,z))) by A1, A2, Lm1;

      s = (1 * s)

      .= (((r " ) * ((r " ) " )) * s) by A1, XCMPLX_0:def 7

      .= ((r " ) * (r * s));

      then ( dist (x,f)) < s by A9, A11, A1, XREAL_1: 68;

      then p in { e where e be Element of ( Euclid n) : ( dist (x,e)) < s };

      then p in A by A3, METRIC_1:def 14;

      hence y in (r * A) by A10;

    end;

    

     Lm3: for n be Nat, r,s,t be Real st 0 < s & s <= t holds for x be Element of ( Euclid n) st x = ( 0* n) holds for BA be Subset of ( TOP-REAL n) st BA = ( Ball (x,r)) holds (s * BA) c= (t * BA)

    proof

      let n be Nat, r,s,t be Real such that

       A1: 0 < s and

       A2: s <= t;

      let x be Element of ( Euclid n) such that

       A3: x = ( 0* n);

      let BA be Subset of ( TOP-REAL n) such that

       A4: BA = ( Ball (x,r));

      let e be object;

      assume e in (s * BA);

      then

      consider w be Element of ( TOP-REAL n) such that

       A5: e = (s * w) and

       A6: w in BA;

      w in { q where q be Element of ( Euclid n) : ( dist (x,q)) < r } by A6, A4, METRIC_1:def 14;

      then

      consider q be Element of ( Euclid n) such that

       A7: w = q and

       A8: ( dist (x,q)) < r;

      set p = ((s / t) * w);

      

       A9: e = (s * w) by A5

      .= ((t * (s / t)) * w) by A1, A2, XCMPLX_1: 87

      .= (t * ((s / t) * w)) by RVSUM_1: 49

      .= (t * p);

      reconsider y = p as Element of ( Euclid n) by EUCLID: 67;

      

       A10: ( dist (x,y)) = ((s / t) * ( dist (x,q))) by A3, A7, Lm1, A1, A2, XREAL_1: 139;

      (s / t) <= 1 by A1, A2, XREAL_1: 183;

      then ( dist (x,y)) <= ( dist (x,q)) by A10, METRIC_1: 5, XREAL_1: 153;

      then ( dist (x,y)) < r by A8, XXREAL_0: 2;

      then p in { f where f be Element of ( Euclid n) : ( dist (x,f)) < r };

      then p in BA by A4, METRIC_1:def 14;

      hence e in (t * BA) by A9;

    end;

    theorem :: JORDAN2C:11

    

     Th5: for n be Nat, A be Subset of ( TOP-REAL n) holds A is bounded iff A is bounded Subset of ( Euclid n)

    proof

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

      reconsider z = ( 0* n) as Element of ( Euclid n);

      thus A is bounded implies A is bounded Subset of ( Euclid n)

      proof

        assume

         A1: A is bounded;

        reconsider B = A as Subset of ( Euclid n) by EUCLID: 67;

        z = ( 0. ( TOP-REAL n)) by EUCLID: 70;

        then

        reconsider V = ( Ball (z,1)) as a_neighborhood of ( 0. ( TOP-REAL n)) by GOBOARD6: 2;

        consider s be Real such that

         A2: s > 0 and

         A3: for t be Real st t > s holds A c= (t * V) by A1;

        set r = (s + 1);

         0 < r by A2;

        then (r * V) = ( Ball (z,(r * 1))) by Lm2;

        then B c= ( Ball (z,r)) by A3, XREAL_1: 29;

        hence A is bounded Subset of ( Euclid n) by A2, METRIC_6:def 3;

      end;

      assume

       A4: A is bounded Subset of ( Euclid n);

      then

      reconsider B = A as Subset of ( Euclid n);

      consider r1 be Real such that

       A5: 0 < r1 and

       A6: B c= ( Ball (z,r1)) by A4, METRIC_6: 29;

      let V be a_neighborhood of ( 0. ( TOP-REAL n));

      ( 0. ( TOP-REAL n)) = ( 0* n) by EUCLID: 70;

      then z in ( Int V) by CONNSP_2:def 1;

      then

      consider r2 be Real such that

       A7: r2 > 0 and

       A8: ( Ball (z,r2)) c= V by GOBOARD6: 5;

      reconsider r2 as Real;

      take s = (r1 / r2);

      thus

       A9: s > 0 by A5, A7, XREAL_1: 139;

      let t;

      reconsider BA = ( Ball (z,r2)) as Subset of ( TOP-REAL n) by EUCLID: 67;

      (s * r2) = r1 by A7, XCMPLX_1: 87;

      then

       A10: A c= (s * BA) by A6, A9, Lm2;

      assume t > s;

      then (s * BA) c= (t * BA) by A9, Lm3;

      then

       A11: A c= (t * BA) by A10;

      (t * BA) c= (t * V) by A8, CONVEX1: 39;

      hence A c= (t * V) by A11;

    end;

    theorem :: JORDAN2C:12

    for A,B be Subset of ( TOP-REAL n) st B is bounded & A c= B holds A is bounded by RLTOPSP1: 42;

    definition

      ::$Canceled

      let n be Nat;

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

      :: JORDAN2C:def2

      pred B is_inside_component_of A means B is_a_component_of (A ` ) & B is bounded;

    end

    registration

      let M be non empty MetrStruct;

      cluster bounded for Subset of M;

      existence

      proof

        take ( {} M), 1;

        thus thesis;

      end;

    end

    theorem :: JORDAN2C:13

    

     Th7: for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) holds B is_inside_component_of A iff ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n)

    proof

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

      

       A1: B is_a_component_of (A ` ) iff ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component by CONNSP_1:def 6;

      thus B is_inside_component_of A implies ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n) by Th5, A1;

      given C be Subset of (( TOP-REAL n) | (A ` )) such that

       A2: C = B & C is a_component & C is bounded Subset of ( Euclid n);

      B is bounded & B is_a_component_of (A ` ) by A2, Th5, CONNSP_1:def 6;

      hence thesis;

    end;

    definition

      let n be Nat;

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

      :: JORDAN2C:def3

      pred B is_outside_component_of A means B is_a_component_of (A ` ) & not B is bounded;

    end

    theorem :: JORDAN2C:14

    

     Th8: for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) holds B is_outside_component_of A iff ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & not C is bounded Subset of ( Euclid n)

    proof

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

      

       A1: B is_a_component_of (A ` ) iff ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component by CONNSP_1:def 6;

      thus B is_outside_component_of A implies ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & not C is bounded Subset of ( Euclid n)

      proof

        reconsider D2 = B as Subset of ( Euclid n) by TOPREAL3: 8;

        assume

         A2: B is_outside_component_of A;

        then

        consider C be Subset of (( TOP-REAL n) | (A ` )) such that

         A3: C = B and

         A4: C is a_component by A1;

        now

          assume for D be Subset of ( Euclid n) st D = C holds D is bounded;

          then D2 is bounded by A3;

          hence contradiction by A2, Th5;

        end;

        hence thesis by A3, A4;

      end;

      given C be Subset of (( TOP-REAL n) | (A ` )) such that

       A5: C = B & C is a_component & not C is bounded Subset of ( Euclid n);

      ( not B is bounded) & B is_a_component_of (A ` ) by A5, Th5, CONNSP_1:def 6;

      hence thesis;

    end;

    theorem :: JORDAN2C:15

    for A,B be Subset of ( TOP-REAL n) st B is_inside_component_of A holds B c= (A ` ) by SPRECT_1: 5;

    theorem :: JORDAN2C:16

    for A,B be Subset of ( TOP-REAL n) st B is_outside_component_of A holds B c= (A ` ) by SPRECT_1: 5;

    definition

      let n be Nat;

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

      :: JORDAN2C:def4

      func BDD A -> Subset of ( TOP-REAL n) equals ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A });

      correctness

      proof

        ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) c= the carrier of ( TOP-REAL n)

        proof

          let x be object;

          assume x in ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A });

          then

          consider y be set such that

           A1: x in y and

           A2: y in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } by TARSKI:def 4;

          ex B be Subset of ( TOP-REAL n) st y = B & B is_inside_component_of A by A2;

          hence thesis by A1;

        end;

        hence thesis;

      end;

    end

    definition

      let n be Nat;

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

      :: JORDAN2C:def5

      func UBD A -> Subset of ( TOP-REAL n) equals ( union { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A });

      correctness

      proof

        ( union { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A }) c= the carrier of ( TOP-REAL n)

        proof

          let x be object;

          assume x in ( union { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A });

          then

          consider y be set such that

           A1: x in y and

           A2: y in { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A } by TARSKI:def 4;

          ex B be Subset of ( TOP-REAL n) st y = B & B is_outside_component_of A by A2;

          hence thesis by A1;

        end;

        hence thesis;

      end;

    end

    registration

      let n be Nat;

      cluster ( [#] ( TOP-REAL n)) -> convex;

      coherence ;

    end

    registration

      let n;

      cluster ( [#] ( TOP-REAL n)) -> a_component;

      coherence

      proof

        set A = ( [#] ( TOP-REAL n));

        for B be Subset of ( TOP-REAL n) st B is connected holds A c= B implies A = B;

        hence thesis by CONNSP_1:def 5;

      end;

    end

    ::$Canceled

    theorem :: JORDAN2C:20

    

     Th11: for A be Subset of ( TOP-REAL n) holds ( BDD A) is a_union_of_components of (( TOP-REAL n) | (A ` ))

    proof

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

      { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } c= ( bool the carrier of (( TOP-REAL n) | (A ` )))

      proof

        let x be object;

        assume x in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A };

        then

        consider B be Subset of ( TOP-REAL n) such that

         A1: x = B and

         A2: B is_inside_component_of A;

        ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n) by A2, Th7;

        hence thesis by A1;

      end;

      then

      reconsider F0 = { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } as Subset-Family of the carrier of (( TOP-REAL n) | (A ` ));

      reconsider F0 as Subset-Family of (( TOP-REAL n) | (A ` ));

      

       A3: for B0 be Subset of (( TOP-REAL n) | (A ` )) st B0 in F0 holds B0 is a_component

      proof

        let B0 be Subset of (( TOP-REAL n) | (A ` ));

        assume B0 in F0;

        then

        consider B be Subset of ( TOP-REAL n) such that

         A4: B = B0 and

         A5: B is_inside_component_of A;

        ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n) by A5, Th7;

        hence thesis by A4;

      end;

      ( BDD A) = ( union F0);

      hence thesis by A3, CONNSP_3:def 2;

    end;

    theorem :: JORDAN2C:21

    

     Th12: for A be Subset of ( TOP-REAL n) holds ( UBD A) is a_union_of_components of (( TOP-REAL n) | (A ` ))

    proof

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

      { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A } c= ( bool the carrier of (( TOP-REAL n) | (A ` )))

      proof

        let x be object;

        assume x in { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A };

        then

        consider B be Subset of ( TOP-REAL n) such that

         A1: x = B and

         A2: B is_outside_component_of A;

        ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & not C is bounded Subset of ( Euclid n) by A2, Th8;

        hence thesis by A1;

      end;

      then

      reconsider F0 = { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A } as Subset-Family of the carrier of (( TOP-REAL n) | (A ` ));

      reconsider F0 as Subset-Family of (( TOP-REAL n) | (A ` ));

      

       A3: for B0 be Subset of (( TOP-REAL n) | (A ` )) st B0 in F0 holds B0 is a_component

      proof

        let B0 be Subset of (( TOP-REAL n) | (A ` ));

        assume B0 in F0;

        then

        consider B be Subset of ( TOP-REAL n) such that

         A4: B = B0 and

         A5: B is_outside_component_of A;

        ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & not C is bounded Subset of ( Euclid n) by A5, Th8;

        hence thesis by A4;

      end;

      ( UBD A) = ( union F0);

      hence thesis by A3, CONNSP_3:def 2;

    end;

    theorem :: JORDAN2C:22

    

     Th13: for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) st B is_inside_component_of A holds B c= ( BDD A)

    proof

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

      assume B is_inside_component_of A;

      then

       A1: B in { B2 where B2 be Subset of ( TOP-REAL n) : B2 is_inside_component_of A };

      let x be object;

      assume x in B;

      hence thesis by A1, TARSKI:def 4;

    end;

    theorem :: JORDAN2C:23

    

     Th14: for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) st B is_outside_component_of A holds B c= ( UBD A)

    proof

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

      assume B is_outside_component_of A;

      then

       A1: B in { B2 where B2 be Subset of ( TOP-REAL n) : B2 is_outside_component_of A };

      let x be object;

      assume x in B;

      hence thesis by A1, TARSKI:def 4;

    end;

    theorem :: JORDAN2C:24

    

     Th15: for A be Subset of ( TOP-REAL n) holds ( BDD A) misses ( UBD A)

    proof

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

      set x = the Element of (( BDD A) /\ ( UBD A));

      assume

       A1: (( BDD A) /\ ( UBD A)) <> {} ;

      then x in ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) by XBOOLE_0:def 4;

      then

      consider y be set such that

       A2: x in y and

       A3: y in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } by TARSKI:def 4;

      x in ( union { B2 where B2 be Subset of ( TOP-REAL n) : B2 is_outside_component_of A }) by A1, XBOOLE_0:def 4;

      then

      consider y2 be set such that

       A4: x in y2 and

       A5: y2 in { B2 where B2 be Subset of ( TOP-REAL n) : B2 is_outside_component_of A } by TARSKI:def 4;

      consider B be Subset of ( TOP-REAL n) such that

       A6: y = B and

       A7: B is_inside_component_of A by A3;

      consider B2 be Subset of ( TOP-REAL n) such that

       A8: y2 = B2 and

       A9: B2 is_outside_component_of A by A5;

      consider C be Subset of (( TOP-REAL n) | (A ` )) such that

       A10: C = B and

       A11: C is a_component & C is bounded Subset of ( Euclid n) by A7, Th7;

      consider C2 be Subset of (( TOP-REAL n) | (A ` )) such that

       A12: C2 = B2 and

       A13: C2 is a_component & not C2 is bounded Subset of ( Euclid n) by A9, Th8;

      (C /\ C2) <> ( {} (( TOP-REAL n) | (A ` ))) by A2, A6, A10, A4, A8, A12, XBOOLE_0:def 4;

      then C meets C2;

      hence contradiction by A11, A13, CONNSP_1: 35;

    end;

    theorem :: JORDAN2C:25

    

     Th16: for A be Subset of ( TOP-REAL n) holds ( BDD A) c= (A ` )

    proof

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

      reconsider D = ( BDD A) as Subset of (( TOP-REAL n) | (A ` )) by Th11;

      D c= the carrier of (( TOP-REAL n) | (A ` ));

      hence thesis by PRE_TOPC: 8;

    end;

    theorem :: JORDAN2C:26

    

     Th17: for A be Subset of ( TOP-REAL n) holds ( UBD A) c= (A ` )

    proof

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

      reconsider D = ( UBD A) as Subset of (( TOP-REAL n) | (A ` )) by Th12;

      D c= the carrier of (( TOP-REAL n) | (A ` ));

      hence thesis by PRE_TOPC: 8;

    end;

    theorem :: JORDAN2C:27

    

     Th18: for A be Subset of ( TOP-REAL n) holds (( BDD A) \/ ( UBD A)) = (A ` )

    proof

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

      

       A1: (A ` ) c= (( BDD A) \/ ( UBD A))

      proof

        let z be object;

        assume

         A2: z in (A ` );

        then

        reconsider p = z as Element of (A ` );

        reconsider B = (A ` ) as non empty Subset of ( TOP-REAL n) by A2;

        reconsider q = p as Point of (( TOP-REAL n) | (A ` )) by PRE_TOPC: 8;

        ( Component_of q) is Subset of ( [#] (( TOP-REAL n) | (A ` )));

        then ( Component_of q) is Subset of (A ` ) by PRE_TOPC:def 5;

        then

        reconsider G = ( Component_of q) as Subset of ( TOP-REAL n) by XBOOLE_1: 1;

        

         A3: (( TOP-REAL n) | B) is non empty;

        then

         A4: q in G by CONNSP_1: 38;

        ( Component_of q) is a_component by A3, CONNSP_1: 40;

        then

         A5: G is_a_component_of (A ` ) by CONNSP_1:def 6;

        per cases ;

          suppose G is bounded;

          then G is_inside_component_of A by A5;

          then G c= ( BDD A) by Th13;

          hence thesis by A4, XBOOLE_0:def 3;

        end;

          suppose not G is bounded;

          then G is_outside_component_of A by A5;

          then G c= ( UBD A) by Th14;

          hence thesis by A4, XBOOLE_0:def 3;

        end;

      end;

      ( BDD A) c= (A ` ) & ( UBD A) c= (A ` ) by Th16, Th17;

      then (( BDD A) \/ ( UBD A)) c= (A ` ) by XBOOLE_1: 8;

      hence thesis by A1;

    end;

    reserve u for Point of ( Euclid n);

    theorem :: JORDAN2C:28

    

     Th19: for P be Subset of ( TOP-REAL n) st P = ( REAL n) holds P is connected

    proof

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

      assume

       A1: P = ( REAL n);

      for p1,p2 be Point of ( TOP-REAL n) st p1 in P & p2 in P holds ( LSeg (p1,p2)) c= P

      proof

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

        assume that p1 in P and p2 in P;

        the carrier of ( TOP-REAL n) = ( REAL n) by EUCLID: 22;

        hence thesis by A1;

      end;

      then P is convex by JORDAN1:def 1;

      hence thesis;

    end;

    ::$Canceled

    theorem :: JORDAN2C:33

    

     Th20: for W be Subset of ( Euclid n) st n >= 1 & W = ( REAL n) holds not W is bounded

    proof

      let W be Subset of ( Euclid n);

      assume that

       A1: n >= 1 and

       A2: W = ( REAL n);

      reconsider y0 = ( 0. ( TOP-REAL n)) as Point of ( Euclid n) by EUCLID: 67;

      assume W is bounded;

      then

      consider r be Real such that

       A3: 0 < r and

       A4: for x,y be Point of ( Euclid n) st x in W & y in W holds ( dist (x,y)) <= r;

      reconsider x0 = ((r + 1) * ( 1.REAL n)) as Point of ( Euclid n) by TOPREAL3: 8;

      ( dist (x0,y0)) <= r by A2, A4;

      then |.(((r + 1) * ( 1.REAL n)) - ( 0. ( TOP-REAL n))).| <= r by JGRAPH_1: 28;

      then |.((r + 1) * ( 1.REAL n)).| <= r by RLVECT_1: 13;

      then ( |.(r + 1).| * |.( 1.REAL n).|) <= r by TOPRNS_1: 7;

      then ( |.(r + 1).| * ( sqrt n)) <= r by EUCLID: 73;

      then

       A5: ((r + 1) * ( sqrt n)) <= r by A3, ABSVALUE:def 1;

      ( sqrt 1) <= ( sqrt n) by A1, SQUARE_1: 26;

      then ((r + 1) * 1) <= ((r + 1) * ( sqrt n)) by A3, SQUARE_1: 18, XREAL_1: 64;

      then ((r + 1) * 1) <= r by A5, XXREAL_0: 2;

      then ((r + 1) - r) <= (r - r) by XREAL_1: 9;

      then 1 <= 0 ;

      hence contradiction;

    end;

    theorem :: JORDAN2C:34

    

     Th21: for A be Subset of ( TOP-REAL n) holds A is bounded iff ex r be Real st for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r

    proof

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

      reconsider C = A as Subset of ( Euclid n) by TOPREAL3: 8;

      hereby

        assume A is bounded;

        then

        reconsider C = A as bounded Subset of ( Euclid n) by Th5;

        per cases ;

          suppose

           A1: C <> {} ;

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

          set x0 = the Element of C;

          x0 in C by A1;

          then

          reconsider x0 as Point of ( Euclid n);

          consider r be Real such that 0 < r and

           A2: for x,y be Point of ( Euclid n) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

          set R0 = ((r + ( dist (o,x0))) + 1);

          for q be Point of ( TOP-REAL n) st q in A holds |.q.| < R0

          proof

            let q1 be Point of ( TOP-REAL n);

            reconsider z = q1 as Point of ( Euclid n) by TOPREAL3: 8;

             |.(q1 - ( 0. ( TOP-REAL n))).| = ( dist (o,z)) by JGRAPH_1: 28;

            then

             A3: |.q1.| = ( dist (o,z)) by RLVECT_1: 13;

            assume q1 in A;

            then ( dist (x0,z)) <= r by A2;

            then ( dist (o,z)) <= (( dist (o,x0)) + ( dist (x0,z))) & (( dist (o,x0)) + ( dist (x0,z))) <= (( dist (o,x0)) + r) by METRIC_1: 4, XREAL_1: 6;

            then

             A4: ( dist (o,z)) <= (( dist (o,x0)) + r) by XXREAL_0: 2;

            (r + ( dist (o,x0))) < ((r + ( dist (o,x0))) + 1) by XREAL_1: 29;

            hence thesis by A3, A4, XXREAL_0: 2;

          end;

          hence ex r2 be Real st for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r2;

        end;

          suppose C = {} ;

          then for q be Point of ( TOP-REAL n) st q in A holds |.q.| < 1;

          hence ex r2 be Real st for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r2;

        end;

      end;

      given r be Real such that

       A5: for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r;

      now

        per cases ;

          suppose

           A6: C <> {} ;

          set x0 = the Element of C;

          x0 in C by A6;

          then

          reconsider x0 as Point of ( Euclid n);

          reconsider q0 = x0 as Point of ( TOP-REAL n) by TOPREAL3: 8;

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

          set R0 = (r + r);

          

           A7: for x,y be Point of ( Euclid n) st x in C & y in C holds ( dist (x,y)) <= R0

          proof

            let x,y be Point of ( Euclid n);

            assume that

             A8: x in C and

             A9: y in C;

            reconsider q2 = y as Point of ( TOP-REAL n) by A9;

            ( dist (o,y)) = |.(q2 - ( 0. ( TOP-REAL n))).| by JGRAPH_1: 28

            .= |.q2.| by RLVECT_1: 13;

            then

             A10: ( dist (o,y)) < r by A5, A9;

            reconsider q1 = x as Point of ( TOP-REAL n) by A8;

            ( dist (x,o)) = |.(q1 - ( 0. ( TOP-REAL n))).| by JGRAPH_1: 28

            .= |.q1.| by RLVECT_1: 13;

            then ( dist (x,o)) < r by A5, A8;

            then ( dist (x,y)) <= (( dist (x,o)) + ( dist (o,y))) & (( dist (x,o)) + ( dist (o,y))) <= (r + r) by A10, METRIC_1: 4, XREAL_1: 7;

            hence thesis by XXREAL_0: 2;

          end;

           |.q0.| < r by A5, A6;

          hence C is bounded by A7;

        end;

          suppose C = {} ;

          hence C is bounded;

        end;

      end;

      hence thesis by Th5;

    end;

    theorem :: JORDAN2C:35

    

     Th22: n >= 1 implies not ( [#] ( TOP-REAL n)) is bounded

    proof

      assume

       A1: n >= 1;

      assume ( [#] ( TOP-REAL n)) is bounded;

      then

      reconsider C = ( [#] ( TOP-REAL n)) as bounded Subset of ( Euclid n) by Th5;

      C = ( REAL n) by EUCLID: 22;

      hence contradiction by A1, Th20;

    end;

    theorem :: JORDAN2C:36

    

     Th23: n >= 1 implies ( UBD ( {} ( TOP-REAL n))) = ( REAL n)

    proof

      set A = ( {} ( TOP-REAL n));

      

       A1: (( TOP-REAL n) | ( [#] ( TOP-REAL n))) = the TopStruct of ( TOP-REAL n) by TSEP_1: 93;

      assume

       A2: n >= 1;

       A3:

      now

        reconsider D1 = ( [#] (( TOP-REAL n) | (A ` ))) as Subset of ( Euclid n) by A1, TOPREAL3: 8;

        assume for D be Subset of ( Euclid n) st D = ( [#] (( TOP-REAL n) | (A ` ))) holds D is bounded;

        then D1 is bounded;

        then ( [#] ( TOP-REAL n)) is bounded by A1, Th5;

        hence contradiction by A2, Th22;

      end;

      ( [#] (( TOP-REAL n) | (A ` ))) is a_component by A1, CONNSP_1: 45;

      then ( [#] ( TOP-REAL n)) is_outside_component_of ( {} ( TOP-REAL n)) by A1, A3, Th8;

      then

       A4: ( [#] ( TOP-REAL n)) in { B2 where B2 be Subset of ( TOP-REAL n) : B2 is_outside_component_of ( {} ( TOP-REAL n)) };

      ( UBD ( {} ( TOP-REAL n))) c= the carrier of ( TOP-REAL n);

      hence ( UBD ( {} ( TOP-REAL n))) c= ( REAL n) by EUCLID: 22;

      let x be object;

      assume x in ( REAL n);

      then x in ( [#] ( TOP-REAL n)) by EUCLID: 22;

      hence thesis by A4, TARSKI:def 4;

    end;

    theorem :: JORDAN2C:37

    

     Th24: for w1,w2,w3 be Point of ( TOP-REAL n), P be non empty Subset of ( TOP-REAL n), h1,h2 be Function of I[01] , (( TOP-REAL n) | P) st h1 is continuous & w1 = (h1 . 0 ) & w2 = (h1 . 1) & h2 is continuous & w2 = (h2 . 0 ) & w3 = (h2 . 1) holds ex h3 be Function of I[01] , (( TOP-REAL n) | P) st h3 is continuous & w1 = (h3 . 0 ) & w3 = (h3 . 1)

    proof

      let w1,w2,w3 be Point of ( TOP-REAL n), P be non empty Subset of ( TOP-REAL n), h1,h2 be Function of I[01] , (( TOP-REAL n) | P);

      assume that

       A1: h1 is continuous and

       A2: w1 = (h1 . 0 ) and

       A3: w2 = (h1 . 1) and

       A4: h2 is continuous and

       A5: w2 = (h2 . 0 ) and

       A6: w3 = (h2 . 1);

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

      then

      reconsider p1 = w1, p2 = w2, p3 = w3 as Point of (( TOP-REAL n) | P) by A2, A3, A6, BORSUK_1: 40, FUNCT_2: 5;

      (p2,p3) are_connected by A4, A5, A6, BORSUK_2:def 1;

      then

      reconsider P2 = h2 as Path of p2, p3 by A4, A5, A6, BORSUK_2:def 2;

      (p1,p2) are_connected by A1, A2, A3, BORSUK_2:def 1;

      then

      reconsider P1 = h1 as Path of p1, p2 by A1, A2, A3, BORSUK_2:def 2;

      ex P0 be Path of p1, p3 st P0 is continuous & (P0 . 0 ) = p1 & (P0 . 1) = p3 & for t be Point of I[01] , t9 be Real st t = t9 holds ( 0 <= t9 & t9 <= (1 / 2) implies (P0 . t) = (P1 . (2 * t9))) & ((1 / 2) <= t9 & t9 <= 1 implies (P0 . t) = (P2 . ((2 * t9) - 1)))

      proof

        (1 / 2) in { r : 0 <= r & r <= 1 };

        then

        reconsider pol = (1 / 2) as Point of I[01] by BORSUK_1: 40, RCOMP_1:def 1;

        reconsider T1 = ( Closed-Interval-TSpace ( 0 ,(1 / 2))), T2 = ( Closed-Interval-TSpace ((1 / 2),1)) as SubSpace of I[01] by TOPMETR: 20, TREAL_1: 3;

        set e2 = ( P[01] ((1 / 2),1,( (#) ( 0 ,1)),(( 0 ,1) (#) )));

        set e1 = ( P[01] ( 0 ,(1 / 2),( (#) ( 0 ,1)),(( 0 ,1) (#) )));

        set E1 = (P1 * e1);

        set E2 = (P2 * e2);

        set f = (E1 +* E2);

        

         A7: ( dom e1) = the carrier of ( Closed-Interval-TSpace ( 0 ,(1 / 2))) by FUNCT_2:def 1

        .= [. 0 , (1 / 2).] by TOPMETR: 18;

        

         A8: ( dom e2) = the carrier of ( Closed-Interval-TSpace ((1 / 2),1)) by FUNCT_2:def 1

        .= [.(1 / 2), 1.] by TOPMETR: 18;

        reconsider gg = E2 as Function of T2, (( TOP-REAL n) | P) by TOPMETR: 20;

        reconsider ff = E1 as Function of T1, (( TOP-REAL n) | P) by TOPMETR: 20;

        reconsider r1 = ( (#) ( 0 ,1)), r2 = (( 0 ,1) (#) ) as Real;

        

         A9: for t9 be Real st (1 / 2) <= t9 & t9 <= 1 holds (E2 . t9) = (P2 . ((2 * t9) - 1))

        proof

          ( dom e2) = the carrier of ( Closed-Interval-TSpace ((1 / 2),1)) by FUNCT_2:def 1;

          

          then

           A10: ( dom e2) = [.(1 / 2), 1.] by TOPMETR: 18

          .= { r : (1 / 2) <= r & r <= 1 } by RCOMP_1:def 1;

          let t9 be Real;

          assume (1 / 2) <= t9 & t9 <= 1;

          then

           A11: t9 in ( dom e2) by A10;

          then

          reconsider s = t9 as Point of ( Closed-Interval-TSpace ((1 / 2),1));

          (e2 . s) = ((((r2 - r1) / (1 - (1 / 2))) * t9) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2)))) by TREAL_1: 11

          .= ((2 * t9) - 1) by BORSUK_1:def 14, BORSUK_1:def 15, TREAL_1: 5;

          hence thesis by A11, FUNCT_1: 13;

        end;

        

         A12: for t9 be Real st 0 <= t9 & t9 <= (1 / 2) holds (E1 . t9) = (P1 . (2 * t9))

        proof

          ( dom e1) = the carrier of ( Closed-Interval-TSpace ( 0 ,(1 / 2))) by FUNCT_2:def 1;

          

          then

           A13: ( dom e1) = [. 0 , (1 / 2).] by TOPMETR: 18

          .= { r : 0 <= r & r <= (1 / 2) } by RCOMP_1:def 1;

          let t9 be Real;

          assume 0 <= t9 & t9 <= (1 / 2);

          then

           A14: t9 in ( dom e1) by A13;

          then

          reconsider s = t9 as Point of ( Closed-Interval-TSpace ( 0 ,(1 / 2)));

          (e1 . s) = ((((r2 - r1) / ((1 / 2) - 0 )) * t9) + ((((1 / 2) * r1) - ( 0 * r2)) / ((1 / 2) - 0 ))) by TREAL_1: 11

          .= (2 * t9) by BORSUK_1:def 14, BORSUK_1:def 15, TREAL_1: 5;

          hence thesis by A14, FUNCT_1: 13;

        end;

        

        then

         A15: (ff . (1 / 2)) = (P2 . ((2 * (1 / 2)) - 1)) by A3, A5

        .= (gg . pol) by A9;

        ( [#] T1) = [. 0 , (1 / 2).] & ( [#] T2) = [.(1 / 2), 1.] by TOPMETR: 18;

        then

         A16: (( [#] T1) \/ ( [#] T2)) = ( [#] I[01] ) & (( [#] T1) /\ ( [#] T2)) = {pol} by BORSUK_1: 40, XXREAL_1: 174, XXREAL_1: 418;

        ( rng f) c= (( rng E1) \/ ( rng E2)) by FUNCT_4: 17;

        then

         A17: ( rng f) c= the carrier of (( TOP-REAL n) | P) by XBOOLE_1: 1;

        

         A18: T1 is compact & T2 is compact by HEINE: 4;

        ( dom P1) = the carrier of I[01] by FUNCT_2:def 1;

        then

         A19: ( rng e1) c= ( dom P1) by TOPMETR: 20;

        ( dom P2) = the carrier of I[01] & ( rng e2) c= the carrier of ( Closed-Interval-TSpace ( 0 ,1)) by FUNCT_2:def 1;

        then

         A20: ( dom E2) = ( dom e2) by RELAT_1: 27, TOPMETR: 20;

         not 0 in { r : (1 / 2) <= r & r <= 1 }

        proof

          assume 0 in { r : (1 / 2) <= r & r <= 1 };

          then ex rr be Real st rr = 0 & (1 / 2) <= rr & rr <= 1;

          hence thesis;

        end;

        then not 0 in ( dom E2) by A8, A20, RCOMP_1:def 1;

        

        then

         A21: (f . 0 ) = (E1 . 0 ) by FUNCT_4: 11

        .= (P1 . (2 * 0 )) by A12

        .= p1 by A2;

        ( dom f) = (( dom E1) \/ ( dom E2)) by FUNCT_4:def 1

        .= ( [. 0 , (1 / 2).] \/ [.(1 / 2), 1.]) by A7, A8, A19, A20, RELAT_1: 27

        .= the carrier of I[01] by BORSUK_1: 40, XXREAL_1: 174;

        then

        reconsider f as Function of I[01] , (( TOP-REAL n) | P) by A17, FUNCT_2:def 1, RELSET_1: 4;

        e1 is continuous & e2 is continuous by TREAL_1: 12;

        then

        reconsider f as continuous Function of I[01] , (( TOP-REAL n) | P) by A1, A4, A15, A16, A18, COMPTS_1: 20, TOPMETR: 20;

        1 in { r : (1 / 2) <= r & r <= 1 };

        then 1 in ( dom E2) by A8, A20, RCOMP_1:def 1;

        

        then

         A22: (f . 1) = (E2 . 1) by FUNCT_4: 13

        .= (P2 . ((2 * 1) - 1)) by A9

        .= p3 by A6;

        then (p1,p3) are_connected by A21, BORSUK_2:def 1;

        then

        reconsider f as Path of p1, p3 by A21, A22, BORSUK_2:def 2;

        for t be Point of I[01] , t9 be Real st t = t9 holds ( 0 <= t9 & t9 <= (1 / 2) implies (f . t) = (P1 . (2 * t9))) & ((1 / 2) <= t9 & t9 <= 1 implies (f . t) = (P2 . ((2 * t9) - 1)))

        proof

          let t be Point of I[01] , t9 be Real;

          assume

           A23: t = t9;

          thus 0 <= t9 & t9 <= (1 / 2) implies (f . t) = (P1 . (2 * t9))

          proof

            assume

             A24: 0 <= t9 & t9 <= (1 / 2);

            then t9 in { r : 0 <= r & r <= (1 / 2) };

            then

             A25: t9 in [. 0 , (1 / 2).] by RCOMP_1:def 1;

            per cases ;

              suppose

               A26: t9 <> (1 / 2);

               not t9 in ( dom E2)

              proof

                assume t9 in ( dom E2);

                then t9 in ( [. 0 , (1 / 2).] /\ [.(1 / 2), 1.]) by A8, A20, A25, XBOOLE_0:def 4;

                then t9 in {(1 / 2)} by XXREAL_1: 418;

                hence thesis by A26, TARSKI:def 1;

              end;

              

              then (f . t) = (E1 . t) by A23, FUNCT_4: 11

              .= (P1 . (2 * t9)) by A12, A23, A24;

              hence thesis;

            end;

              suppose

               A27: t9 = (1 / 2);

              (1 / 2) in { r : (1 / 2) <= r & r <= 1 };

              then (1 / 2) in [.(1 / 2), 1.] by RCOMP_1:def 1;

              then (1 / 2) in the carrier of ( Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR: 18;

              then t in ( dom E2) by A23, A27, FUNCT_2:def 1, TOPMETR: 20;

              

              then (f . t) = (E2 . (1 / 2)) by A23, A27, FUNCT_4: 13

              .= (P1 . (2 * t9)) by A12, A15, A27;

              hence thesis;

            end;

          end;

          thus (1 / 2) <= t9 & t9 <= 1 implies (f . t) = (P2 . ((2 * t9) - 1))

          proof

            assume

             A28: (1 / 2) <= t9 & t9 <= 1;

            then t9 in { r : (1 / 2) <= r & r <= 1 };

            then t9 in [.(1 / 2), 1.] by RCOMP_1:def 1;

            

            then (f . t) = (E2 . t) by A8, A20, A23, FUNCT_4: 13

            .= (P2 . ((2 * t9) - 1)) by A9, A23, A28;

            hence thesis;

          end;

        end;

        hence thesis by A21, A22;

      end;

      hence thesis;

    end;

    theorem :: JORDAN2C:38

    

     Th25: for P be Subset of ( TOP-REAL n), w1,w2,w3 be Point of ( TOP-REAL n) st w1 in P & w2 in P & w3 in P & ( LSeg (w1,w2)) c= P & ( LSeg (w2,w3)) c= P holds ex h be Function of I[01] , (( TOP-REAL n) | P) st h is continuous & w1 = (h . 0 ) & w3 = (h . 1)

    proof

      let P be Subset of ( TOP-REAL n), w1,w2,w3 be Point of ( TOP-REAL n);

      assume that

       A1: w1 in P and

       A2: w2 in P and

       A3: w3 in P and

       A4: ( LSeg (w1,w2)) c= P and

       A5: ( LSeg (w2,w3)) c= P;

      reconsider Y = P as non empty Subset of ( TOP-REAL n) by A1;

      per cases ;

        suppose

         A6: w1 <> w2;

        then ( LSeg (w1,w2)) is_an_arc_of (w1,w2) by TOPREAL1: 9;

        then

        consider f be Function of I[01] , (( TOP-REAL n) | ( LSeg (w1,w2))) such that

         A7: f is being_homeomorphism and

         A8: (f . 0 ) = w1 and

         A9: (f . 1) = w2 by TOPREAL1:def 1;

        

         A10: ( rng f) = ( [#] (( TOP-REAL n) | ( LSeg (w1,w2)))) by A7;

        then

         A11: ( rng f) c= P by A4, PRE_TOPC:def 5;

        then ( [#] (( TOP-REAL n) | ( LSeg (w1,w2)))) c= ( [#] (( TOP-REAL n) | P)) by A10, PRE_TOPC:def 5;

        then

         A12: (( TOP-REAL n) | ( LSeg (w1,w2))) is SubSpace of (( TOP-REAL n) | P) by TOPMETR: 3;

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

        then

        reconsider g = f as Function of [. 0 , 1.], P by A11, FUNCT_2: 2;

        reconsider gt = g as Function of I[01] , (( TOP-REAL n) | Y) by BORSUK_1: 40, PRE_TOPC: 8;

        

         A13: f is continuous by A7;

        now

          per cases ;

            suppose w2 <> w3;

            then ( LSeg (w2,w3)) is_an_arc_of (w2,w3) by TOPREAL1: 9;

            then

            consider f2 be Function of I[01] , (( TOP-REAL n) | ( LSeg (w2,w3))) such that

             A14: f2 is being_homeomorphism and

             A15: (f2 . 0 ) = w2 & (f2 . 1) = w3 by TOPREAL1:def 1;

            

             A16: ( rng f2) = ( [#] (( TOP-REAL n) | ( LSeg (w2,w3)))) by A14;

            then

             A17: ( rng f2) c= P by A5, PRE_TOPC:def 5;

            then ( [#] (( TOP-REAL n) | ( LSeg (w2,w3)))) c= ( [#] (( TOP-REAL n) | P)) by A16, PRE_TOPC:def 5;

            then

             A18: (( TOP-REAL n) | ( LSeg (w2,w3))) is SubSpace of (( TOP-REAL n) | P) by TOPMETR: 3;

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

            then

            reconsider w19 = w1, w29 = w2, w39 = w3 as Point of (( TOP-REAL n) | P) by A1, A2, A3;

            

             A19: gt is continuous & w29 = (gt . 1) by A9, A13, A12, PRE_TOPC: 26;

            ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

            then

            reconsider g2 = f2 as Function of [. 0 , 1.], P by A17, FUNCT_2: 2;

            reconsider gt2 = g2 as Function of I[01] , (( TOP-REAL n) | Y) by BORSUK_1: 40, PRE_TOPC: 8;

            f2 is continuous by A14;

            then gt2 is continuous by A18, PRE_TOPC: 26;

            then ex h be Function of I[01] , (( TOP-REAL n) | Y) st h is continuous & w19 = (h . 0 ) & w39 = (h . 1) & ( rng h) c= (( rng gt) \/ ( rng gt2)) by A8, A15, A19, BORSUK_2: 13;

            hence thesis;

          end;

            suppose

             A20: w2 = w3;

            then ( LSeg (w1,w3)) is_an_arc_of (w1,w3) by A6, TOPREAL1: 9;

            then

            consider f be Function of I[01] , (( TOP-REAL n) | ( LSeg (w1,w3))) such that

             A21: f is being_homeomorphism and

             A22: (f . 0 ) = w1 & (f . 1) = w3 by TOPREAL1:def 1;

            

             A23: ( rng f) = ( [#] (( TOP-REAL n) | ( LSeg (w1,w3)))) by A21;

            then

             A24: ( rng f) c= P by A4, A20, PRE_TOPC:def 5;

            then ( [#] (( TOP-REAL n) | ( LSeg (w1,w3)))) c= ( [#] (( TOP-REAL n) | P)) by A23, PRE_TOPC:def 5;

            then

             A25: (( TOP-REAL n) | ( LSeg (w1,w3))) is SubSpace of (( TOP-REAL n) | P) by TOPMETR: 3;

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

            then

            reconsider g = f as Function of [. 0 , 1.], P by A24, FUNCT_2: 2;

            reconsider gt = g as Function of I[01] , (( TOP-REAL n) | Y) by BORSUK_1: 40, PRE_TOPC: 8;

            f is continuous by A21;

            then gt is continuous by A25, PRE_TOPC: 26;

            hence thesis by A22;

          end;

        end;

        hence thesis;

      end;

        suppose

         A26: w1 = w2;

        now

          per cases ;

            case w2 <> w3;

            then ( LSeg (w1,w3)) is_an_arc_of (w1,w3) by A26, TOPREAL1: 9;

            then

            consider f be Function of I[01] , (( TOP-REAL n) | ( LSeg (w1,w3))) such that

             A27: f is being_homeomorphism and

             A28: (f . 0 ) = w1 & (f . 1) = w3 by TOPREAL1:def 1;

            

             A29: ( rng f) = ( [#] (( TOP-REAL n) | ( LSeg (w1,w3)))) by A27;

            then

             A30: ( rng f) c= P by A5, A26, PRE_TOPC:def 5;

            then ( [#] (( TOP-REAL n) | ( LSeg (w1,w3)))) c= ( [#] (( TOP-REAL n) | P)) by A29, PRE_TOPC:def 5;

            then

             A31: (( TOP-REAL n) | ( LSeg (w1,w3))) is SubSpace of (( TOP-REAL n) | P) by TOPMETR: 3;

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

            then

            reconsider g = f as Function of [. 0 , 1.], P by A30, FUNCT_2: 2;

            reconsider gt = g as Function of I[01] , (( TOP-REAL n) | Y) by BORSUK_1: 40, PRE_TOPC: 8;

            f is continuous by A27;

            then gt is continuous by A31, PRE_TOPC: 26;

            hence thesis by A28;

          end;

            case

             A32: w2 = w3;

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

            then

            reconsider w19 = w1, w39 = w3 as Point of (( TOP-REAL n) | P) by A1, A3;

            ex f be Function of I[01] , (( TOP-REAL n) | Y) st f is continuous & (f . 0 ) = w19 & (f . 1) = w39 by A26, A32, BORSUK_2: 3;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN2C:39

    

     Th26: for P be Subset of ( TOP-REAL n), w1,w2,w3,w4 be Point of ( TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & ( LSeg (w1,w2)) c= P & ( LSeg (w2,w3)) c= P & ( LSeg (w3,w4)) c= P holds ex h be Function of I[01] , (( TOP-REAL n) | P) st h is continuous & w1 = (h . 0 ) & w4 = (h . 1)

    proof

      let P be Subset of ( TOP-REAL n), w1,w2,w3,w4 be Point of ( TOP-REAL n);

      assume that

       A1: w1 in P and

       A2: w2 in P and

       A3: w3 in P and

       A4: w4 in P and

       A5: ( LSeg (w1,w2)) c= P & ( LSeg (w2,w3)) c= P and

       A6: ( LSeg (w3,w4)) c= P;

      reconsider Y = P as non empty Subset of ( TOP-REAL n) by A1;

      consider h2 be Function of I[01] , (( TOP-REAL n) | P) such that

       A7: h2 is continuous & w1 = (h2 . 0 ) and

       A8: w3 = (h2 . 1) by A1, A2, A3, A5, Th25;

      per cases ;

        suppose w3 <> w4;

        then ( LSeg (w3,w4)) is_an_arc_of (w3,w4) by TOPREAL1: 9;

        then

        consider f be Function of I[01] , (( TOP-REAL n) | ( LSeg (w3,w4))) such that

         A9: f is being_homeomorphism and

         A10: (f . 0 ) = w3 & (f . 1) = w4 by TOPREAL1:def 1;

        

         A11: ( rng f) = ( [#] (( TOP-REAL n) | ( LSeg (w3,w4)))) by A9;

        then

         A12: ( rng f) c= P by A6, PRE_TOPC:def 5;

        then ( [#] (( TOP-REAL n) | ( LSeg (w3,w4)))) c= ( [#] (( TOP-REAL n) | P)) by A11, PRE_TOPC:def 5;

        then

         A13: (( TOP-REAL n) | ( LSeg (w3,w4))) is SubSpace of (( TOP-REAL n) | P) by TOPMETR: 3;

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

        then

        reconsider w19 = w1, w39 = w3, w49 = w4 as Point of (( TOP-REAL n) | P) by A1, A3, A4;

        

         A14: w39 = (h2 . 1) by A8;

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

        then

        reconsider g = f as Function of [. 0 , 1.], P by A12, FUNCT_2: 2;

        reconsider gt = g as Function of I[01] , (( TOP-REAL n) | Y) by BORSUK_1: 40, PRE_TOPC: 8;

        f is continuous by A9;

        then gt is continuous by A13, PRE_TOPC: 26;

        then ex h be Function of I[01] , (( TOP-REAL n) | Y) st h is continuous & w19 = (h . 0 ) & w49 = (h . 1) & ( rng h) c= (( rng h2) \/ ( rng gt)) by A7, A10, A14, BORSUK_2: 13;

        hence thesis;

      end;

        suppose w3 = w4;

        hence thesis by A7, A8;

      end;

    end;

    theorem :: JORDAN2C:40

    

     Th27: for P be Subset of ( TOP-REAL n), w1,w2,w3,w4,w5,w6,w7 be Point of ( TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & ( LSeg (w1,w2)) c= P & ( LSeg (w2,w3)) c= P & ( LSeg (w3,w4)) c= P & ( LSeg (w4,w5)) c= P & ( LSeg (w5,w6)) c= P & ( LSeg (w6,w7)) c= P holds ex h be Function of I[01] , (( TOP-REAL n) | P) st h is continuous & w1 = (h . 0 ) & w7 = (h . 1)

    proof

      let P be Subset of ( TOP-REAL n), w1,w2,w3,w4,w5,w6,w7 be Point of ( TOP-REAL n);

      assume that

       A1: w1 in P and

       A2: w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & ( LSeg (w1,w2)) c= P & ( LSeg (w2,w3)) c= P & ( LSeg (w3,w4)) c= P & ( LSeg (w4,w5)) c= P & ( LSeg (w5,w6)) c= P & ( LSeg (w6,w7)) c= P;

      (ex h2 be Function of I[01] , (( TOP-REAL n) | P) st h2 is continuous & w1 = (h2 . 0 ) & w4 = (h2 . 1)) & ex h4 be Function of I[01] , (( TOP-REAL n) | P) st h4 is continuous & w4 = (h4 . 0 ) & w7 = (h4 . 1) by A1, A2, Th26;

      hence thesis by A1, Th24;

    end;

    theorem :: JORDAN2C:41

    

     Th28: for w1,w2 be Point of ( TOP-REAL n), P be Subset of ( TopSpaceMetr ( Euclid n)) st P = ( LSeg (w1,w2)) & not ( 0. ( TOP-REAL n)) in ( LSeg (w1,w2)) holds ex w0 be Point of ( TOP-REAL n) st w0 in ( LSeg (w1,w2)) & |.w0.| > 0 & |.w0.| = (( dist_min P) . ( 0. ( TOP-REAL n)))

    proof

      let w1,w2 be Point of ( TOP-REAL n), P be Subset of ( TopSpaceMetr ( Euclid n));

      assume that

       A1: P = ( LSeg (w1,w2)) and

       A2: not ( 0. ( TOP-REAL n)) in ( LSeg (w1,w2));

      set M = ( Euclid n);

      reconsider P0 = P as Subset of ( TopSpaceMetr M);

      

       A3: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr M) by EUCLID:def 8;

      then

      reconsider Q = {( 0. ( TOP-REAL n))} as Subset of ( TopSpaceMetr M);

      P0 is compact by A1, A3, COMPTS_1: 23;

      then

      consider x1,x2 be Point of M such that

       A4: x1 in P0 and

       A5: x2 in Q and

       A6: ( dist (x1,x2)) = ( min_dist_min (P0,Q)) by A1, A3, WEIERSTR: 30;

      reconsider w01 = x1 as Point of ( TOP-REAL n) by EUCLID: 67;

      

       A7: x2 = ( 0. ( TOP-REAL n)) by A5, TARSKI:def 1;

      reconsider o = ( 0. ( TOP-REAL n)) as Point of M by EUCLID: 67;

      reconsider o2 = ( 0. ( TOP-REAL n)) as Point of ( TopSpaceMetr M) by A3;

      for x be object holds x in (( dist_min P0) .: Q) iff x = (( dist_min P0) . o)

      proof

        let x be object;

        hereby

          assume x in (( dist_min P0) .: Q);

          then ex y be object st y in ( dom ( dist_min P0)) & y in Q & x = (( dist_min P0) . y) by FUNCT_1:def 6;

          hence x = (( dist_min P0) . o) by TARSKI:def 1;

        end;

        o2 in the carrier of ( TopSpaceMetr M) by A3;

        then

         A8: o in Q & o in ( dom ( dist_min P0)) by FUNCT_2:def 1, TARSKI:def 1;

        assume x = (( dist_min P0) . o);

        hence thesis by A8, FUNCT_1:def 6;

      end;

      then

       A9: (( dist_min P0) .: Q) = {(( dist_min P0) . o)} by TARSKI:def 1;

      ( [#] (( dist_min P0) .: Q)) = (( dist_min P0) .: Q) & ( lower_bound ( [#] (( dist_min P0) .: Q))) = ( lower_bound (( dist_min P0) .: Q)) by WEIERSTR:def 1, WEIERSTR:def 3;

      then

       A10: ( lower_bound (( dist_min P0) .: Q)) = (( dist_min P0) . o) by A9, SEQ_4: 9;

      

       A11: |.w01.| = |.(w01 - ( 0. ( TOP-REAL n))).| by RLVECT_1: 13

      .= ( dist (x1,x2)) by A7, JGRAPH_1: 28;

       |.w01.| <> 0 by A1, A2, A4, TOPRNS_1: 24;

      hence thesis by A1, A4, A6, A10, A11, WEIERSTR:def 7;

    end;

    theorem :: JORDAN2C:42

    

     Th29: for a be Real, Q be Subset of ( TOP-REAL n), w1,w4 be Point of ( TOP-REAL n) st Q = { q : |.q.| > a } & w1 in Q & w4 in Q & not (ex r be Real st w1 = (r * w4) or w4 = (r * w1)) holds ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q

    proof

      let a be Real, Q be Subset of ( TOP-REAL n), w1,w4 be Point of ( TOP-REAL n);

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

      then

      reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

      assume

       A1: Q = { q : |.q.| > a } & w1 in Q & w4 in Q & not (ex r be Real st w1 = (r * w4) or w4 = (r * w1));

      then not ( 0. ( TOP-REAL n)) in ( LSeg (w1,w4)) by RLTOPSP1: 71;

      then

      consider w0 be Point of ( TOP-REAL n) such that w0 in ( LSeg (w1,w4)) and

       A2: |.w0.| > 0 and

       A3: |.w0.| = (( dist_min P) . ( 0. ( TOP-REAL n))) by Th28;

      set l9 = ((a + 1) / |.w0.|);

      set w2 = (l9 * w1), w3 = (l9 * w4);

      

       A4: ( LSeg (w2,w3)) c= Q

      proof

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

        then

        reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

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

        let x be object;

        

         A5: |.l9.| = ( |.(a + 1).| / |. |.w0.|.|) by COMPLEX1: 67

        .= ( |.(a + 1).| / |.w0.|) by ABSVALUE:def 1;

        (( dist o) .: P) c= REAL by XREAL_0:def 1;

        then

        reconsider F = (( dist o) .: P) as Subset of REAL ;

        assume x in ( LSeg (w2,w3));

        then

        consider r such that

         A6: x = (((1 - r) * w2) + (r * w3)) and

         A7: 0 <= r & r <= 1;

        reconsider w5 = (((1 - r) * w1) + (r * w4)) as Point of ( TOP-REAL n);

        reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

        

         A8: ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

         0 is LowerBound of (( dist o) .: P)

        proof

          let r be ExtReal;

          assume r in (( dist o) .: P);

          then

          consider x be object such that x in ( dom ( dist o)) and

           A9: x in P and

           A10: r = (( dist o) . x) by FUNCT_1:def 6;

          reconsider w0 = x as Point of ( Euclid n) by A9, TOPREAL3: 8;

          r = ( dist (w0,o)) by A10, WEIERSTR:def 4;

          hence thesis by METRIC_1: 5;

        end;

        then

         A11: F is bounded_below;

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

        then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

        then

         A12: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

        w5 in ( LSeg (w1,w4)) by A7;

        then ( dist (w59,o)) in (( dist o) .: P) by A12, A8, FUNCT_1:def 6;

        then ( lower_bound F) <= ( dist (w59,o)) by A11, SEQ_4:def 2;

        then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

        then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

        then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

        then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

        then |.w5.| >= |.w0.| by RLVECT_1: 13;

        then |.(a + 1).| >= 0 & ( |.w5.| / |.w0.|) >= 1 by A2, COMPLEX1: 46, XREAL_1: 181;

        then ( |.(a + 1).| * ( |.w5.| / |.w0.|)) >= ( |.(a + 1).| * 1) by XREAL_1: 66;

        then ( |.(a + 1).| * (( |.w0.| " ) * |.w5.|)) >= |.(a + 1).| by XCMPLX_0:def 9;

        then (( |.(a + 1).| * ( |.w0.| " )) * |.w5.|) >= |.(a + 1).|;

        then

         A13: (( |.(a + 1).| / |.w0.|) * |.w5.|) >= |.(a + 1).| by XCMPLX_0:def 9;

        (a + 1) > a & |.(a + 1).| >= (a + 1) by ABSVALUE: 4, XREAL_1: 29;

        then |.(a + 1).| > a by XXREAL_0: 2;

        then (( |.(a + 1).| / |.w0.|) * |.w5.|) > a by A13, XXREAL_0: 2;

        then |.(l9 * (((1 - r) * w1) + (r * w4))).| > a by A5, TOPRNS_1: 7;

        then |.((l9 * ((1 - r) * w1)) + (l9 * (r * w4))).| > a by RLVECT_1:def 5;

        then |.((l9 * ((1 - r) * w1)) + ((l9 * r) * w4)).| > a by RLVECT_1:def 7;

        then |.(((l9 * (1 - r)) * w1) + ((l9 * r) * w4)).| > a by RLVECT_1:def 7;

        then |.((((1 - r) * l9) * w1) + (r * (l9 * w4))).| > a by RLVECT_1:def 7;

        then |.(((1 - r) * w2) + (r * w3)).| > a by RLVECT_1:def 7;

        hence thesis by A1, A6;

      end;

      

       A14: w3 in ( LSeg (w2,w3)) by RLTOPSP1: 68;

      then

       A15: w3 in Q by A4;

      

       A16: ( LSeg (w4,w3)) c= Q

      proof

        let x be object;

        assume x in ( LSeg (w4,w3));

        then

        consider r such that

         A17: x = (((1 - r) * w4) + (r * w3)) and

         A18: 0 <= r and

         A19: r <= 1;

        now

          per cases ;

            case

             A20: a >= 0 ;

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

            then

            reconsider P = ( LSeg (w4,w1)) as Subset of ( TopSpaceMetr ( Euclid n));

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

            reconsider w5 = (((1 - 0 ) * w4) + ( 0 * w1)) as Point of ( TOP-REAL n);

            

             A21: (((1 - 0 ) * w4) + ( 0 * w1)) = (((1 - 0 ) * w4) + ( 0. ( TOP-REAL n))) by RLVECT_1: 10

            .= ((1 - 0 ) * w4) by RLVECT_1: 4

            .= w4 by RLVECT_1:def 8;

            (( dist o) .: P) c= REAL by XREAL_0:def 1;

            then

            reconsider F = (( dist o) .: P) as Subset of REAL ;

            reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

            

             A22: ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

             0 is LowerBound of (( dist o) .: P)

            proof

              let r be ExtReal;

              assume r in (( dist o) .: P);

              then

              consider x be object such that x in ( dom ( dist o)) and

               A23: x in P and

               A24: r = (( dist o) . x) by FUNCT_1:def 6;

              reconsider w0 = x as Point of ( Euclid n) by A23, TOPREAL3: 8;

              r = ( dist (w0,o)) by A24, WEIERSTR:def 4;

              hence thesis by METRIC_1: 5;

            end;

            then

             A25: F is bounded_below;

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

            then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

            then

             A26: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

            w5 in { (((1 - r1) * w4) + (r1 * w1)) : 0 <= r1 & r1 <= 1 };

            then ( dist (w59,o)) in (( dist o) .: P) by A26, A22, FUNCT_1:def 6;

            then ( lower_bound F) <= ( dist (w59,o)) by A25, SEQ_4:def 2;

            then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

            then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

            then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

            then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

            then

             A27: |.w5.| >= |.w0.| by RLVECT_1: 13;

            ((r * l9) * |.w0.|) = (((r * (a + 1)) / |.w0.|) * |.w0.|) by XCMPLX_1: 74

            .= (r * (a + 1)) by A2, XCMPLX_1: 87;

            then

             A28: ((r * l9) * |.w4.|) >= (r * (a + 1)) by A18, A20, A21, A27, XREAL_1: 64;

            

             A29: (1 - r) >= 0 by A19, XREAL_1: 48;

            

             A30: (a + r) >= (a + 0 ) by A18, XREAL_1: 6;

            

             A31: ex q1 be Point of ( TOP-REAL n) st q1 = w4 & |.q1.| > a by A1;

            now

              per cases ;

                case (1 - r) > 0 ;

                then

                 A32: ((1 - r) * |.w4.|) > ((1 - r) * a) by A31, XREAL_1: 68;

                ( |.((1 - r) + (r * l9)).| * |.w4.|) = (((1 - r) + (r * l9)) * |.w4.|) by A18, A20, A29, ABSVALUE:def 1

                .= (((1 - r) * |.w4.|) + ((r * l9) * |.w4.|));

                then ( |.((1 - r) + (r * l9)).| * |.w4.|) > ((r * (a + 1)) + ((1 - r) * a)) by A28, A32, XREAL_1: 8;

                then ( |.((1 - r) + (r * l9)).| * |.w4.|) > a by A30, XXREAL_0: 2;

                then |.(((1 - r) + (r * l9)) * w4).| > a by TOPRNS_1: 7;

                then |.(((1 - r) * w4) + ((r * l9) * w4)).| > a by RLVECT_1:def 6;

                hence |.(((1 - r) * w4) + (r * w3)).| > a by RLVECT_1:def 7;

              end;

                case (1 - r) <= 0 ;

                then ((1 - r) + r) <= ( 0 + r) by XREAL_1: 6;

                then r = 1 by A19, XXREAL_0: 1;

                

                then

                 A33: (((1 - r) * w4) + (r * w3)) = (( 0. ( TOP-REAL n)) + (1 * w3)) by RLVECT_1: 10

                .= (( 0. ( TOP-REAL n)) + w3) by RLVECT_1:def 8

                .= w3 by RLVECT_1: 4;

                ex q3 be Point of ( TOP-REAL n) st q3 = w3 & |.q3.| > a by A1, A15;

                hence |.(((1 - r) * w4) + (r * w3)).| > a by A33;

              end;

            end;

            hence |.(((1 - r) * w4) + (r * w3)).| > a;

          end;

            case a < 0 ;

            hence |.(((1 - r) * w4) + (r * w3)).| > a;

          end;

        end;

        hence thesis by A1, A17;

      end;

      

       A34: w2 in ( LSeg (w2,w3)) by RLTOPSP1: 68;

      then

       A35: w2 in Q by A4;

      ( LSeg (w1,w2)) c= Q

      proof

        let x be object;

        assume x in ( LSeg (w1,w2));

        then

        consider r such that

         A36: x = (((1 - r) * w1) + (r * w2)) and

         A37: 0 <= r and

         A38: r <= 1;

        now

          per cases ;

            case

             A39: a >= 0 ;

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

            then

            reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

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

            reconsider w5 = (((1 - 0 ) * w1) + ( 0 * w4)) as Point of ( TOP-REAL n);

            

             A40: (((1 - 0 ) * w1) + ( 0 * w4)) = (((1 - 0 ) * w1) + ( 0. ( TOP-REAL n))) by RLVECT_1: 10

            .= ((1 - 0 ) * w1) by RLVECT_1: 4

            .= w1 by RLVECT_1:def 8;

            (( dist o) .: P) c= REAL by XREAL_0:def 1;

            then

            reconsider F = (( dist o) .: P) as Subset of REAL ;

            reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

             0 is LowerBound of (( dist o) .: P)

            proof

              let r be ExtReal;

              assume r in (( dist o) .: P);

              then

              consider x be object such that x in ( dom ( dist o)) and

               A41: x in P and

               A42: r = (( dist o) . x) by FUNCT_1:def 6;

              reconsider w0 = x as Point of ( Euclid n) by A41, TOPREAL3: 8;

              r = ( dist (w0,o)) by A42, WEIERSTR:def 4;

              hence thesis by METRIC_1: 5;

            end;

            then

             A43: F is bounded_below;

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

            then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

            then

             A44: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

            w5 in ( LSeg (w1,w4)) & ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

            then ( dist (w59,o)) in (( dist o) .: P) by A44, FUNCT_1:def 6;

            then ( lower_bound F) <= ( dist (w59,o)) by A43, SEQ_4:def 2;

            then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

            then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

            then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

            then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

            then

             A45: |.w5.| >= |.w0.| by RLVECT_1: 13;

            ((r * l9) * |.w0.|) = (((r * (a + 1)) / |.w0.|) * |.w0.|) by XCMPLX_1: 74

            .= (r * (a + 1)) by A2, XCMPLX_1: 87;

            then

             A46: ((r * l9) * |.w1.|) >= (r * (a + 1)) by A37, A39, A40, A45, XREAL_1: 64;

            

             A47: ex q1 be Point of ( TOP-REAL n) st q1 = w1 & |.q1.| > a by A1;

            

             A48: (a + r) >= (a + 0 ) by A37, XREAL_1: 6;

            

             A49: (1 - r) >= 0 by A38, XREAL_1: 48;

            

             A50: ex q2 be Point of ( TOP-REAL n) st q2 = w2 & |.q2.| > a by A1, A35;

            now

              per cases ;

                case (1 - r) > 0 ;

                then

                 A51: ((1 - r) * |.w1.|) > ((1 - r) * a) by A47, XREAL_1: 68;

                ( |.((1 - r) + (r * l9)).| * |.w1.|) = (((1 - r) + (r * l9)) * |.w1.|) by A37, A39, A49, ABSVALUE:def 1

                .= (((1 - r) * |.w1.|) + ((r * l9) * |.w1.|));

                then ( |.((1 - r) + (r * l9)).| * |.w1.|) > ((r * (a + 1)) + ((1 - r) * a)) by A46, A51, XREAL_1: 8;

                then ( |.((1 - r) + (r * l9)).| * |.w1.|) > a by A48, XXREAL_0: 2;

                then |.(((1 - r) + (r * l9)) * w1).| > a by TOPRNS_1: 7;

                then |.(((1 - r) * w1) + ((r * l9) * w1)).| > a by RLVECT_1:def 6;

                hence |.(((1 - r) * w1) + (r * w2)).| > a by RLVECT_1:def 7;

              end;

                case (1 - r) <= 0 ;

                then ((1 - r) + r) <= ( 0 + r) by XREAL_1: 6;

                then r = 1 by A38, XXREAL_0: 1;

                

                then (((1 - r) * w1) + (r * w2)) = (( 0. ( TOP-REAL n)) + (1 * w2)) by RLVECT_1: 10

                .= (( 0. ( TOP-REAL n)) + w2) by RLVECT_1:def 8

                .= w2 by RLVECT_1: 4;

                hence |.(((1 - r) * w1) + (r * w2)).| > a by A50;

              end;

            end;

            hence |.(((1 - r) * w1) + (r * w2)).| > a;

          end;

            case a < 0 ;

            hence |.(((1 - r) * w1) + (r * w2)).| > a;

          end;

        end;

        hence thesis by A1, A36;

      end;

      hence thesis by A4, A34, A14, A16;

    end;

    theorem :: JORDAN2C:43

    

     Th30: for a be Real, Q be Subset of ( TOP-REAL n), w1,w4 be Point of ( TOP-REAL n) st Q = (( REAL n) \ { q : |.q.| < a }) & w1 in Q & w4 in Q & not (ex r be Real st w1 = (r * w4) or w4 = (r * w1)) holds ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q

    proof

      let a be Real, Q be Subset of ( TOP-REAL n), w1,w4 be Point of ( TOP-REAL n);

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

      then

      reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

      assume

       A1: Q = (( REAL n) \ { q : |.q.| < a }) & w1 in Q & w4 in Q & not (ex r be Real st w1 = (r * w4) or w4 = (r * w1));

      then not ( 0. ( TOP-REAL n)) in ( LSeg (w1,w4)) by RLTOPSP1: 71;

      then

      consider w0 be Point of ( TOP-REAL n) such that w0 in ( LSeg (w1,w4)) and

       A2: |.w0.| > 0 and

       A3: |.w0.| = (( dist_min P) . ( 0. ( TOP-REAL n))) by Th28;

      set l9 = (a / |.w0.|);

      set w2 = (l9 * w1), w3 = (l9 * w4);

      

       A4: (( REAL n) \ { q : |.q.| < a }) = { q1 : |.q1.| >= a }

      proof

        thus (( REAL n) \ { q : |.q.| < a }) c= { q1 : |.q1.| >= a }

        proof

          let z be object;

          assume

           A5: z in (( REAL n) \ { q : |.q.| < a });

          then

          reconsider q2 = z as Point of ( TOP-REAL n) by EUCLID: 22;

           not z in { q : |.q.| < a } by A5, XBOOLE_0:def 5;

          then |.q2.| >= a;

          hence thesis;

        end;

        let z be object;

        assume z in { q1 : |.q1.| >= a };

        then

        consider q1 such that

         A6: z = q1 and

         A7: |.q1.| >= a;

        q1 in the carrier of ( TOP-REAL n);

        then

         A8: z in ( REAL n) by A6, EUCLID: 22;

        for q st q = z holds |.q.| >= a by A6, A7;

        then not z in { q : |.q.| < a };

        hence thesis by A8, XBOOLE_0:def 5;

      end;

      

       A9: ( LSeg (w1,w2)) c= Q

      proof

        let x be object;

        assume x in ( LSeg (w1,w2));

        then

        consider r such that

         A10: x = (((1 - r) * w1) + (r * w2)) and

         A11: 0 <= r and

         A12: r <= 1;

        now

          per cases ;

            case

             A13: a > 0 ;

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

            then

            reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

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

            reconsider w5 = (((1 - 0 ) * w1) + ( 0 * w4)) as Point of ( TOP-REAL n);

            

             A14: (((1 - 0 ) * w1) + ( 0 * w4)) = (((1 - 0 ) * w1) + ( 0. ( TOP-REAL n))) by RLVECT_1: 10

            .= ((1 - 0 ) * w1) by RLVECT_1: 4

            .= w1 by RLVECT_1:def 8;

            (( dist o) .: P) c= REAL by XREAL_0:def 1;

            then

            reconsider F = (( dist o) .: P) as Subset of REAL ;

            reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

             0 is LowerBound of (( dist o) .: P)

            proof

              let r be ExtReal;

              assume r in (( dist o) .: P);

              then

              consider x be object such that x in ( dom ( dist o)) and

               A15: x in P and

               A16: r = (( dist o) . x) by FUNCT_1:def 6;

              reconsider w0 = x as Point of ( Euclid n) by A15, TOPREAL3: 8;

              r = ( dist (w0,o)) by A16, WEIERSTR:def 4;

              hence thesis by METRIC_1: 5;

            end;

            then

             A17: F is bounded_below;

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

            then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

            then

             A18: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

            w5 in ( LSeg (w1,w4)) & ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

            then ( dist (w59,o)) in (( dist o) .: P) by A18, FUNCT_1:def 6;

            then ( lower_bound F) <= ( dist (w59,o)) by A17, SEQ_4:def 2;

            then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

            then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

            then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

            then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

            then

             A19: |.w5.| >= |.w0.| by RLVECT_1: 13;

            

             A20: (1 - r) >= 0 by A12, XREAL_1: 48;

            

            then

             A21: ( |.((1 - r) + (r * l9)).| * |.w1.|) = (((1 - r) + (r * l9)) * |.w1.|) by A11, A13, ABSVALUE:def 1

            .= (((1 - r) * |.w1.|) + ((r * l9) * |.w1.|));

            ex q1 be Point of ( TOP-REAL n) st q1 = w1 & |.q1.| >= a by A1, A4;

            then

             A22: ((1 - r) * |.w1.|) >= ((1 - r) * a) by A20, XREAL_1: 64;

            ((r * l9) * |.w0.|) = (((r * a) / |.w0.|) * |.w0.|) by XCMPLX_1: 74

            .= (r * a) by A2, XCMPLX_1: 87;

            then ((r * l9) * |.w1.|) >= (r * a) by A11, A13, A14, A19, XREAL_1: 64;

            then ( |.((1 - r) + (r * l9)).| * |.w1.|) >= ((r * a) + ((1 - r) * a)) by A22, A21, XREAL_1: 7;

            then |.(((1 - r) + (r * l9)) * w1).| >= a by TOPRNS_1: 7;

            then |.(((1 - r) * w1) + ((r * l9) * w1)).| >= a by RLVECT_1:def 6;

            hence |.(((1 - r) * w1) + (r * w2)).| >= a by RLVECT_1:def 7;

          end;

            case a <= 0 ;

            hence |.(((1 - r) * w1) + (r * w2)).| >= a;

          end;

        end;

        hence thesis by A1, A4, A10;

      end;

      

       A23: ( LSeg (w4,w3)) c= Q

      proof

        let x be object;

        assume x in ( LSeg (w4,w3));

        then

        consider r such that

         A24: x = (((1 - r) * w4) + (r * w3)) and

         A25: 0 <= r and

         A26: r <= 1;

        now

          per cases ;

            case

             A27: a > 0 ;

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

            then

            reconsider P = ( LSeg (w4,w1)) as Subset of ( TopSpaceMetr ( Euclid n));

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

            reconsider w5 = (((1 - 0 ) * w4) + ( 0 * w1)) as Point of ( TOP-REAL n);

            

             A28: (((1 - 0 ) * w4) + ( 0 * w1)) = (((1 - 0 ) * w4) + ( 0. ( TOP-REAL n))) by RLVECT_1: 10

            .= ((1 - 0 ) * w4) by RLVECT_1: 4

            .= w4 by RLVECT_1:def 8;

            (( dist o) .: P) c= REAL by XREAL_0:def 1;

            then

            reconsider F = (( dist o) .: P) as Subset of REAL ;

            reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

            

             A29: ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

             0 is LowerBound of (( dist o) .: P)

            proof

              let r be ExtReal;

              assume r in (( dist o) .: P);

              then

              consider x be object such that x in ( dom ( dist o)) and

               A30: x in P and

               A31: r = (( dist o) . x) by FUNCT_1:def 6;

              reconsider w0 = x as Point of ( Euclid n) by A30, TOPREAL3: 8;

              r = ( dist (w0,o)) by A31, WEIERSTR:def 4;

              hence thesis by METRIC_1: 5;

            end;

            then

             A32: F is bounded_below;

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

            then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

            then

             A33: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

            w5 in { (((1 - r1) * w4) + (r1 * w1)) : 0 <= r1 & r1 <= 1 };

            then ( dist (w59,o)) in (( dist o) .: P) by A33, A29, FUNCT_1:def 6;

            then ( lower_bound F) <= ( dist (w59,o)) by A32, SEQ_4:def 2;

            then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

            then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

            then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

            then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

            then

             A34: |.w5.| >= |.w0.| by RLVECT_1: 13;

            

             A35: (1 - r) >= 0 by A26, XREAL_1: 48;

            

            then

             A36: ( |.((1 - r) + (r * l9)).| * |.w4.|) = (((1 - r) + (r * l9)) * |.w4.|) by A25, A27, ABSVALUE:def 1

            .= (((1 - r) * |.w4.|) + ((r * l9) * |.w4.|));

            ex q1 be Point of ( TOP-REAL n) st q1 = w4 & |.q1.| >= a by A1, A4;

            then

             A37: ((1 - r) * |.w4.|) >= ((1 - r) * a) by A35, XREAL_1: 64;

            ((r * l9) * |.w0.|) = (((r * a) / |.w0.|) * |.w0.|) by XCMPLX_1: 74

            .= (r * a) by A2, XCMPLX_1: 87;

            then ((r * l9) * |.w4.|) >= (r * a) by A25, A27, A28, A34, XREAL_1: 64;

            then ( |.((1 - r) + (r * l9)).| * |.w4.|) >= ((r * a) + ((1 - r) * a)) by A37, A36, XREAL_1: 7;

            then |.(((1 - r) + (r * l9)) * w4).| >= a by TOPRNS_1: 7;

            then |.(((1 - r) * w4) + ((r * l9) * w4)).| >= a by RLVECT_1:def 6;

            hence |.(((1 - r) * w4) + (r * w3)).| >= a by RLVECT_1:def 7;

          end;

            case a <= 0 ;

            hence |.(((1 - r) * w4) + (r * w3)).| >= a;

          end;

        end;

        hence thesis by A1, A4, A24;

      end;

      

       A38: ( LSeg (w2,w3)) c= Q

      proof

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

        then

        reconsider P = ( LSeg (w1,w4)) as Subset of ( TopSpaceMetr ( Euclid n));

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

        let x be object;

        

         A39: |.l9.| = ( |.a.| / |. |.w0.|.|) by COMPLEX1: 67

        .= ( |.a.| / |.w0.|) by ABSVALUE:def 1;

        (( dist o) .: P) c= REAL by XREAL_0:def 1;

        then

        reconsider F = (( dist o) .: P) as Subset of REAL ;

        assume x in ( LSeg (w2,w3));

        then

        consider r such that

         A40: x = (((1 - r) * w2) + (r * w3)) and

         A41: 0 <= r & r <= 1;

        reconsider w5 = (((1 - r) * w1) + (r * w4)) as Point of ( TOP-REAL n);

        reconsider w59 = w5 as Point of ( Euclid n) by TOPREAL3: 8;

        

         A42: ( dist (w59,o)) = (( dist o) . w59) by WEIERSTR:def 4;

         0 is LowerBound of (( dist o) .: P)

        proof

          let r be ExtReal;

          assume r in (( dist o) .: P);

          then

          consider x be object such that x in ( dom ( dist o)) and

           A43: x in P and

           A44: r = (( dist o) . x) by FUNCT_1:def 6;

          reconsider w0 = x as Point of ( Euclid n) by A43, TOPREAL3: 8;

          r = ( dist (w0,o)) by A44, WEIERSTR:def 4;

          hence thesis by METRIC_1: 5;

        end;

        then

         A45: F is bounded_below;

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

        then w59 in the carrier of ( TopSpaceMetr ( Euclid n));

        then

         A46: w59 in ( dom ( dist o)) by FUNCT_2:def 1;

        w5 in ( LSeg (w1,w4)) by A41;

        then ( dist (w59,o)) in (( dist o) .: P) by A46, A42, FUNCT_1:def 6;

        then ( lower_bound F) <= ( dist (w59,o)) by A45, SEQ_4:def 2;

        then ( dist (w59,o)) >= ( lower_bound ( [#] (( dist o) .: P))) by WEIERSTR:def 1;

        then ( dist (w59,o)) >= ( lower_bound (( dist o) .: P)) by WEIERSTR:def 3;

        then ( dist (w59,o)) >= |.w0.| by A3, WEIERSTR:def 6;

        then |.(w5 - ( 0. ( TOP-REAL n))).| >= |.w0.| by JGRAPH_1: 28;

        then |.w5.| >= |.w0.| by RLVECT_1: 13;

        then |.a.| >= 0 & ( |.w5.| / |.w0.|) >= 1 by A2, COMPLEX1: 46, XREAL_1: 181;

        then ( |.a.| * ( |.w5.| / |.w0.|)) >= ( |.a.| * 1) by XREAL_1: 66;

        then ( |.a.| * ( |.w5.| * ( |.w0.| " ))) >= |.a.| by XCMPLX_0:def 9;

        then (( |.a.| * ( |.w0.| " )) * |.w5.|) >= |.a.|;

        then

         A47: (( |.a.| / |.w0.|) * |.w5.|) >= |.a.| by XCMPLX_0:def 9;

         |.a.| >= a by ABSVALUE: 4;

        then (( |.a.| / |.w0.|) * |.w5.|) >= a by A47, XXREAL_0: 2;

        then |.(l9 * (((1 - r) * w1) + (r * w4))).| >= a by A39, TOPRNS_1: 7;

        then |.((l9 * ((1 - r) * w1)) + (l9 * (r * w4))).| >= a by RLVECT_1:def 5;

        then |.((l9 * ((1 - r) * w1)) + ((l9 * r) * w4)).| >= a by RLVECT_1:def 7;

        then |.(((l9 * (1 - r)) * w1) + ((l9 * r) * w4)).| >= a by RLVECT_1:def 7;

        then |.((((1 - r) * l9) * w1) + (r * (l9 * w4))).| >= a by RLVECT_1:def 7;

        then |.(((1 - r) * w2) + (r * w3)).| >= a by RLVECT_1:def 7;

        hence thesis by A1, A4, A40;

      end;

      w2 in ( LSeg (w2,w3)) & w3 in ( LSeg (w2,w3)) by RLTOPSP1: 68;

      hence thesis by A38, A9, A23;

    end;

    theorem :: JORDAN2C:44

    for f be FinSequence of REAL holds f is Element of ( REAL ( len f)) & f is Point of ( TOP-REAL ( len f)) by EUCLID: 76;

    theorem :: JORDAN2C:45

    

     Th32: for x be Element of ( REAL n), f,g be FinSequence of REAL , r be Real st f = x & g = (r * x) holds ( len f) = ( len g) & for i be Element of NAT st 1 <= i & i <= ( len f) holds (g /. i) = (r * (f /. i))

    proof

      reconsider h2 = ( id REAL ) as Function;

      let x be Element of ( REAL n), f,g be FinSequence of REAL , r be Real;

      assume that

       A1: f = x and

       A2: g = (r * x);

      

       A3: ( len f) = n by A1, CARD_1:def 7;

      set h1 = (( dom ( id REAL )) --> r);

      

       A4: ( dom <:h1, h2:>) = (( dom h1) /\ ( dom ( id REAL ))) by FUNCT_3:def 7;

      

       A5: ( len g) = n by A2, CARD_1:def 7;

      

       A6: g = (( multreal * <:h1, h2:>) * x) by A2, FUNCOP_1:def 5;

      for i be Element of NAT st 1 <= i & i <= ( len f) holds (g /. i) = (r * (f /. i))

      proof

        let i be Element of NAT ;

        

         A7: ( dom h1) = ( dom ( id REAL )) by FUNCOP_1: 13

        .= REAL by FUNCT_1: 17;

        reconsider xi = (x . i) as Element of REAL by XREAL_0:def 1;

        ( dom h2) = REAL by FUNCT_1: 17;

        then

         A8: (h1 . xi) = r by FUNCOP_1: 7;

        assume

         A9: 1 <= i & i <= ( len f);

        then

         A10: (f . i) = (f /. i) by FINSEQ_4: 15;

        i in ( Seg ( len f)) by A9, FINSEQ_1: 1;

        then i in ( dom g) by A3, A5, FINSEQ_1:def 3;

        then

         A11: (g . i) = (( multreal * <:h1, h2:>) . (x . i)) by A6, FUNCT_1: 12;

        

         A12: ( dom <:h1, h2:>) = (( dom h1) /\ REAL ) by A4, FUNCT_1: 17;

        then ( <:h1, h2:> . (x . i)) = [(h1 . xi), (h2 . xi)] by A7, FUNCT_3:def 7;

        then (g . i) = ( multreal . (r,(f . i))) by A1, A11, A12, A7, A8, FUNCT_1: 13;

        then (g . i) = (r * (f /. i)) by A10, BINOP_2:def 11;

        hence thesis by A3, A5, A9, FINSEQ_4: 15;

      end;

      hence thesis by A2, A3, CARD_1:def 7;

    end;

    theorem :: JORDAN2C:46

    

     Th33: for x be Element of ( REAL n), f be FinSequence st x <> ( 0* n) & x = f holds ex i be Element of NAT st 1 <= i & i <= n & (f . i) <> 0

    proof

      let x be Element of ( REAL n), f be FinSequence;

      assume that

       A1: x <> ( 0* n) and

       A2: x = f;

      

       A3: ( len f) = n by A2, CARD_1:def 7;

      assume

       A4: not ex i be Element of NAT st 1 <= i & i <= n & (f . i) <> 0 ;

      for z be object holds z in f iff ex x,y be object st x in ( Seg n) & y in { 0 } & z = [x, y]

      proof

        let z be object;

        hereby

          assume

           A5: z in f;

          then

          consider x0,y0 be object such that

           A6: z = [x0, y0] by RELAT_1:def 1;

          

           A7: y0 = (f . x0) by A5, A6, FUNCT_1: 1;

          

           A8: x0 in ( dom f) by A5, A6, XTUPLE_0:def 12;

          then

          reconsider n1 = x0 as Element of NAT ;

          

           A9: x0 in ( Seg ( len f)) by A8, FINSEQ_1:def 3;

          then 1 <= n1 & n1 <= ( len f) by FINSEQ_1: 1;

          then (f . n1) = 0 by A3, A4;

          then y0 in { 0 } by A7, TARSKI:def 1;

          hence ex x,y be object st x in ( Seg n) & y in { 0 } & z = [x, y] by A3, A6, A9;

        end;

        given x,y be object such that

         A10: x in ( Seg n) and

         A11: y in { 0 } and

         A12: z = [x, y];

        reconsider n1 = x as Element of NAT by A10;

        

         A13: n1 <= n by A10, FINSEQ_1: 1;

        

         A14: x in ( dom f) by A3, A10, FINSEQ_1:def 3;

        y = 0 & 1 <= n1 by A10, A11, FINSEQ_1: 1, TARSKI:def 1;

        then y = (f . x) by A4, A13;

        hence thesis by A12, A14, FUNCT_1: 1;

      end;

      then f = [:( Seg n), { 0 }:] by ZFMISC_1:def 2;

      hence contradiction by A1, A2, FUNCOP_1:def 2;

    end;

    theorem :: JORDAN2C:47

    

     Th34: for x be Element of ( REAL n) st n >= 2 & x <> ( 0* n) holds ex y be Element of ( REAL n) st not ex r be Real st y = (r * x) or x = (r * y)

    proof

      let x be Element of ( REAL n);

      assume that

       A1: n >= 2 and

       A2: x <> ( 0* n);

      reconsider f = x as FinSequence of REAL ;

      consider i2 be Element of NAT such that

       A3: 1 <= i2 and

       A4: i2 <= n and

       A5: (f . i2) <> 0 by A2, Th33;

      

       A6: ( len f) = n by CARD_1:def 7;

      then

       A7: 1 <= ( len f) by A1, XXREAL_0: 2;

      per cases ;

        suppose

         A8: i2 > 1;

        reconsider f11 = ((f /. 1) + 1) as Element of REAL by XREAL_0:def 1;

        reconsider g = ( <*f11*> ^ ( mid (f,2,( len f)))) as FinSequence of REAL ;

        

         A9: ( len ( mid (f,2,( len f)))) = ((( len f) -' 2) + 1) by A1, A6, A7, FINSEQ_6: 118

        .= ((( len f) - 2) + 1) by A1, A6, XREAL_1: 233;

        ( len g) = (( len <*((f /. 1) + 1)*>) + ( len ( mid (f,2,( len f))))) by FINSEQ_1: 22;

        

        then

         A10: ( len g) = (1 + ((( len f) - 2) + 1)) by A9, FINSEQ_1: 39

        .= ( len f);

        then

        reconsider y2 = g as Element of ( REAL n) by A6, EUCLID: 76;

        

         A11: ( len <*((f /. 1) + 1)*>) = 1 by FINSEQ_1: 39;

        now

          given r be Real such that

           A12: y2 = (r * x) or x = (r * y2);

          per cases by A12;

            suppose

             A13: y2 = (r * x);

            i2 <= ((( len f) - (1 + 1)) + (1 + 1)) by A4, CARD_1:def 7;

            then

             A14: (i2 - 1) <= ((((( len f) - (1 + 1)) + 1) + 1) - 1) by XREAL_1: 9;

            

             A15: (i2 -' 1) = (i2 - 1) & 1 <= (i2 -' 1) by A8, NAT_D: 49, XREAL_1: 233;

            

             A16: 1 <= ( len f) by A1, A6, XXREAL_0: 2;

            then

             A17: (g /. 1) = (g . 1) by A10, FINSEQ_4: 15;

            

             A18: (g /. i2) = (g . i2) by A3, A4, A6, A10, FINSEQ_4: 15;

            

             A19: (((i2 -' 1) + 2) -' 1) = ((((i2 -' 1) + 1) + 1) -' 1)

            .= ((i2 -' 1) + 1) by NAT_D: 34

            .= ((i2 - 1) + 1) by A3, XREAL_1: 233

            .= i2;

            

             A20: (f /. i2) = (f . i2) by A3, A4, A6, FINSEQ_4: 15;

            (1 + 1) <= i2 & i2 <= (1 + ( len ( mid (f,2,( len f))))) by A4, A8, A9, CARD_1:def 7, NAT_1: 13;

            

            then (g . i2) = (( mid (f,2,( len f))) . (i2 - 1)) by A11, FINSEQ_1: 23

            .= (f . i2) by A1, A6, A9, A16, A15, A14, A19, FINSEQ_6: 118;

            then (1 * (f /. i2)) = (r * (f /. i2)) by A3, A4, A6, A13, A18, A20, Th32;

            then

             A21: 1 = r by A5, A20, XCMPLX_1: 5;

            (g /. 1) = (r * (f /. 1)) by A13, A16, Th32;

            then ((f /. 1) + 1) = (1 * (f /. 1)) by A21, A17, FINSEQ_1: 41;

            hence contradiction;

          end;

            suppose

             A22: x = (r * y2);

            i2 <= ((( len f) - (1 + 1)) + (1 + 1)) by A4, CARD_1:def 7;

            then

             A23: (i2 - 1) <= ((((( len f) - (1 + 1)) + 1) + 1) - 1) by XREAL_1: 9;

            

             A24: (i2 -' 1) = (i2 - 1) & 1 <= (i2 -' 1) by A8, NAT_D: 49, XREAL_1: 233;

            

             A25: 1 <= ( len f) by A1, A6, XXREAL_0: 2;

            then

             A26: (g /. 1) = (g . 1) by A10, FINSEQ_4: 15;

            

             A27: (g /. i2) = (g . i2) by A3, A4, A6, A10, FINSEQ_4: 15;

            

             A28: (((i2 -' 1) + 2) -' 1) = ((((i2 -' 1) + 1) + 1) -' 1)

            .= ((i2 -' 1) + 1) by NAT_D: 34

            .= ((i2 - 1) + 1) by A3, XREAL_1: 233

            .= i2;

            

             A29: (f /. i2) = (f . i2) by A3, A4, A6, FINSEQ_4: 15;

            (1 + 1) <= i2 & i2 <= (1 + ( len ( mid (f,2,( len f))))) by A4, A8, A9, CARD_1:def 7, NAT_1: 13;

            

            then (g . i2) = (( mid (f,2,( len f))) . (i2 - 1)) by A11, FINSEQ_1: 23

            .= (f . i2) by A1, A6, A9, A25, A24, A23, A28, FINSEQ_6: 118;

            then (1 * (f /. i2)) = (r * (f /. i2)) by A3, A4, A6, A10, A22, A27, A29, Th32;

            then

             A30: 1 = r by A5, A29, XCMPLX_1: 5;

            (f /. 1) = (r * (g /. 1)) by A10, A22, A25, Th32;

            then ((f /. 1) + 1) = (1 * (f /. 1)) by A30, A26, FINSEQ_1: 41;

            hence contradiction;

          end;

        end;

        hence thesis;

      end;

        suppose

         A31: i2 <= 1;

        reconsider ff1 = ((f /. ( len f)) + 1) as Element of REAL by XREAL_0:def 1;

        reconsider g = (( mid (f,1,(( len f) -' 1))) ^ <*ff1*>) as FinSequence of REAL ;

        

         A32: (( len f) -' 1) <= ( len f) by NAT_D: 44;

        

         A33: ((1 + 1) - 1) <= (( len f) - 1) by A1, A6, XREAL_1: 9;

        

         A34: (( len f) -' 1) = (( len f) - 1) by A1, A6, XREAL_1: 233, XXREAL_0: 2;

        

        then

         A35: (((( len f) -' 1) -' 1) + 1) = (((( len f) - 1) - 1) + 1) by A33, XREAL_1: 233

        .= ((( len f) - (1 + 1)) + 1);

        then

         A36: ( len ( mid (f,1,(( len f) -' 1)))) = (( len f) - 1) by A7, A34, A32, A33, FINSEQ_6: 118;

        ( len <*((f /. ( len f)) + 1)*>) = 1 & ( len ( mid (f,1,(( len f) -' 1)))) = ((( len f) - 2) + 1) by A7, A34, A32, A33, A35, FINSEQ_1: 39, FINSEQ_6: 118;

        

        then

         A37: ( len g) = (((( len f) - 2) + 1) + 1) by FINSEQ_1: 22

        .= ( len f);

        then

        reconsider y2 = g as Element of ( REAL n) by A6, EUCLID: 76;

        

         A38: i2 = 1 by A3, A31, XXREAL_0: 1;

        now

          given r be Real such that

           A39: y2 = (r * x) or x = (r * y2);

          per cases by A39;

            suppose

             A40: y2 = (r * x);

            

             A41: (g /. i2) = (g . i2) by A3, A4, A6, A37, FINSEQ_4: 15;

            

             A42: (f /. i2) = (f . i2) by A3, A4, A6, FINSEQ_4: 15;

            (g . i2) = (( mid (f,1,(( len f) -' 1))) . i2) by A38, A33, A36, FINSEQ_6: 109

            .= (f . i2) by A38, A34, A32, A33, FINSEQ_6: 123;

            then (1 * (f /. i2)) = (r * (f /. i2)) by A3, A4, A6, A40, A41, A42, Th32;

            then

             A43: 1 = r by A5, A42, XCMPLX_1: 5;

            

             A44: (g . ( len f)) = (g . ((( len f) - 1) + 1))

            .= ((f /. ( len f)) + 1) by A36, FINSEQ_1: 42;

            

             A45: 1 <= ( len f) by A1, A6, XXREAL_0: 2;

            then

             A46: (g /. ( len f)) = (g . ( len f)) by A37, FINSEQ_4: 15;

            (g /. ( len f)) = (r * (f /. ( len f))) by A40, A45, Th32;

            hence contradiction by A43, A46, A44;

          end;

            suppose

             A47: x = (r * y2);

            

             A48: (g /. i2) = (g . i2) by A3, A4, A6, A37, FINSEQ_4: 15;

            

             A49: (f /. i2) = (f . i2) by A3, A4, A6, FINSEQ_4: 15;

            (g . i2) = (( mid (f,1,(( len f) -' 1))) . i2) by A38, A33, A36, FINSEQ_6: 109

            .= (f . i2) by A38, A34, A32, A33, FINSEQ_6: 123;

            then (1 * (f /. i2)) = (r * (f /. i2)) by A3, A4, A6, A37, A47, A48, A49, Th32;

            then

             A50: 1 = r by A5, A49, XCMPLX_1: 5;

            

             A51: (g . ( len f)) = (g . ((( len f) - 1) + 1))

            .= ((f /. ( len f)) + 1) by A36, FINSEQ_1: 42;

            

             A52: 1 <= ( len f) by A1, A6, XXREAL_0: 2;

            then

             A53: (g /. ( len f)) = (g . ( len f)) by A37, FINSEQ_4: 15;

            (f /. ( len f)) = (r * (g /. ( len f))) by A37, A47, A52, Th32;

            hence contradiction by A50, A53, A51;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN2C:48

    

     Th35: for a be Real, Q be Subset of ( TOP-REAL n), w1,w7 be Point of ( TOP-REAL n) st n >= 2 & Q = { q : |.q.| > a } & w1 in Q & w7 in Q & (ex r be Real st w1 = (r * w7) or w7 = (r * w1)) holds ex w2,w3,w4,w5,w6 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q & ( LSeg (w4,w5)) c= Q & ( LSeg (w5,w6)) c= Q & ( LSeg (w6,w7)) c= Q

    proof

      let a be Real, Q be Subset of ( TOP-REAL n), w1,w7 be Point of ( TOP-REAL n);

      assume

       A1: n >= 2 & Q = { q : |.q.| > a } & w1 in Q & w7 in Q;

      reconsider y1 = w1 as Element of ( REAL n) by EUCLID: 22;

      given r8 be Real such that

       A2: w1 = (r8 * w7) or w7 = (r8 * w1);

      per cases ;

        suppose

         A3: a >= 0 ;

        now

          assume

           A4: w1 = ( 0. ( TOP-REAL n));

          ex q st q = w1 & |.q.| > a by A1;

          hence contradiction by A3, A4, TOPRNS_1: 23;

        end;

        then w1 <> ( 0* n) by EUCLID: 70;

        then

        consider y be Element of ( REAL n) such that

         A5: not ex r be Real st y = (r * y1) or y1 = (r * y) by A1, Th34;

        set y4 = (((a + 1) / |.y.|) * y);

        reconsider w4 = y4 as Point of ( TOP-REAL n) by EUCLID: 22;

         A6:

        now

          

           A7: ( 0 * y1) = ( 0 * w1)

          .= ( 0. ( TOP-REAL n)) by RLVECT_1: 10

          .= ( 0* n) by EUCLID: 70;

          assume |.y.| = 0 ;

          hence contradiction by A5, A7, EUCLID: 8;

        end;

        then

         A8: ((a + 1) / |.y.|) > 0 by A3, XREAL_1: 139;

         A9:

        now

          reconsider y9 = y, y19 = y1 as Element of (n -tuples_on REAL );

          given r be Real such that

           A10: w1 = (r * w4) or w4 = (r * w1);

          per cases by A10;

            suppose w1 = (r * w4);

            then y1 = ((r * ((a + 1) / |.y.|)) * y) by RVSUM_1: 49;

            hence contradiction by A5;

          end;

            suppose w4 = (r * w1);

            then (((((a + 1) / |.y.|) " ) * ((a + 1) / |.y.|)) * y9) = ((((a + 1) / |.y.|) " ) * (r * y1)) by RVSUM_1: 49;

            then (((((a + 1) / |.y.|) " ) * ((a + 1) / |.y.|)) * y) = (((((a + 1) / |.y.|) " ) * r) * y19) by RVSUM_1: 49;

            then (1 * y) = (((((a + 1) / |.y.|) " ) * r) * y1) by A8, XCMPLX_0:def 7;

            then y = (((((a + 1) / |.y.|) " ) * r) * y1) by RVSUM_1: 52;

            hence contradiction by A5;

          end;

        end;

        

         A11: |.w4.| = ( |.((a + 1) / |.y.|).| * |.y.|) by EUCLID: 11

        .= (((a + 1) / |.y.|) * |.y.|) by A3, ABSVALUE:def 1

        .= (a + 1) by A6, XCMPLX_1: 87;

        then |.w4.| > a by XREAL_1: 29;

        then

         A12: w4 in Q by A1;

        now

          given r1 be Real such that

           A13: w4 = (r1 * w7) or w7 = (r1 * w4);

           A14:

          now

            assume r1 = 0 ;

            then

             A15: w4 = ( 0. ( TOP-REAL n)) or w7 = ( 0. ( TOP-REAL n)) by A13, RLVECT_1: 10;

            ex q7 be Point of ( TOP-REAL n) st q7 = w7 & |.q7.| > a by A1;

            hence contradiction by A3, A11, A15, TOPRNS_1: 23;

          end;

          per cases by A2;

            suppose

             A16: w1 = (r8 * w7);

            now

              per cases by A13;

                case w4 = (r1 * w7);

                then ((r1 " ) * w4) = (((r1 " ) * r1) * w7) by RLVECT_1:def 7;

                then ((r1 " ) * w4) = (1 * w7) by A14, XCMPLX_0:def 7;

                then ((r1 " ) * w4) = w7 by RLVECT_1:def 8;

                then w1 = ((r8 * (r1 " )) * w4) by A16, RLVECT_1:def 7;

                hence contradiction by A9;

              end;

                case w7 = (r1 * w4);

                then ((r1 " ) * w7) = (((r1 " ) * r1) * w4) by RLVECT_1:def 7;

                then ((r1 " ) * w7) = (1 * w4) by A14, XCMPLX_0:def 7;

                then ((r1 " ) * w7) = w4 by RLVECT_1:def 8;

                then (((r1 " ) " ) * w4) = ((((r1 " ) " ) * (r1 " )) * w7) by RLVECT_1:def 7;

                then (((r1 " ) " ) * w4) = (1 * w7) by A14, XCMPLX_0:def 7;

                then (((r1 " ) " ) * w4) = w7 by RLVECT_1:def 8;

                then w1 = ((r8 * ((r1 " ) " )) * w4) by A16, RLVECT_1:def 7;

                hence contradiction by A9;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A17: w7 = (r8 * w1);

             A18:

            now

              assume r8 = 0 ;

              then

               A19: w7 = ( 0. ( TOP-REAL n)) by A17, RLVECT_1: 10;

              ex q7 be Point of ( TOP-REAL n) st q7 = w7 & |.q7.| > a by A1;

              hence contradiction by A3, A19, TOPRNS_1: 23;

            end;

            ((r8 " ) * w7) = (((r8 " ) * r8) * w1) by A17, RLVECT_1:def 7;

            then ((r8 " ) * w7) = (1 * w1) by A18, XCMPLX_0:def 7;

            then

             A20: ((r8 " ) * w7) = w1 by RLVECT_1:def 8;

            now

              per cases by A13;

                case w4 = (r1 * w7);

                then ((r1 " ) * w4) = (((r1 " ) * r1) * w7) by RLVECT_1:def 7;

                then ((r1 " ) * w4) = (1 * w7) by A14, XCMPLX_0:def 7;

                then ((r1 " ) * w4) = w7 by RLVECT_1:def 8;

                then w1 = (((r8 " ) * (r1 " )) * w4) by A20, RLVECT_1:def 7;

                hence contradiction by A9;

              end;

                case w7 = (r1 * w4);

                then ((r1 " ) * w7) = (((r1 " ) * r1) * w4) by RLVECT_1:def 7;

                then ((r1 " ) * w7) = (1 * w4) by A14, XCMPLX_0:def 7;

                then ((r1 " ) * w7) = w4 by RLVECT_1:def 8;

                then (((r1 " ) " ) * w4) = ((((r1 " ) " ) * (r1 " )) * w7) by RLVECT_1:def 7;

                then (((r1 " ) " ) * w4) = (1 * w7) by A14, XCMPLX_0:def 7;

                then (((r1 " ) " ) * w4) = w7 by RLVECT_1:def 8;

                then w1 = (((r8 " ) * ((r1 " ) " )) * w4) by A20, RLVECT_1:def 7;

                hence contradiction by A9;

              end;

            end;

            hence contradiction;

          end;

        end;

        then

         A21: ex w29,w39 be Point of ( TOP-REAL n) st w29 in Q & w39 in Q & ( LSeg (w4,w29)) c= Q & ( LSeg (w29,w39)) c= Q & ( LSeg (w39,w7)) c= Q by A1, A12, Th29;

        ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q by A1, A12, A9, Th29;

        hence thesis by A12, A21;

      end;

        suppose

         A22: a < 0 ;

        set w2 = ( 0. ( TOP-REAL n));

        

         A23: ( REAL n) c= Q

        proof

          let x be object;

          assume x in ( REAL n);

          then

          reconsider w = x as Point of ( TOP-REAL n) by EUCLID: 22;

           |.w.| >= 0 ;

          hence thesis by A1, A22;

        end;

        the carrier of ( TOP-REAL n) = ( REAL n) by EUCLID: 22;

        then

         A24: Q = the carrier of ( TOP-REAL n) by A23;

        then ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w7)) c= Q;

        hence thesis by A24;

      end;

    end;

    theorem :: JORDAN2C:49

    

     Th36: for a be Real, Q be Subset of ( TOP-REAL n), w1,w7 be Point of ( TOP-REAL n) st n >= 2 & Q = (( REAL n) \ { q : |.q.| < a }) & w1 in Q & w7 in Q & (ex r be Real st w1 = (r * w7) or w7 = (r * w1)) holds ex w2,w3,w4,w5,w6 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q & ( LSeg (w4,w5)) c= Q & ( LSeg (w5,w6)) c= Q & ( LSeg (w6,w7)) c= Q

    proof

      let a be Real, Q be Subset of ( TOP-REAL n), w1,w7 be Point of ( TOP-REAL n);

      reconsider y1 = w1 as Element of ( REAL n) by EUCLID: 22;

      assume

       A1: n >= 2 & Q = (( REAL n) \ { q : |.q.| < a }) & w1 in Q & w7 in Q & ex r be Real st w1 = (r * w7) or w7 = (r * w1);

      then

      consider r8 be Real such that

       A2: w1 = (r8 * w7) or w7 = (r8 * w1);

      per cases ;

        suppose

         A3: a > 0 ;

        now

          assume w1 = ( 0. ( TOP-REAL n));

          then |.w1.| = 0 by TOPRNS_1: 23;

          then w1 in { q : |.q.| < a } by A3;

          hence contradiction by A1, XBOOLE_0:def 5;

        end;

        then w1 <> ( 0* n) by EUCLID: 70;

        then

        consider y be Element of ( REAL n) such that

         A4: not ex r be Real st y = (r * y1) or y1 = (r * y) by A1, Th34;

        set y4 = ((a / |.y.|) * y);

        reconsider w4 = y4 as Point of ( TOP-REAL n) by EUCLID: 22;

         A5:

        now

          

           A6: ( 0 * y1) = ( 0 * w1)

          .= ( 0. ( TOP-REAL n)) by RLVECT_1: 10

          .= ( 0* n) by EUCLID: 70;

          assume |.y.| = 0 ;

          hence contradiction by A4, A6, EUCLID: 8;

        end;

        then

         A7: (a / |.y.|) > 0 by A3, XREAL_1: 139;

         A8:

        now

          reconsider y9 = y, y19 = y1 as Element of (n -tuples_on REAL );

          given r be Real such that

           A9: w1 = (r * w4) or w4 = (r * w1);

          y1 = ((r * (a / |.y.|)) * y) or ((((a / |.y.|) " ) * (a / |.y.|)) * y9) = (((a / |.y.|) " ) * (r * y1)) by A9, RVSUM_1: 49;

          then y1 = ((r * (a / |.y.|)) * y) or ((((a / |.y.|) " ) * (a / |.y.|)) * y) = ((((a / |.y.|) " ) * r) * y19) by RVSUM_1: 49;

          then

           A10: y1 = ((r * (a / |.y.|)) * y9) or (1 * y) = ((((a / |.y.|) " ) * r) * y1) by A7, XCMPLX_0:def 7;

          per cases by A10, RVSUM_1: 52;

            suppose y1 = ((r * (a / |.y.|)) * y);

            hence contradiction by A4;

          end;

            suppose y = ((((a / |.y.|) " ) * r) * y1);

            hence contradiction by A4;

          end;

        end;

        

         A11: |.w4.| = ( |.(a / |.y.|).| * |.y.|) by EUCLID: 11

        .= ((a / |.y.|) * |.y.|) by A3, ABSVALUE:def 1

        .= a by A5, XCMPLX_1: 87;

         A12:

        now

          assume w4 in { q : |.q.| < a };

          then ex q st q = w4 & |.q.| < a;

          hence contradiction by A11;

        end;

        then

         A13: w4 in Q by A1, XBOOLE_0:def 5;

        now

          given r1 be Real such that

           A14: w4 = (r1 * w7) or w7 = (r1 * w4);

           A15:

          now

            assume r1 = 0 ;

            then w4 = ( 0. ( TOP-REAL n)) or w7 = ( 0. ( TOP-REAL n)) by A14, RLVECT_1: 10;

            then |.w4.| = 0 or |.w7.| = 0 by TOPRNS_1: 23;

            then w4 in { q : |.q.| < a } or w7 in { q2 where q2 be Point of ( TOP-REAL n) : |.q2.| < a } by A3;

            hence contradiction by A1, A12, XBOOLE_0:def 5;

          end;

          now

            per cases by A2;

              case

               A16: w1 = (r8 * w7);

              now

                per cases by A14;

                  case w4 = (r1 * w7);

                  then ((r1 " ) * w4) = (((r1 " ) * r1) * w7) by RLVECT_1:def 7;

                  then ((r1 " ) * w4) = (1 * w7) by A15, XCMPLX_0:def 7;

                  then ((r1 " ) * w4) = w7 by RLVECT_1:def 8;

                  then w1 = ((r8 * (r1 " )) * w4) by A16, RLVECT_1:def 7;

                  hence contradiction by A8;

                end;

                  case w7 = (r1 * w4);

                  then ((r1 " ) * w7) = (((r1 " ) * r1) * w4) by RLVECT_1:def 7;

                  then ((r1 " ) * w7) = (1 * w4) by A15, XCMPLX_0:def 7;

                  then ((r1 " ) * w7) = w4 by RLVECT_1:def 8;

                  then (((r1 " ) " ) * w4) = ((((r1 " ) " ) * (r1 " )) * w7) by RLVECT_1:def 7;

                  then (((r1 " ) " ) * w4) = (1 * w7) by A15, XCMPLX_0:def 7;

                  then (((r1 " ) " ) * w4) = w7 by RLVECT_1:def 8;

                  then w1 = ((r8 * ((r1 " ) " )) * w4) by A16, RLVECT_1:def 7;

                  hence contradiction by A8;

                end;

              end;

              hence contradiction;

            end;

              case

               A17: w7 = (r8 * w1);

               A18:

              now

                assume r8 = 0 ;

                then w7 = ( 0. ( TOP-REAL n)) by A17, RLVECT_1: 10;

                then |.w7.| = 0 by TOPRNS_1: 23;

                then w7 in { q : |.q.| < a } by A3;

                hence contradiction by A1, XBOOLE_0:def 5;

              end;

              ((r8 " ) * w7) = (((r8 " ) * r8) * w1) by A17, RLVECT_1:def 7;

              then ((r8 " ) * w7) = (1 * w1) by A18, XCMPLX_0:def 7;

              then

               A19: ((r8 " ) * w7) = w1 by RLVECT_1:def 8;

              now

                per cases by A14;

                  case w4 = (r1 * w7);

                  then ((r1 " ) * w4) = (((r1 " ) * r1) * w7) by RLVECT_1:def 7;

                  then ((r1 " ) * w4) = (1 * w7) by A15, XCMPLX_0:def 7;

                  then ((r1 " ) * w4) = w7 by RLVECT_1:def 8;

                  then w1 = (((r8 " ) * (r1 " )) * w4) by A19, RLVECT_1:def 7;

                  hence contradiction by A8;

                end;

                  case w7 = (r1 * w4);

                  then ((r1 " ) * w7) = (((r1 " ) * r1) * w4) by RLVECT_1:def 7;

                  then ((r1 " ) * w7) = (1 * w4) by A15, XCMPLX_0:def 7;

                  then ((r1 " ) * w7) = w4 by RLVECT_1:def 8;

                  then (((r1 " ) " ) * w4) = ((((r1 " ) " ) * (r1 " )) * w7) by RLVECT_1:def 7;

                  then (((r1 " ) " ) * w4) = (1 * w7) by A15, XCMPLX_0:def 7;

                  then (((r1 " ) " ) * w4) = w7 by RLVECT_1:def 8;

                  then w1 = (((r8 " ) * ((r1 " ) " )) * w4) by A19, RLVECT_1:def 7;

                  hence contradiction by A8;

                end;

              end;

              hence contradiction;

            end;

          end;

          hence contradiction;

        end;

        then

         A20: ex w29,w39 be Point of ( TOP-REAL n) st w29 in Q & w39 in Q & ( LSeg (w4,w29)) c= Q & ( LSeg (w29,w39)) c= Q & ( LSeg (w39,w7)) c= Q by A1, A13, Th30;

        ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q by A1, A13, A8, Th30;

        hence thesis by A13, A20;

      end;

        suppose

         A21: a <= 0 ;

        set w2 = ( 0. ( TOP-REAL n));

        

         A22: ( REAL n) c= Q

        proof

          let x be object;

           A23:

          now

            assume x in { q : |.q.| < a };

            then ex q be Point of ( TOP-REAL n) st q = x & |.q.| < a;

            hence contradiction by A21;

          end;

          assume x in ( REAL n);

          hence thesis by A1, A23, XBOOLE_0:def 5;

        end;

        the carrier of ( TOP-REAL n) = ( REAL n) by EUCLID: 22;

        then

         A24: Q = the carrier of ( TOP-REAL n) by A22;

        then ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w7)) c= Q;

        hence thesis by A24;

      end;

    end;

    theorem :: JORDAN2C:50

    

     Th37: for a be Real st n >= 1 holds { q : |.q.| > a } <> {}

    proof

      let a be Real;

      assume

       A1: n >= 1;

      now

         |.(a + 1).| >= 0 & ( sqrt 1) <= ( sqrt n) by A1, COMPLEX1: 46, SQUARE_1: 26;

        then

         A2: ( |.(a + 1).| * 1) <= ( |.(a + 1).| * ( sqrt n)) by SQUARE_1: 18, XREAL_1: 64;

        

         A3: (a + 1) <= |.(a + 1).| by ABSVALUE: 4;

        assume not ((a + 1) * ( 1.REAL n)) in { q : |.q.| > a };

        then

         A4: |.((a + 1) * ( 1.REAL n)).| <= a;

        

         A5: a < (a + 1) by XREAL_1: 29;

         |.((a + 1) * ( 1.REAL n)).| = ( |.(a + 1).| * |.( 1.REAL n).|) by TOPRNS_1: 7

        .= ( |.(a + 1).| * ( sqrt n)) by EUCLID: 73;

        then (a + 1) <= |.((a + 1) * ( 1.REAL n)).| by A2, A3, XXREAL_0: 2;

        hence contradiction by A4, A5, XXREAL_0: 2;

      end;

      hence thesis;

    end;

    theorem :: JORDAN2C:51

    

     Th38: for a be Real, P be Subset of ( TOP-REAL n) st n >= 2 & P = { q : |.q.| > a } holds P is connected

    proof

      let a be Real, P be Subset of ( TOP-REAL n);

      assume

       A1: n >= 2 & P = { q : |.q.| > a };

      then

      reconsider Q = P as non empty Subset of ( TOP-REAL n) by Th37, XXREAL_0: 2;

      for w1,w7 be Point of ( TOP-REAL n) st w1 in Q & w7 in Q & w1 <> w7 holds ex f be Function of I[01] , (( TOP-REAL n) | Q) st f is continuous & w1 = (f . 0 ) & w7 = (f . 1)

      proof

        let w1,w7 be Point of ( TOP-REAL n);

        assume that

         A2: w1 in Q & w7 in Q and w1 <> w7;

        per cases ;

          suppose not (ex r be Real st w1 = (r * w7) or w7 = (r * w1));

          then ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w7)) c= Q by A1, A2, Th29;

          hence thesis by A2, Th26;

        end;

          suppose ex r be Real st w1 = (r * w7) or w7 = (r * w1);

          then ex w2,w3,w4,w5,w6 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q & ( LSeg (w4,w5)) c= Q & ( LSeg (w5,w6)) c= Q & ( LSeg (w6,w7)) c= Q by A1, A2, Th35;

          hence thesis by A2, Th27;

        end;

      end;

      hence thesis by JORDAN1: 2;

    end;

    theorem :: JORDAN2C:52

    

     Th39: for a be Real st n >= 1 holds (( REAL n) \ { q : |.q.| < a }) <> {}

    proof

      let a be Real;

      

       A1: { q : |.q.| > a } c= (( REAL n) \ { q2 : |.q2.| < a })

      proof

        let x be object;

        assume x in { q : |.q.| > a };

        then

        consider q such that

         A2: q = x and

         A3: |.q.| > a;

         A4:

        now

          assume x in { q2 : |.q2.| < a };

          then ex q2 st q2 = x & |.q2.| < a;

          hence contradiction by A2, A3;

        end;

        q in the carrier of ( TOP-REAL n);

        then q in ( REAL n) by EUCLID: 22;

        hence thesis by A2, A4, XBOOLE_0:def 5;

      end;

      assume n >= 1;

      hence thesis by A1, Th37, XBOOLE_1: 3;

    end;

    theorem :: JORDAN2C:53

    

     Th40: for a be Real, P be Subset of ( TOP-REAL n) st n >= 2 & P = (( REAL n) \ { q : |.q.| < a }) holds P is connected

    proof

      let a be Real, P be Subset of ( TOP-REAL n);

      assume

       A1: n >= 2 & P = (( REAL n) \ { q : |.q.| < a });

      then

      reconsider Q = P as non empty Subset of ( TOP-REAL n) by Th39, XXREAL_0: 2;

      for w1,w7 be Point of ( TOP-REAL n) st w1 in Q & w7 in Q & w1 <> w7 holds ex f be Function of I[01] , (( TOP-REAL n) | Q) st f is continuous & w1 = (f . 0 ) & w7 = (f . 1)

      proof

        let w1,w7 be Point of ( TOP-REAL n);

        assume that

         A2: w1 in Q & w7 in Q and w1 <> w7;

        per cases ;

          suppose not (ex r be Real st w1 = (r * w7) or w7 = (r * w1));

          then ex w2,w3 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w7)) c= Q by A1, A2, Th30;

          hence thesis by A2, Th26;

        end;

          suppose ex r be Real st w1 = (r * w7) or w7 = (r * w1);

          then ex w2,w3,w4,w5,w6 be Point of ( TOP-REAL n) st w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & ( LSeg (w1,w2)) c= Q & ( LSeg (w2,w3)) c= Q & ( LSeg (w3,w4)) c= Q & ( LSeg (w4,w5)) c= Q & ( LSeg (w5,w6)) c= Q & ( LSeg (w6,w7)) c= Q by A1, A2, Th36;

          hence thesis by A2, Th27;

        end;

      end;

      hence thesis by JORDAN1: 2;

    end;

    theorem :: JORDAN2C:54

    

     Th41: for a be Real, n be Nat, P be Subset of ( TOP-REAL n) st n >= 1 & P = (( REAL n) \ { q where q be Point of ( TOP-REAL n) : |.q.| < a }) holds not P is bounded

    proof

      let a be Real, n be Nat, P be Subset of ( TOP-REAL n);

      assume that

       A1: n >= 1 and

       A2: P = (( REAL n) \ { q where q be Point of ( TOP-REAL n) : |.q.| < a });

      per cases ;

        suppose

         A3: a > 0 ;

        now

          set p = the Element of P;

          assume P is bounded;

          then

          consider r be Real such that

           A4: for q be Point of ( TOP-REAL n) st q in P holds |.q.| < r by Th21;

          

           A5: P <> {} by A1, A2, Th39;

          then p in P;

          then

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

          

           A6: |.p.| < r by A4, A5;

           A7:

          now

            assume not (((a + r) + 1) * ( 1.REAL n)) in (( REAL n) \ { q where q be Point of ( TOP-REAL n) : |.q.| < a });

            then

             A8: not ((((a + r) + 1) * ( 1.REAL n)) in ( REAL n) & not (((a + r) + 1) * ( 1.REAL n)) in { q where q be Point of ( TOP-REAL n) : |.q.| < a }) by XBOOLE_0:def 5;

            (((a + r) + 1) * ( 1.REAL n)) in the carrier of ( TOP-REAL n);

            then

             A9: ex q be Point of ( TOP-REAL n) st q = (((a + r) + 1) * ( 1.REAL n)) & |.q.| < a by A8, EUCLID: 22;

            

             A10: ((a + r) + 1) <= |.((a + r) + 1).| by ABSVALUE: 4;

            (a + r) < ((a + r) + 1) & a < (a + r) by A6, XREAL_1: 29;

            then

             A11: a < ((a + r) + 1) by XXREAL_0: 2;

             |.((a + r) + 1).| >= 0 & ( sqrt 1) <= ( sqrt n) by A1, COMPLEX1: 46, SQUARE_1: 26;

            then

             A12: ( |.((a + r) + 1).| * 1) <= ( |.((a + r) + 1).| * ( sqrt n)) by SQUARE_1: 18, XREAL_1: 64;

             |.(((a + r) + 1) * ( 1.REAL n)).| = ( |.((a + r) + 1).| * |.( 1.REAL n).|) by TOPRNS_1: 7

            .= ( |.((a + r) + 1).| * ( sqrt n)) by EUCLID: 73;

            then ((a + r) + 1) <= |.(((a + r) + 1) * ( 1.REAL n)).| by A12, A10, XXREAL_0: 2;

            hence contradiction by A9, A11, XXREAL_0: 2;

          end;

          

           A13: ((a + r) + 1) <= |.((a + r) + 1).| by ABSVALUE: 4;

           |.((a + r) + 1).| >= 0 & ( sqrt 1) <= ( sqrt n) by A1, COMPLEX1: 46, SQUARE_1: 26;

          then

           A14: ( |.((a + r) + 1).| * 1) <= ( |.((a + r) + 1).| * ( sqrt n)) by SQUARE_1: 18, XREAL_1: 64;

          

           A15: (a + r) < ((a + r) + 1) by XREAL_1: 29;

           |.(((a + r) + 1) * ( 1.REAL n)).| = ( |.((a + r) + 1).| * |.( 1.REAL n).|) by TOPRNS_1: 7

          .= ( |.((a + r) + 1).| * ( sqrt n)) by EUCLID: 73;

          then ((a + r) + 1) <= |.(((a + r) + 1) * ( 1.REAL n)).| by A14, A13, XXREAL_0: 2;

          then

           A16: (a + r) < |.(((a + r) + 1) * ( 1.REAL n)).| by A15, XXREAL_0: 2;

          r < (r + a) by A3, XREAL_1: 29;

          hence contradiction by A2, A4, A7, A16, XXREAL_0: 2;

        end;

        hence thesis;

      end;

        suppose

         A17: a <= 0 ;

        now

          { q where q be Point of ( TOP-REAL n) : |.q.| < a } c= the carrier of ( TOP-REAL n)

          proof

            let z be object;

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

            then ex q be Point of ( TOP-REAL n) st q = z & |.q.| < a;

            hence thesis;

          end;

          then

          reconsider Q = { q where q be Point of ( TOP-REAL n) : |.q.| < a } as Subset of ( TOP-REAL n);

          set d = the Element of Q;

          assume { q where q be Point of ( TOP-REAL n) : |.q.| < a } <> {} ;

          then d in { q where q be Point of ( TOP-REAL n) : |.q.| < a };

          then ex q be Point of ( TOP-REAL n) st q = d & |.q.| < a;

          hence contradiction by A17;

        end;

        then P = ( [#] ( TOP-REAL n)) by A2, EUCLID: 22;

        hence thesis by A1, Th22;

      end;

    end;

    theorem :: JORDAN2C:55

    

     Th42: for a be Real, P be Subset of ( TOP-REAL 1) st P = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r > a } holds P is convex

    proof

      let a be Real, P be Subset of ( TOP-REAL 1);

      assume

       A1: P = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r > a };

      for w1,w2 be Point of ( TOP-REAL 1) st w1 in P & w2 in P holds ( LSeg (w1,w2)) c= P

      proof

        let w1,w2 be Point of ( TOP-REAL 1);

        assume that

         A2: w1 in P and

         A3: w2 in P;

        consider q2 be Point of ( TOP-REAL 1) such that

         A4: q2 = w2 and

         A5: ex r st q2 = <*r*> & r > a by A1, A3;

        consider q1 be Point of ( TOP-REAL 1) such that

         A6: q1 = w1 and

         A7: ex r st q1 = <*r*> & r > a by A1, A2;

        consider r2 such that

         A8: q2 = <*r2*> and

         A9: r2 > a by A5;

        consider r1 such that

         A10: q1 = <*r1*> and

         A11: r1 > a by A7;

        thus ( LSeg (w1,w2)) c= P

        proof

          let x be object;

          assume x in ( LSeg (w1,w2));

          then

          consider r3 be Real such that

           A12: x = (((1 - r3) * w1) + (r3 * w2)) and

           A13: 0 <= r3 and

           A14: r3 <= 1;

          

           A15: (1 - r3) >= 0 by A14, XREAL_1: 48;

          per cases ;

            suppose

             A16: r3 > 0 ;

            

             A17: ((1 - r3) * r1) >= ((1 - r3) * a) & (((1 - r3) * a) + (r3 * a)) = a by A11, A15, XREAL_1: 64;

            (r3 * r2) > (r3 * a) by A9, A16, XREAL_1: 68;

            then

             A18: (((1 - r3) * r1) + (r3 * r2)) > a by A17, XREAL_1: 8;

             <*(((1 - r3) * r1) + (r3 * r2))*> = ( |[((1 - r3) * r1)]| + |[(r3 * r2)]|) by JORDAN2B: 22

            .= (((1 - r3) * |[r1]|) + |[(r3 * r2)]|) by JORDAN2B: 21

            .= (((1 - r3) * |[r1]|) + (r3 * |[r2]|)) by JORDAN2B: 21;

            hence thesis by A1, A6, A10, A4, A8, A12, A18;

          end;

            suppose r3 <= 0 ;

            then r3 = 0 by A13;

            

            then x = (w1 + ( 0 * w2)) by A12, RLVECT_1:def 8

            .= (w1 + ( 0. ( TOP-REAL 1))) by RLVECT_1: 10

            .= w1 by RLVECT_1: 4;

            hence thesis by A2;

          end;

        end;

      end;

      hence thesis by JORDAN1:def 1;

    end;

    theorem :: JORDAN2C:56

    

     Th43: for a be Real, P be Subset of ( TOP-REAL 1) st P = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r < ( - a) } holds P is convex

    proof

      let a be Real, P be Subset of ( TOP-REAL 1);

      assume

       A1: P = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r < ( - a) };

      for w1,w2 be Point of ( TOP-REAL 1) st w1 in P & w2 in P holds ( LSeg (w1,w2)) c= P

      proof

        let w1,w2 be Point of ( TOP-REAL 1);

        assume that

         A2: w1 in P and

         A3: w2 in P;

        consider q2 be Point of ( TOP-REAL 1) such that

         A4: q2 = w2 and

         A5: ex r st q2 = <*r*> & r < ( - a) by A1, A3;

        consider q1 be Point of ( TOP-REAL 1) such that

         A6: q1 = w1 and

         A7: ex r st q1 = <*r*> & r < ( - a) by A1, A2;

        consider r2 such that

         A8: q2 = <*r2*> and

         A9: r2 < ( - a) by A5;

        consider r1 such that

         A10: q1 = <*r1*> and

         A11: r1 < ( - a) by A7;

        thus ( LSeg (w1,w2)) c= P

        proof

          let x be object;

          assume x in ( LSeg (w1,w2));

          then

          consider r3 be Real such that

           A12: x = (((1 - r3) * w1) + (r3 * w2)) and

           A13: 0 <= r3 and

           A14: r3 <= 1;

          

           A15: (1 - r3) >= 0 by A14, XREAL_1: 48;

          per cases ;

            suppose

             A16: r3 > 0 ;

            

             A17: ((1 - r3) * r1) <= ((1 - r3) * ( - a)) & (((1 - r3) * ( - a)) + (r3 * ( - a))) = ( - a) by A11, A15, XREAL_1: 64;

            (r3 * r2) < (r3 * ( - a)) by A9, A16, XREAL_1: 68;

            then

             A18: (((1 - r3) * r1) + (r3 * r2)) < ( - a) by A17, XREAL_1: 8;

             <*(((1 - r3) * r1) + (r3 * r2))*> = ( |[((1 - r3) * r1)]| + |[(r3 * r2)]|) by JORDAN2B: 22

            .= (((1 - r3) * |[r1]|) + |[(r3 * r2)]|) by JORDAN2B: 21

            .= (((1 - r3) * |[r1]|) + (r3 * |[r2]|)) by JORDAN2B: 21;

            hence thesis by A1, A6, A10, A4, A8, A12, A18;

          end;

            suppose r3 <= 0 ;

            then r3 = 0 by A13;

            

            then x = (w1 + ( 0 * w2)) by A12, RLVECT_1:def 8

            .= (w1 + ( 0. ( TOP-REAL 1))) by RLVECT_1: 10

            .= w1 by RLVECT_1: 4;

            hence thesis by A2;

          end;

        end;

      end;

      hence thesis by JORDAN1:def 1;

    end;

    ::$Canceled

    theorem :: JORDAN2C:59

    

     Th44: for W be Subset of ( Euclid 1), a be Real st W = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r > a } holds not W is bounded

    proof

      let W be Subset of ( Euclid 1), a be Real;

      assume

       A1: W = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r > a };

       |.a.| >= 0 by COMPLEX1: 46;

      then

       A2: (( |.a.| + |.a.|) + |.a.|) >= ( 0 + |.a.|) by XREAL_1: 6;

      assume W is bounded;

      then

      consider r such that

       A3: 0 < r and

       A4: for x,y be Point of ( Euclid 1) st x in W & y in W holds ( dist (x,y)) <= r;

      

       A5: ((r + |.a.|) * ( 1.REAL 1)) = ((r + |.a.|) * <*1*>) by FINSEQ_2: 59

      .= <*((r + |.a.|) * 1)*> by RVSUM_1: 47;

      reconsider z2 = ((r + |.a.|) * ( 1.REAL 1)) as Point of ( Euclid 1) by FINSEQ_2: 131;

      a <= |.a.| & ( 0 + |.a.|) < (r + |.a.|) by A3, ABSVALUE: 4, XREAL_1: 6;

      then a < (r + |.a.|) by XXREAL_0: 2;

      then

       A6: ((r + |.a.|) * ( 1.REAL 1)) in W by A1, A5;

      

       A7: ((3 * (r + |.a.|)) * ( 1.REAL 1)) = ((3 * (r + |.a.|)) * <*1*>) by FINSEQ_2: 59

      .= <*((3 * (r + |.a.|)) * 1)*> by RVSUM_1: 47;

      reconsider z1 = ((3 * (r + |.a.|)) * ( 1.REAL 1)) as Point of ( Euclid 1) by FINSEQ_2: 131;

      ( dist (z1,z2)) = |.(((3 * (r + |.a.|)) * ( 1.REAL 1)) - ((r + |.a.|) * ( 1.REAL 1))).| by JGRAPH_1: 28

      .= |.(((((r + |.a.|) + (r + |.a.|)) + (r + |.a.|)) - (r + |.a.|)) * ( 1.REAL 1)).| by RLVECT_1: 35

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * |.( 1.REAL 1).|) by TOPRNS_1: 7

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt 1)) by EUCLID: 73;

      then

       A8: ((r + |.a.|) + (r + |.a.|)) <= ( dist (z1,z2)) by ABSVALUE: 4, SQUARE_1: 18;

      

       A9: 0 <= |.a.| by COMPLEX1: 46;

      then ((r + |.a.|) + 0 ) < ((r + |.a.|) + (r + |.a.|)) by A3, XREAL_1: 6;

      then

       A10: (r + |.a.|) < ( dist (z1,z2)) by A8, XXREAL_0: 2;

      (r + 0 ) <= (r + |.a.|) by A9, XREAL_1: 6;

      then

       A11: r < ( dist (z1,z2)) by A10, XXREAL_0: 2;

      (3 * r) > 0 by A3, XREAL_1: 129;

      then a <= |.a.| & ( 0 + |.a.|) < ((3 * r) + (3 * |.a.|)) by A2, ABSVALUE: 4, XREAL_1: 8;

      then a < (3 * (r + |.a.|)) by XXREAL_0: 2;

      then ((3 * (r + |.a.|)) * ( 1.REAL 1)) in W by A1, A7;

      hence contradiction by A4, A6, A11;

    end;

    theorem :: JORDAN2C:60

    

     Th45: for W be Subset of ( Euclid 1), a be Real st W = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r < ( - a) } holds not W is bounded

    proof

      let W be Subset of ( Euclid 1), a be Real;

       |.a.| >= 0 by COMPLEX1: 46;

      then

       A1: (( |.a.| + |.a.|) + |.a.|) >= ( 0 + |.a.|) by XREAL_1: 6;

      assume

       A2: W = { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r < ( - a) };

      assume W is bounded;

      then

      consider r such that

       A3: 0 < r and

       A4: for x,y be Point of ( Euclid 1) st x in W & y in W holds ( dist (x,y)) <= r;

      

       A5: (( - (3 * (r + |.a.|))) * ( 1.REAL 1)) = (( - (3 * (r + |.a.|))) * <*1*>) by FINSEQ_2: 59

      .= <*(( - (3 * (r + |.a.|))) * 1)*> by RVSUM_1: 47;

      reconsider z1 = (( - (3 * (r + |.a.|))) * ( 1.REAL 1)) as Point of ( Euclid 1) by FINSEQ_2: 131;

      (3 * r) > 0 by A3, XREAL_1: 129;

      then a <= |.a.| & ( 0 + |.a.|) < ((3 * r) + (3 * |.a.|)) by A1, ABSVALUE: 4, XREAL_1: 8;

      then a < (3 * (r + |.a.|)) by XXREAL_0: 2;

      then ( - a) > ( - (3 * (r + |.a.|))) by XREAL_1: 24;

      then

       A6: (( - (3 * (r + |.a.|))) * ( 1.REAL 1)) in { q where q be Point of ( TOP-REAL 1) : ex r st q = <*r*> & r < ( - a) } by A5;

      

       A7: (( - (r + |.a.|)) * ( 1.REAL 1)) = (( - (r + |.a.|)) * <*1*>) by FINSEQ_2: 59

      .= <*(( - (r + |.a.|)) * 1)*> by RVSUM_1: 47;

      reconsider z2 = (( - (r + |.a.|)) * ( 1.REAL 1)) as Point of ( Euclid 1) by FINSEQ_2: 131;

      ( dist (z1,z2)) = |.((( - (3 * (r + |.a.|))) * ( 1.REAL 1)) - (( - (r + |.a.|)) * ( 1.REAL 1))).| by JGRAPH_1: 28

      .= |.((( - (3 * (r + |.a.|))) - ( - (r + |.a.|))) * ( 1.REAL 1)).| by RLVECT_1: 35

      .= |.( - ((( - (3 * (r + |.a.|))) - ( - (r + |.a.|))) * ( 1.REAL 1))).| by TOPRNS_1: 26

      .= |.(( - (( - (3 * (r + |.a.|))) + ( - ( - (r + |.a.|))))) * ( 1.REAL 1)).| by RLVECT_1: 79

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * |.( 1.REAL 1).|) by TOPRNS_1: 7

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt 1)) by EUCLID: 73;

      then

       A8: ((r + |.a.|) + (r + |.a.|)) <= ( dist (z1,z2)) by ABSVALUE: 4, SQUARE_1: 18;

      

       A9: 0 <= |.a.| by COMPLEX1: 46;

      then ((r + |.a.|) + 0 ) < ((r + |.a.|) + (r + |.a.|)) by A3, XREAL_1: 6;

      then

       A10: (r + |.a.|) < ( dist (z1,z2)) by A8, XXREAL_0: 2;

      (r + 0 ) <= (r + |.a.|) by A9, XREAL_1: 6;

      then

       A11: r < ( dist (z1,z2)) by A10, XXREAL_0: 2;

      a <= |.a.| & ( 0 + |.a.|) < (r + |.a.|) by A3, ABSVALUE: 4, XREAL_1: 6;

      then a < (r + |.a.|) by XXREAL_0: 2;

      then ( - a) > ( - (r + |.a.|)) by XREAL_1: 24;

      then (( - (r + |.a.|)) * ( 1.REAL 1)) in W by A2, A7;

      hence contradiction by A2, A4, A6, A11;

    end;

    theorem :: JORDAN2C:61

    

     Th46: for W be Subset of ( Euclid n), a be Real st n >= 2 & W = { q : |.q.| > a } holds not W is bounded

    proof

      let W be Subset of ( Euclid n), a be Real;

      assume

       A1: n >= 2 & W = { q : |.q.| > a };

      

       A2: 1 <= n by A1, XXREAL_0: 2;

      then

       A3: 1 <= ( sqrt n) by SQUARE_1: 18, SQUARE_1: 26;

      assume W is bounded;

      then

      consider r such that

       A4: 0 < r and

       A5: for x,y be Point of ( Euclid n) st x in W & y in W holds ( dist (x,y)) <= r;

      

       A6: (r + |.a.|) <= |.(r + |.a.|).| by ABSVALUE: 4;

       |.(r + |.a.|).| >= 0 & 1 <= ( sqrt n) by A2, COMPLEX1: 46, SQUARE_1: 18, SQUARE_1: 26;

      then

       A7: ( |.(r + |.a.|).| * 1) <= ( |.(r + |.a.|).| * ( sqrt n)) by XREAL_1: 64;

      a <= |.a.| & |.a.| < (r + |.a.|) by A4, ABSVALUE: 4, XREAL_1: 29;

      then

       A8: a < (r + |.a.|) by XXREAL_0: 2;

       |.( - ((r + |.a.|) * ( 1.REAL n))).| = |.((r + |.a.|) * ( 1.REAL n)).| by TOPRNS_1: 26

      .= ( |.(r + |.a.|).| * |.( 1.REAL n).|) by TOPRNS_1: 7

      .= ( |.(r + |.a.|).| * ( sqrt n)) by EUCLID: 73;

      then (r + |.a.|) <= |.( - ((r + |.a.|) * ( 1.REAL n))).| by A7, A6, XXREAL_0: 2;

      then a < |.( - ((r + |.a.|) * ( 1.REAL n))).| by A8, XXREAL_0: 2;

      then

       A9: ( - ((r + |.a.|) * ( 1.REAL n))) in W by A1;

      then

      reconsider z2 = ( - ((r + |.a.|) * ( 1.REAL n))) as Point of ( Euclid n);

      

       A10: (r + |.a.|) <= |.(r + |.a.|).| by ABSVALUE: 4;

       |.(r + |.a.|).| >= 0 by COMPLEX1: 46;

      then

       A11: ( |.(r + |.a.|).| * 1) <= ( |.(r + |.a.|).| * ( sqrt n)) by A3, XREAL_1: 64;

       |.((r + |.a.|) * ( 1.REAL n)).| = ( |.(r + |.a.|).| * |.( 1.REAL n).|) by TOPRNS_1: 7

      .= ( |.(r + |.a.|).| * ( sqrt n)) by EUCLID: 73;

      then (r + |.a.|) <= |.((r + |.a.|) * ( 1.REAL n)).| by A11, A10, XXREAL_0: 2;

      then a < |.((r + |.a.|) * ( 1.REAL n)).| by A8, XXREAL_0: 2;

      then

       A12: ((r + |.a.|) * ( 1.REAL n)) in W by A1;

      then

      reconsider z1 = ((r + |.a.|) * ( 1.REAL n)) as Point of ( Euclid n);

      

       A13: ((r + |.a.|) + (r + |.a.|)) <= |.((r + |.a.|) + (r + |.a.|)).| by ABSVALUE: 4;

       |.((r + |.a.|) + (r + |.a.|)).| >= 0 by COMPLEX1: 46;

      then

       A14: ( |.((r + |.a.|) + (r + |.a.|)).| * 1) <= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt n)) by A3, XREAL_1: 64;

      

       A15: 0 <= |.a.| by COMPLEX1: 46;

      then

       A16: (r + 0 ) <= (r + |.a.|) by XREAL_1: 6;

      

       A17: ((r + |.a.|) + 0 ) < ((r + |.a.|) + (r + |.a.|)) by A4, A15, XREAL_1: 6;

      ( dist (z1,z2)) = |.(((r + |.a.|) * ( 1.REAL n)) - ( - ((r + |.a.|) * ( 1.REAL n)))).| by JGRAPH_1: 28

      .= |.(((r + |.a.|) * ( 1.REAL n)) + ((r + |.a.|) * ( 1.REAL n))).|

      .= |.(((r + |.a.|) + (r + |.a.|)) * ( 1.REAL n)).| by RLVECT_1:def 6

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * |.( 1.REAL n).|) by TOPRNS_1: 7

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt n)) by EUCLID: 73;

      then ((r + |.a.|) + (r + |.a.|)) <= ( dist (z1,z2)) by A14, A13, XXREAL_0: 2;

      then (r + |.a.|) < ( dist (z1,z2)) by A17, XXREAL_0: 2;

      then r < ( dist (z1,z2)) by A16, XXREAL_0: 2;

      hence contradiction by A5, A12, A9;

    end;

    theorem :: JORDAN2C:62

    

     Th47: for W be Subset of ( Euclid n), a be Real st n >= 2 & W = (( REAL n) \ { q : |.q.| < a }) holds not W is bounded

    proof

      let W be Subset of ( Euclid n), a be Real;

      reconsider 1R = ( 1.REAL n) as Point of ( TOP-REAL n);

      assume

       A1: n >= 2 & W = (( REAL n) \ { q : |.q.| < a });

      assume W is bounded;

      then

      consider r such that

       A2: 0 < r and

       A3: for x,y be Point of ( Euclid n) st x in W & y in W holds ( dist (x,y)) <= r;

      

       A4: 0 <= |.a.| by COMPLEX1: 46;

      then

       A5: ((r + |.a.|) + 0 ) < ((r + |.a.|) + (r + |.a.|)) by A2, XREAL_1: 6;

      n >= 1 by A1, XXREAL_0: 2;

      then

       A6: 1 <= ( sqrt n) by SQUARE_1: 18, SQUARE_1: 26;

       A7:

      now

        a <= |.a.| & |.a.| < (r + |.a.|) by A2, ABSVALUE: 4, XREAL_1: 29;

        then

         A8: a < (r + |.a.|) by XXREAL_0: 2;

        assume ( - ((r + |.a.|) * ( 1.REAL n))) in { q : |.q.| < a };

        then

         A9: ex q be Point of ( TOP-REAL n) st q = ( - ((r + |.a.|) * ( 1.REAL n))) & |.q.| < a;

         |.(r + |.a.|).| >= 0 by COMPLEX1: 46;

        then

         A10: ( |.(r + |.a.|).| * 1) <= ( |.(r + |.a.|).| * ( sqrt n)) by A6, XREAL_1: 64;

        

         A11: (r + |.a.|) <= |.(r + |.a.|).| by ABSVALUE: 4;

         |.( - ((r + |.a.|) * ( 1.REAL n))).| = |.((r + |.a.|) * ( 1.REAL n)).| by TOPRNS_1: 26

        .= ( |.(r + |.a.|).| * |.( 1.REAL n).|) by TOPRNS_1: 7

        .= ( |.(r + |.a.|).| * ( sqrt n)) by EUCLID: 73;

        then (r + |.a.|) <= |.( - ((r + |.a.|) * ( 1.REAL n))).| by A10, A11, XXREAL_0: 2;

        hence contradiction by A9, A8, XXREAL_0: 2;

      end;

       A12:

      now

        a <= |.a.| & |.a.| < (r + |.a.|) by A2, ABSVALUE: 4, XREAL_1: 29;

        then

         A13: a < (r + |.a.|) by XXREAL_0: 2;

        assume ((r + |.a.|) * ( 1.REAL n)) in { q : |.q.| < a };

        then

         A14: ex q be Point of ( TOP-REAL n) st q = ((r + |.a.|) * ( 1.REAL n)) & |.q.| < a;

         |.(r + |.a.|).| >= 0 by COMPLEX1: 46;

        then

         A15: ( |.(r + |.a.|).| * 1) <= ( |.(r + |.a.|).| * ( sqrt n)) by A6, XREAL_1: 64;

        

         A16: (r + |.a.|) <= |.(r + |.a.|).| by ABSVALUE: 4;

         |.((r + |.a.|) * ( 1.REAL n)).| = ( |.(r + |.a.|).| * |.( 1.REAL n).|) by TOPRNS_1: 7

        .= ( |.(r + |.a.|).| * ( sqrt n)) by EUCLID: 73;

        then (r + |.a.|) <= |.((r + |.a.|) * ( 1.REAL n)).| by A15, A16, XXREAL_0: 2;

        hence contradiction by A14, A13, XXREAL_0: 2;

      end;

      reconsider z2 = ( - ((r + |.a.|) * ( 1.REAL n))) as Point of ( Euclid n) by EUCLID: 22;

      reconsider z1 = ((r + |.a.|) * ( 1.REAL n)) as Point of ( Euclid n) by EUCLID: 22;

      

       A17: ((r + |.a.|) + (r + |.a.|)) <= |.((r + |.a.|) + (r + |.a.|)).| by ABSVALUE: 4;

       |.((r + |.a.|) + (r + |.a.|)).| >= 0 by COMPLEX1: 46;

      then

       A18: ( |.((r + |.a.|) + (r + |.a.|)).| * 1) <= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt n)) by A6, XREAL_1: 64;

      ( dist (z1,z2)) = |.(((r + |.a.|) * ( 1.REAL n)) - ( - ((r + |.a.|) * ( 1.REAL n)))).| by JGRAPH_1: 28

      .= |.(((r + |.a.|) * 1R) + ( - ( - ((r + |.a.|) * 1R)))).|

      .= |.(((r + |.a.|) * 1R) + ((r + |.a.|) * 1R)).|

      .= |.(((r + |.a.|) + (r + |.a.|)) * 1R).| by RLVECT_1:def 6

      .= |.(((r + |.a.|) + (r + |.a.|)) * ( 1.REAL n)).|

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * |.( 1.REAL n).|) by TOPRNS_1: 7

      .= ( |.((r + |.a.|) + (r + |.a.|)).| * ( sqrt n)) by EUCLID: 73;

      then ((r + |.a.|) + (r + |.a.|)) <= ( dist (z1,z2)) by A18, A17, XXREAL_0: 2;

      then

       A19: (r + |.a.|) < ( dist (z1,z2)) by A5, XXREAL_0: 2;

      (r + 0 ) <= (r + |.a.|) by A4, XREAL_1: 6;

      then

       A20: r < ( dist (z1,z2)) by A19, XXREAL_0: 2;

      ( - ((r + |.a.|) * ( 1.REAL n))) in the carrier of ( TOP-REAL n);

      then ( - ((r + |.a.|) * ( 1.REAL n))) in ( REAL n) by EUCLID: 22;

      then

       A21: ( - ((r + |.a.|) * ( 1.REAL n))) in W by A1, A7, XBOOLE_0:def 5;

      ((r + |.a.|) * ( 1.REAL n)) in the carrier of ( TOP-REAL n);

      then ((r + |.a.|) * ( 1.REAL n)) in ( REAL n) by EUCLID: 22;

      then ((r + |.a.|) * ( 1.REAL n)) in W by A1, A12, XBOOLE_0:def 5;

      hence contradiction by A3, A21, A20;

    end;

    theorem :: JORDAN2C:63

    

     Th48: for P,P1 be Subset of ( TOP-REAL n), Q be Subset of ( TOP-REAL n), W be Subset of ( Euclid n) st P = W & P is connected & not W is bounded & P1 = ( Component_of ( Down (P,(Q ` )))) & W misses Q holds P1 is_outside_component_of Q

    proof

      let P,P1 be Subset of ( TOP-REAL n), Q be Subset of ( TOP-REAL n), W be Subset of ( Euclid n);

      assume that

       A1: P = W and

       A2: P is connected and

       A3: not W is bounded and

       A4: P1 = ( Component_of ( Down (P,(Q ` )))) and

       A5: W misses Q;

      

       A6: (( TOP-REAL n) | P) is connected by A2, CONNSP_1:def 3;

      

       A7: ( Down (P,(Q ` ))) = (P \ Q) by SUBSET_1: 13

      .= P by A1, A5, XBOOLE_1: 83;

      then

      reconsider P0 = P as Subset of (( TOP-REAL n) | (Q ` ));

      reconsider W0 = ( Component_of P0) as Subset of ( Euclid n) by A4, A7, TOPREAL3: 8;

      P0 c= (Q ` ) by A1, A5, SUBSET_1: 23;

      then ((( TOP-REAL n) | (Q ` )) | P0) = (( TOP-REAL n) | P) by PRE_TOPC: 7;

      then

       A8: P0 is connected by A6, CONNSP_1:def 3;

       A9:

      now

        assume for D be Subset of ( Euclid n) st D = P1 holds D is bounded;

        then W0 is bounded by A4, A7;

        hence contradiction by A1, A3, A8, CONNSP_3: 1, TBSP_1: 14;

      end;

      

       A10: W <> ( {} ( Euclid n)) by A3;

      

       A11: (W /\ Q) = {} by A5;

      now

        assume (Q ` ) = {} ;

        then Q = (( {} the carrier of ( TOP-REAL n)) ` );

        hence contradiction by A1, A10, A11, XBOOLE_1: 28;

      end;

      then

      reconsider Q1 = (Q ` ) as non empty Subset of ( TOP-REAL n);

      (( TOP-REAL n) | Q1) is non empty;

      then ( Component_of P0) is a_component by A1, A10, A8, CONNSP_3: 9;

      hence thesis by A4, A7, A9, Th8;

    end;

    theorem :: JORDAN2C:64

    

     Th49: for A be Subset of ( Euclid n), B be non empty Subset of ( Euclid n), C be Subset of (( Euclid n) | B) st A = C & C is bounded holds A is bounded

    proof

      let A be Subset of ( Euclid n), B be non empty Subset of ( Euclid n), C be Subset of (( Euclid n) | B);

      assume that

       A1: A = C and

       A2: C is bounded;

      consider r0 be Real such that

       A3: 0 < r0 and

       A4: for x,y be Point of (( Euclid n) | B) st x in C & y in C holds ( dist (x,y)) <= r0 by A2;

      for x,y be Point of ( Euclid n) st x in A & y in A holds ( dist (x,y)) <= r0

      proof

        let x,y be Point of ( Euclid n);

        assume

         A5: x in A & y in A;

        then

        reconsider x0 = x, y0 = y as Point of (( Euclid n) | B) by A1;

        (the distance of (( Euclid n) | B) . (x0,y0)) = (the distance of ( Euclid n) . (x,y)) & (the distance of (( Euclid n) | B) . (x0,y0)) = ( dist (x0,y0)) by TOPMETR:def 1;

        hence thesis by A1, A4, A5;

      end;

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:65

    

     Th50: for A be Subset of ( TOP-REAL n) st A is compact holds A is bounded

    proof

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

      assume

       A1: A is compact;

      A c= the carrier of (( TOP-REAL n) | A) by PRE_TOPC: 8;

      then

      reconsider A2 = A as Subset of (( TOP-REAL n) | A);

      per cases ;

        suppose A <> {} ;

        then

        reconsider A1 = A as non empty Subset of ( Euclid n) by TOPREAL3: 8;

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

        then ( [#] (( TOP-REAL n) | A)) is compact by A1, COMPTS_1: 2;

        then

         A2: (( TOP-REAL n) | A) is compact by COMPTS_1: 1;

        ( TopSpaceMetr (( Euclid n) | A1)) = (( TOP-REAL n) | A) by EUCLID: 63;

        then (( Euclid n) | A1) is totally_bounded by A2, TBSP_1: 9;

        then

         A3: (( Euclid n) | A1) is bounded by TBSP_1: 19;

        ( [#] (( Euclid n) | A1)) = A1 by TOPMETR:def 2;

        then A1 is bounded by A3, Th49;

        hence thesis by Th5;

      end;

        suppose A = {} ;

        hence thesis;

      end;

    end;

    registration

      let n be Element of NAT ;

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

      coherence by Th50;

    end

    theorem :: JORDAN2C:66

    

     Th51: for A be Subset of ( TOP-REAL n) st 1 <= n & A is bounded holds (A ` ) <> {}

    proof

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

      assume that

       A1: 1 <= n and

       A2: A is bounded;

      consider r be Real such that

       A3: for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r by A2, Th21;

       |.r.| >= 0 by COMPLEX1: 46;

      then

       A4: ( |.r.| * |.( 1.REAL n).|) >= ( |.r.| * 1) by A1, EUCLID: 75, XREAL_1: 64;

       |.(r * ( 1.REAL n)).| = ( |.r.| * |.( 1.REAL n).|) & r <= |.r.| by ABSVALUE: 4, TOPRNS_1: 7;

      then not (r * ( 1.REAL n)) in A by A3, A4, XXREAL_0: 2;

      hence thesis by XBOOLE_0:def 5;

    end;

    theorem :: JORDAN2C:67

    

     Th52: for r be Real holds (ex B be Subset of ( Euclid n) st B = { q : |.q.| < r }) & for A be Subset of ( Euclid n) st A = { q1 : |.q1.| < r } holds A is bounded

    proof

      let r be Real;

      

       A1: { q : |.q.| < r } c= the carrier of ( Euclid n)

      proof

        let x be object;

        assume x in { q : |.q.| < r };

        then ex q be Point of ( TOP-REAL n) st q = x & |.q.| < r;

        then x in the carrier of ( TOP-REAL n);

        hence thesis by TOPREAL3: 8;

      end;

      hence ex B be Subset of ( Euclid n) st B = { q : |.q.| < r };

      reconsider C = { q1 : |.q1.| < r } as Subset of ( TOP-REAL n) by A1, TOPREAL3: 8;

      let A be Subset of ( Euclid n);

      for q be Point of ( TOP-REAL n) st q in C holds |.q.| < r

      proof

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

        assume q in C;

        then ex q1 be Point of ( TOP-REAL n) st q1 = q & |.q1.| < r;

        hence thesis;

      end;

      then

       A2: C is bounded by Th21;

      assume A = { q1 : |.q1.| < r };

      hence thesis by A2, Th5;

    end;

    theorem :: JORDAN2C:68

    

     Th53: for A be Subset of ( TOP-REAL n) st n >= 2 & A is bounded holds ( UBD A) is_outside_component_of A

    proof

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

      assume that

       A1: n >= 2 and

       A2: A is bounded;

      reconsider C = A as bounded Subset of ( Euclid n) by A2, Th5;

      per cases ;

        suppose

         A3: C <> {} ;

        set x0 = the Element of C;

        

         A4: x0 in C by A3;

        then

        reconsider q1 = x0 as Point of ( TOP-REAL n);

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

        reconsider x0 as Point of ( Euclid n) by A4;

        consider r be Real such that 0 < r and

         A5: for x,y be Point of ( Euclid n) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

        set R0 = ((r + ( dist (o,x0))) + 1);

        reconsider W = (( REAL n) \ { q where q be Point of ( TOP-REAL n) : |.q.| < R0 }) as Subset of ( Euclid n);

         A6:

        now

          assume W meets A;

          then

          consider z be object such that

           A7: z in W and

           A8: z in A by XBOOLE_0: 3;

          

           A9: not z in { q where q be Point of ( TOP-REAL n) : |.q.| < R0 } by A7, XBOOLE_0:def 5;

          reconsider z as Point of ( Euclid n) by A7;

          ( dist (x0,z)) <= r by A5, A8;

          then ( dist (o,z)) <= (( dist (o,x0)) + ( dist (x0,z))) & (( dist (o,x0)) + ( dist (x0,z))) <= (( dist (o,x0)) + r) by METRIC_1: 4, XREAL_1: 6;

          then

           A10: ( dist (o,z)) <= (( dist (o,x0)) + r) by XXREAL_0: 2;

          reconsider q1 = z as Point of ( TOP-REAL n) by TOPREAL3: 8;

          

           A11: |.(q1 - ( 0. ( TOP-REAL n))).| = ( dist (o,z)) by JGRAPH_1: 28;

           |.q1.| >= ((r + ( dist (o,x0))) + 1) by A9;

          then ( dist (o,z)) >= ((r + ( dist (o,x0))) + 1) by A11, RLVECT_1: 13;

          then (r + (( dist (o,x0)) + 1)) <= (r + ( dist (o,x0))) by A10, XXREAL_0: 2;

          then (( dist (o,x0)) + 1) <= (( dist (o,x0)) + 0 ) by XREAL_1: 6;

          hence contradiction by XREAL_1: 6;

        end;

        reconsider P = W as Subset of ( TOP-REAL n) by TOPREAL3: 8;

        reconsider P as Subset of ( TOP-REAL n);

        the carrier of (( TOP-REAL n) | (A ` )) = (A ` ) by PRE_TOPC: 8;

        then

        reconsider P1 = ( Component_of ( Down (P,(A ` )))) as Subset of ( TOP-REAL n) by XBOOLE_1: 1;

        

         A12: P is connected by A1, Th40;

        

         A13: ( UBD A) c= P1

        proof

          

           A14: (( TOP-REAL n) | P) is connected by A12, CONNSP_1:def 3;

          

           A15: P c= (A ` ) by A6, SUBSET_1: 23;

          then ( Down (P,(A ` ))) = P by XBOOLE_1: 28;

          then ((( TOP-REAL n) | (A ` )) | ( Down (P,(A ` )))) is connected by A15, A14, PRE_TOPC: 7;

          then

           A16: ( Down (P,(A ` ))) is connected by CONNSP_1:def 3;

          reconsider G = (A ` ) as non empty Subset of ( TOP-REAL n) by A1, A2, Th51, XXREAL_0: 2;

          let z be object;

          assume z in ( UBD A);

          then

          consider y be set such that

           A17: z in y and

           A18: y in { B where B be Subset of ( TOP-REAL n) : B is_outside_component_of A } by TARSKI:def 4;

          consider B be Subset of ( TOP-REAL n) such that

           A19: y = B and

           A20: B is_outside_component_of A by A18;

          consider C2 be Subset of (( TOP-REAL n) | (A ` )) such that

           A21: C2 = B and

           A22: C2 is a_component and

           A23: not C2 is bounded Subset of ( Euclid n) by A20, Th8;

          consider D2 be Subset of ( Euclid n) such that

           A24: D2 = { q : |.q.| < R0 } by Th52;

          reconsider D2 as Subset of ( Euclid n);

          

           A25: A c= D2

          proof

            let z be object;

            

             A26: |.q1.| = |.(q1 - ( 0. ( TOP-REAL n))).| by RLVECT_1: 13

            .= ( dist (x0,o)) by JGRAPH_1: 28;

            assume

             A27: z in A;

            then

            reconsider q2 = z as Point of ( TOP-REAL n);

            reconsider x1 = q2 as Point of ( Euclid n) by TOPREAL3: 8;

             |.(q2 - q1).| = ( dist (x1,x0)) & ( dist (x1,x0)) <= r by A5, A27, JGRAPH_1: 28;

            then

             A28: ( |.(q2 - q1).| + |.q1.|) <= (r + ( dist (o,x0))) by A26, XREAL_1: 6;

            

             A29: (r + ( dist (o,x0))) < ((r + ( dist (o,x0))) + 1) by XREAL_1: 29;

             |.q2.| = |.((q2 - q1) + q1).| & |.((q2 - q1) + q1).| <= ( |.(q2 - q1).| + |.q1.|) by RLVECT_4: 1, TOPRNS_1: 29;

            then |.q2.| <= (r + ( dist (o,x0))) by A28, XXREAL_0: 2;

            then |.q2.| < ((r + ( dist (o,x0))) + 1) by A29, XXREAL_0: 2;

            hence thesis by A24;

          end;

          the carrier of ( Euclid n) = the carrier of ( TOP-REAL n) by TOPREAL3: 8;

          then (D2 ` ) c= (the carrier of ( TOP-REAL n) \ A) by A25, XBOOLE_1: 34;

          then

           A30: (P /\ (D2 ` )) c= (P /\ (A ` )) by XBOOLE_1: 26;

          now

            reconsider D = C2 as Subset of ( Euclid n) by A21, TOPREAL3: 8;

            assume

             A31: (W /\ C2) = {} ;

            

             A32: C2 c= { q : |.q.| < R0 }

            proof

              let x8 be object;

              assume

               A33: x8 in C2;

              assume not x8 in { q : |.q.| < R0 };

              then x8 in W by A21, A23, A33, EUCLID: 22, XBOOLE_0:def 5;

              hence contradiction by A31, A33, XBOOLE_0:def 4;

            end;

             not D is bounded by A23;

            hence contradiction by A24, A32, Th52, TBSP_1: 14;

          end;

          then (( Down (P,(A ` ))) /\ C2) <> {} by A24, A30, XBOOLE_1: 3, XBOOLE_1: 26;

          then

           A34: ( Down (P,(A ` ))) meets C2;

          C2 is connected by A22, CONNSP_1:def 5;

          then C2 c= ( Component_of ( Down (P,(A ` )))) by A16, A34, CONNSP_3: 16;

          hence thesis by A17, A19, A21;

        end;

         not W is bounded by A1, Th47;

        then P1 is_outside_component_of A & P1 c= ( UBD A) by A12, A6, Th14, Th48;

        hence thesis by A13, XBOOLE_0:def 10;

      end;

        suppose

         A35: C = {} ;

        ( REAL n) c= the carrier of ( Euclid n);

        then

        reconsider W = ( REAL n) as Subset of ( Euclid n);

        (W /\ A) = {} by A35;

        then

         A36: W misses A;

        reconsider P = W as Subset of ( TOP-REAL n) by TOPREAL3: 8;

        reconsider P as Subset of ( TOP-REAL n);

        the carrier of (( TOP-REAL n) | (A ` )) = (A ` ) by PRE_TOPC: 8;

        then

        reconsider P1 = ( Component_of ( Down (P,(A ` )))) as Subset of ( TOP-REAL n) by XBOOLE_1: 1;

        ( [#] ( TOP-REAL n)) is a_component;

        then

         A37: ( [#] the TopStruct of ( TOP-REAL n)) is a_component by CONNSP_1: 45;

         not W is bounded by A1, Th20, XXREAL_0: 2;

        then

         A38: P1 is_outside_component_of A by A36, Th19, Th48;

        A = ( {} ( TOP-REAL n)) by A35;

        then

         A39: ( UBD A) = ( REAL n) by A1, Th23, XXREAL_0: 2;

        ( [#] ( TOP-REAL n)) = ( REAL n) & (( TOP-REAL n) | ( [#] ( TOP-REAL n))) = the TopStruct of ( TOP-REAL n) by EUCLID: 22, TSEP_1: 93;

        hence ( UBD A) is_outside_component_of A by A35, A38, A39, A37, CONNSP_3: 7;

      end;

    end;

    theorem :: JORDAN2C:69

    

     Th54: for a be Real, P be Subset of ( TOP-REAL n) st P = { q : |.q.| < a } holds P is convex

    proof

      let a be Real, P be Subset of ( TOP-REAL n);

      assume

       A1: P = { q : |.q.| < a };

      for p1,p2 be Point of ( TOP-REAL n) st p1 in P & p2 in P holds ( LSeg (p1,p2)) c= P

      proof

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

        assume that

         A2: p1 in P and

         A3: p2 in P;

        

         A4: ex q2 st q2 = p2 & |.q2.| < a by A1, A3;

        

         A5: ex q1 st q1 = p1 & |.q1.| < a by A1, A2;

        ( LSeg (p1,p2)) c= P

        proof

          let x be object;

          assume

           A6: x in ( LSeg (p1,p2));

          then

          consider r such that

           A7: x = (((1 - r) * p1) + (r * p2)) and

           A8: 0 <= r and

           A9: r <= 1;

          

           A10: |.((1 - r) * p1).| = ( |.(1 - r).| * |.p1.|) by TOPRNS_1: 7;

          reconsider q = x as Point of ( TOP-REAL n) by A6;

          

           A11: |.(((1 - r) * p1) + (r * p2)).| <= ( |.((1 - r) * p1).| + |.(r * p2).|) by TOPRNS_1: 29;

          

           A12: (1 - r) >= 0 by A9, XREAL_1: 48;

          then

           A13: |.(1 - r).| = (1 - r) by ABSVALUE:def 1;

          per cases ;

            suppose

             A14: (1 - r) > 0 ;

            

             A15: |.(r * p2).| = ( |.r.| * |.p2.|) & r = |.r.| by A8, ABSVALUE:def 1, TOPRNS_1: 7;

             0 <= |.r.| by COMPLEX1: 46;

            then

             A16: ( |.r.| * |.p2.|) <= ( |.r.| * a) by A4, XREAL_1: 64;

            ( |.(1 - r).| * |.p1.|) < ( |.(1 - r).| * a) by A5, A13, A14, XREAL_1: 68;

            then ( |.((1 - r) * p1).| + |.(r * p2).|) < (((1 - r) * a) + (r * a)) by A10, A13, A16, A15, XREAL_1: 8;

            then |.q.| < a by A7, A11, XXREAL_0: 2;

            hence thesis by A1;

          end;

            suppose (1 - r) <= 0 ;

            then ((1 - r) + r) = ( 0 + r) by A12;

            then 0 < |.r.| by ABSVALUE:def 1;

            then

             A17: ( |.r.| * |.p2.|) < ( |.r.| * a) by A4, XREAL_1: 68;

            

             A18: r = |.r.| by A8, ABSVALUE:def 1;

            ( |.(1 - r).| * |.p1.|) <= ( |.(1 - r).| * a) & |.(r * p2).| = ( |.r.| * |.p2.|) by A5, A12, A13, TOPRNS_1: 7, XREAL_1: 64;

            then ( |.((1 - r) * p1).| + |.(r * p2).|) < (((1 - r) * a) + (r * a)) by A10, A13, A17, A18, XREAL_1: 8;

            then |.q.| < a by A7, A11, XXREAL_0: 2;

            hence thesis by A1;

          end;

        end;

        hence thesis;

      end;

      hence thesis by JORDAN1:def 1;

    end;

    theorem :: JORDAN2C:70

    

     Th55: for a be Real, P be Subset of ( TOP-REAL n) st P = ( Ball (u,a)) holds P is convex

    proof

      let a be Real, P be Subset of ( TOP-REAL n);

      assume

       A1: P = ( Ball (u,a));

      for p1,p2 be Point of ( TOP-REAL n) st p1 in P & p2 in P holds ( LSeg (p1,p2)) c= P

      proof

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

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

        assume that

         A2: p1 in P and

         A3: p2 in P;

        

         A4: P = { q where q be Element of ( Euclid n) : ( dist (u,q)) < a } by A1, METRIC_1: 17;

        then ex q2 be Point of ( Euclid n) st q2 = p2 & ( dist (u,q2)) < a by A3;

        then

         A5: |.(p - p2).| < a by JGRAPH_1: 28;

        

         A6: for p3 be Point of ( TOP-REAL n) st |.(p - p3).| < a holds p3 in P

        proof

          let p3 be Point of ( TOP-REAL n);

          reconsider u3 = p3 as Point of ( Euclid n) by TOPREAL3: 8;

          assume |.(p - p3).| < a;

          then ( dist (u,u3)) < a by JGRAPH_1: 28;

          hence thesis by A4;

        end;

        ex q1 be Point of ( Euclid n) st q1 = p1 & ( dist (u,q1)) < a by A2, A4;

        then

         A7: |.(p - p1).| < a by JGRAPH_1: 28;

        ( LSeg (p1,p2)) c= P

        proof

          let x be object;

          assume

           A8: x in ( LSeg (p1,p2));

          then

          consider r such that

           A9: x = (((1 - r) * p1) + (r * p2)) and

           A10: 0 <= r and

           A11: r <= 1;

          reconsider q = x as Point of ( TOP-REAL n) by A8;

          

           A12: |.((1 - r) * (p - p1)).| = ( |.(1 - r).| * |.(p - p1).|) by TOPRNS_1: 7;

          (((1 - r) * p) + (r * p)) = (((1 - r) + r) * p) by RLVECT_1:def 6

          .= p by RLVECT_1:def 8;

          

          then |.(p - (((1 - r) * p1) + (r * p2))).| = |.(((((1 - r) * p) + (r * p)) - ((1 - r) * p1)) - (r * p2)).| by RLVECT_1: 27

          .= |.(((((1 - r) * p) + ( - ((1 - r) * p1))) + (r * p)) + ( - (r * p2))).| by RLVECT_1:def 3

          .= |.((((1 - r) * p) + ( - ((1 - r) * p1))) + ((r * p) + ( - (r * p2)))).| by RLVECT_1:def 3

          .= |.((((1 - r) * p) + ((1 - r) * ( - p1))) + ((r * p) + ( - (r * p2)))).| by RLVECT_1: 25

          .= |.(((1 - r) * (p - p1)) + ((r * p) + ( - (r * p2)))).| by RLVECT_1:def 5

          .= |.(((1 - r) * (p - p1)) + ((r * p) + (r * ( - p2)))).| by RLVECT_1: 25

          .= |.(((1 - r) * (p - p1)) + (r * (p - p2))).| by RLVECT_1:def 5;

          then

           A13: |.(p - (((1 - r) * p1) + (r * p2))).| <= ( |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).|) by TOPRNS_1: 29;

          

           A14: (1 - r) >= 0 by A11, XREAL_1: 48;

          then

           A15: |.(1 - r).| = (1 - r) by ABSVALUE:def 1;

          per cases ;

            suppose

             A16: (1 - r) > 0 ;

            

             A17: |.(r * (p - p2)).| = ( |.r.| * |.(p - p2).|) & r = |.r.| by A10, ABSVALUE:def 1, TOPRNS_1: 7;

             0 <= |.r.| by COMPLEX1: 46;

            then

             A18: ( |.r.| * |.(p - p2).|) <= ( |.r.| * a) by A5, XREAL_1: 64;

            ( |.(1 - r).| * |.(p - p1).|) < ( |.(1 - r).| * a) by A7, A15, A16, XREAL_1: 68;

            then ( |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).|) < (((1 - r) * a) + (r * a)) by A12, A15, A18, A17, XREAL_1: 8;

            then |.(p - q).| < a by A9, A13, XXREAL_0: 2;

            hence thesis by A6;

          end;

            suppose (1 - r) <= 0 ;

            then ((1 - r) + r) = ( 0 + r) by A14;

            then 0 < |.r.| by ABSVALUE:def 1;

            then

             A19: ( |.r.| * |.(p - p2).|) < ( |.r.| * a) by A5, XREAL_1: 68;

            

             A20: r = |.r.| by A10, ABSVALUE:def 1;

            ( |.(1 - r).| * |.(p - p1).|) <= ( |.(1 - r).| * a) & |.(r * (p - p2)).| = ( |.r.| * |.(p - p2).|) by A7, A14, A15, TOPRNS_1: 7, XREAL_1: 64;

            then ( |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).|) < (((1 - r) * a) + (r * a)) by A12, A15, A19, A20, XREAL_1: 8;

            then |.(p - q).| < a by A9, A13, XXREAL_0: 2;

            hence thesis by A6;

          end;

        end;

        hence thesis;

      end;

      hence thesis by JORDAN1:def 1;

    end;

    reserve R for Subset of ( TOP-REAL n);

    reserve P,Q for Subset of ( TOP-REAL n);

    ::$Canceled

    theorem :: JORDAN2C:72

    

     Th56: p <> q & p in ( Ball (u,r)) & q in ( Ball (u,r)) implies ex h be Function of I[01] , ( TOP-REAL n) st h is continuous & (h . 0 ) = p & (h . 1) = q & ( rng h) c= ( Ball (u,r))

    proof

      assume that

       A1: p <> q and

       A2: p in ( Ball (u,r)) & q in ( Ball (u,r));

      reconsider Q = ( Ball (u,r)) as Subset of ( TOP-REAL n) by TOPREAL3: 8;

      Q is convex by Th55;

      then

       A3: ( LSeg (p,q)) c= ( Ball (u,r)) by A2, JORDAN1:def 1;

      reconsider P = ( LSeg (p,q)) as Subset of ( TOP-REAL n);

      ( LSeg (p,q)) is_an_arc_of (p,q) by A1, TOPREAL1: 9;

      then

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

       A4: f is being_homeomorphism and

       A5: (f . 0 ) = p & (f . 1) = q by TOPREAL1:def 1;

      reconsider h = f as Function of I[01] , ( TOP-REAL n) by JORDAN6: 3;

      take h;

      ( rng f) = ( [#] (( TOP-REAL n) | P)) & f is continuous by A4;

      hence thesis by A3, A5, JORDAN6: 3, PRE_TOPC:def 5;

    end;

    theorem :: JORDAN2C:73

    

     Th57: for f be Function of I[01] , ( TOP-REAL n) st f is continuous & (f . 0 ) = p1 & (f . 1) = p2 & p in ( Ball (u,r)) & p2 in ( Ball (u,r)) holds ex h be Function of I[01] , ( TOP-REAL n) st h is continuous & (h . 0 ) = p1 & (h . 1) = p & ( rng h) c= (( rng f) \/ ( Ball (u,r)))

    proof

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

      assume that

       A1: f is continuous & (f . 0 ) = p1 & (f . 1) = p2 and

       A2: p in ( Ball (u,r)) & p2 in ( Ball (u,r));

      per cases ;

        suppose p2 <> p;

        then ( LSeg (p2,p)) is_an_arc_of (p2,p) by TOPREAL1: 9;

        then

        consider f1 be Function of I[01] , (( TOP-REAL n) | ( LSeg (p2,p))) such that

         A3: f1 is being_homeomorphism and

         A4: (f1 . 0 ) = p2 & (f1 . 1) = p by TOPREAL1:def 1;

        reconsider f2 = f1 as Function of I[01] , ( TOP-REAL n) by JORDAN6: 3;

        ( rng f1) = ( [#] (( TOP-REAL n) | ( LSeg (p2,p)))) by A3;

        then ( rng f2) = ( LSeg (p2,p)) by PRE_TOPC:def 5;

        then

         A5: (( rng f) \/ ( rng f2)) c= (( rng f) \/ ( Ball (u,r))) by A2, TOPREAL3: 21, XBOOLE_1: 9;

        f1 is continuous by A3;

        then f2 is continuous by JORDAN6: 3;

        then ex h3 be Function of I[01] , ( TOP-REAL n) st h3 is continuous & p1 = (h3 . 0 ) & p = (h3 . 1) & ( rng h3) c= (( rng f) \/ ( rng f2)) by A1, A4, BORSUK_2: 13;

        hence thesis by A5, XBOOLE_1: 1;

      end;

        suppose p2 = p;

        hence thesis by A1, XBOOLE_1: 7;

      end;

    end;

    theorem :: JORDAN2C:74

    

     Th58: for f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= P & (f . 0 ) = p1 & (f . 1) = p2 & p in ( Ball (u,r)) & p2 in ( Ball (u,r)) & ( Ball (u,r)) c= P holds ex f1 be Function of I[01] , ( TOP-REAL n) st f1 is continuous & ( rng f1) c= P & (f1 . 0 ) = p1 & (f1 . 1) = p

    proof

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

      assume f is continuous & ( rng f) c= P & (f . 0 ) = p1 & (f . 1) = p2 & p in ( Ball (u,r)) & p2 in ( Ball (u,r)) & ( Ball (u,r)) c= P;

      then (ex f3 be Function of I[01] , ( TOP-REAL n) st f3 is continuous & (f3 . 0 ) = p1 & (f3 . 1) = p & ( rng f3) c= (( rng f) \/ ( Ball (u,r)))) & (( rng f) \/ ( Ball (u,r))) c= P by Th57, XBOOLE_1: 8;

      hence thesis by XBOOLE_1: 1;

    end;

    theorem :: JORDAN2C:75

    

     Th59: for p holds for P be Subset of ( TOP-REAL n) st R is connected & R is open & P = { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } holds P is open

    proof

      let p;

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

      assume that

       A1: R is connected & R is open and

       A2: P = { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q };

      

       A3: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider P9 = P as Subset of ( TopSpaceMetr ( Euclid n));

      

       A4: P c= R

      proof

        let x be object;

        assume x in P;

        then ex q st q = x & q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q by A2;

        hence thesis;

      end;

      now

        

         A5: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

        then

        reconsider R9 = R as Subset of ( TopSpaceMetr ( Euclid n));

        let u;

        reconsider p2 = u as Point of ( TOP-REAL n) by TOPREAL3: 8;

        assume

         A6: u in P;

        R9 is open by A1, A5, PRE_TOPC: 30;

        then

        consider r be Real such that

         A7: r > 0 and

         A8: ( Ball (u,r)) c= R9 by A4, A6, TOPMETR: 15;

        take r;

        thus r > 0 by A7;

        reconsider r9 = r as Real;

        

         A9: p2 in ( Ball (u,r9)) by A7, TBSP_1: 11;

        ( Ball (u,r)) c= P9

        proof

          assume not thesis;

          then

          consider x be object such that

           A10: x in ( Ball (u,r)) and

           A11: not x in P;

          x in R by A8, A10;

          then

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

          per cases by A2, A8, A10, A11;

            suppose

             A12: q = p;

             A13:

            now

              assume

               A14: q = p2;

              ex p3 st p3 = p2 & p3 <> p & p3 in R & not ex f1 be Function of I[01] , ( TOP-REAL n) st f1 is continuous & ( rng f1) c= R & (f1 . 0 ) = p & (f1 . 1) = p3 by A2, A6;

              hence contradiction by A12, A14;

            end;

            u in ( Ball (u,r9)) by A7, TBSP_1: 11;

            then

             A15: ex f2 be Function of I[01] , ( TOP-REAL n) st f2 is continuous & (f2 . 0 ) = q & (f2 . 1) = p2 & ( rng f2) c= ( Ball (u,r9)) by A10, A13, Th56;

             not p2 in P

            proof

              assume p2 in P;

              then ex q4 st q4 = p2 & q4 <> p & q4 in R & not ex f1 be Function of I[01] , ( TOP-REAL n) st f1 is continuous & ( rng f1) c= R & (f1 . 0 ) = p & (f1 . 1) = q4 by A2;

              hence contradiction by A8, A12, A15, XBOOLE_1: 1;

            end;

            hence contradiction by A6;

          end;

            suppose

             A16: ex f1 be Function of I[01] , ( TOP-REAL n) st f1 is continuous & ( rng f1) c= R & (f1 . 0 ) = p & (f1 . 1) = q;

             not p2 in P

            proof

              assume p2 in P;

              then ex q4 st q4 = p2 & q4 <> p & q4 in R & not ex f1 be Function of I[01] , ( TOP-REAL n) st f1 is continuous & ( rng f1) c= R & (f1 . 0 ) = p & (f1 . 1) = q4 by A2;

              hence contradiction by A8, A9, A10, A16, Th58;

            end;

            hence contradiction by A6;

          end;

        end;

        hence ( Ball (u,r)) c= P9;

      end;

      then P9 is open by TOPMETR: 15;

      hence thesis by A3, PRE_TOPC: 30;

    end;

    theorem :: JORDAN2C:76

    

     Th60: for P be Subset of ( TOP-REAL n) st R is connected & R is open & p in R & P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } holds P is open

    proof

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

      assume that

       A1: R is connected & R is open and

       A2: p in R and

       A3: P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q };

      

       A4: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider P9 = P as Subset of ( TopSpaceMetr ( Euclid n));

      reconsider RR = R as Subset of ( TopSpaceMetr ( Euclid n)) by A4;

      now

        let u;

        reconsider p2 = u as Point of ( TOP-REAL n) by TOPREAL3: 8;

        assume u in P9;

        then

        consider q1 such that

         A5: q1 = u and

         A6: q1 = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q1 by A3;

         A7:

        now

          per cases by A6;

            suppose q1 = p;

            hence p2 in R by A2, A5;

          end;

            suppose q1 <> p & ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q1;

            then

            consider f2 be Function of I[01] , ( TOP-REAL n) such that f2 is continuous and

             A8: ( rng f2) c= R and (f2 . 0 ) = p and

             A9: (f2 . 1) = q1;

            ( dom f2) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

            then 1 in ( dom f2) by XXREAL_1: 1;

            then u in ( rng f2) by A5, A9, FUNCT_1:def 3;

            hence p2 in R by A8;

          end;

        end;

        RR is open by A1, A4, PRE_TOPC: 30;

        then

        consider r be Real such that

         A10: r > 0 and

         A11: ( Ball (u,r)) c= R by A7, TOPMETR: 15;

        take r;

        thus r > 0 by A10;

        reconsider r9 = r as Real;

        

         A12: p2 in ( Ball (u,r9)) by A10, TBSP_1: 11;

        thus ( Ball (u,r)) c= P

        proof

          let x be object;

          assume

           A13: x in ( Ball (u,r));

          then

          reconsider q = x as Point of ( TOP-REAL n) by A11, TARSKI:def 3;

          per cases ;

            suppose q = p;

            hence thesis by A3;

          end;

            suppose

             A14: q <> p;

             A15:

            now

              assume q1 = p;

              then p in ( Ball (u,r9)) by A5, A10, TBSP_1: 11;

              then

              consider f2 be Function of I[01] , ( TOP-REAL n) such that

               A16: f2 is continuous & (f2 . 0 ) = p & (f2 . 1) = q and

               A17: ( rng f2) c= ( Ball (u,r9)) by A13, A14, Th56;

              ( rng f2) c= R by A11, A17;

              hence thesis by A3, A16;

            end;

            now

              assume q1 <> p;

              then ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q by A5, A6, A11, A12, A13, Th58;

              hence thesis by A3;

            end;

            hence thesis by A15;

          end;

        end;

      end;

      then P9 is open by TOPMETR: 15;

      hence thesis by A4, PRE_TOPC: 30;

    end;

    theorem :: JORDAN2C:77

    

     Th61: for R be Subset of ( TOP-REAL n) holds p in R & P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } implies P c= R

    proof

      let R be Subset of ( TOP-REAL n);

      assume that

       A1: p in R and

       A2: P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q };

      let x be object;

      assume x in P;

      then

      consider q such that

       A3: q = x and

       A4: q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q by A2;

      per cases by A4;

        suppose q = p;

        hence thesis by A1, A3;

      end;

        suppose ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q;

        then

        consider f1 be Function of I[01] , ( TOP-REAL n) such that f1 is continuous and

         A5: ( rng f1) c= R and (f1 . 0 ) = p and

         A6: (f1 . 1) = q;

        reconsider P1 = ( rng f1) as Subset of ( TOP-REAL n);

        ( dom f1) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

        then 1 in ( dom f1) by XXREAL_1: 1;

        then q in P1 by A6, FUNCT_1:def 3;

        hence thesis by A3, A5;

      end;

    end;

    theorem :: JORDAN2C:78

    

     Th62: for R be Subset of ( TOP-REAL n), p be Point of ( TOP-REAL n) st R is connected & R is open & p in R & P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } holds R c= P

    proof

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

      assume that

       A1: R is connected & R is open and

       A2: p in R and

       A3: P = { q : q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q };

      reconsider R9 = R as non empty Subset of ( TOP-REAL n) by A2;

      

       A4: p in P by A3;

      set P2 = (R \ P);

      reconsider P22 = P2 as Subset of ( TOP-REAL n);

      

       A5: ( [#] (( TOP-REAL n) | R)) = R by PRE_TOPC:def 5;

      then

      reconsider P11 = P, P12 = P22 as Subset of (( TOP-REAL n) | R) by A2, A3, Th61, XBOOLE_1: 36;

      reconsider P11, P12 as Subset of (( TOP-REAL n) | R);

      (P \/ (R \ P)) = (P \/ R) by XBOOLE_1: 39;

      then

       A6: P11 misses P12 & ( [#] (( TOP-REAL n) | R)) = (P11 \/ P12) by A5, XBOOLE_1: 12, XBOOLE_1: 79;

      now

        let x be object;

         A7:

        now

          assume

           A8: x in P2;

          then

          reconsider q2 = x as Point of ( TOP-REAL n);

           not x in P by A8, XBOOLE_0:def 5;

          then

           A9: q2 <> p & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q2 by A3;

          q2 in R by A8, XBOOLE_0:def 5;

          hence x in { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } by A9;

        end;

        now

          assume x in { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q };

          then

           A10: ex q3 st q3 = x & q3 <> p & q3 in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q3;

          then not ex q st q = x & (q = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q);

          then not x in P by A3;

          hence x in P2 by A10, XBOOLE_0:def 5;

        end;

        hence x in P2 iff x in { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } by A7;

      end;

      then P2 = { q : q <> p & q in R & not ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q } by TARSKI: 2;

      then P22 is open by A1, Th59;

      then

       A11: P22 in the topology of ( TOP-REAL n) by PRE_TOPC:def 2;

      reconsider PPP = P as Subset of ( TOP-REAL n);

      PPP is open by A1, A2, A3, Th60;

      then

       A12: P in the topology of ( TOP-REAL n) by PRE_TOPC:def 2;

      P11 = (P /\ ( [#] (( TOP-REAL n) | R))) by XBOOLE_1: 28;

      then P11 in the topology of (( TOP-REAL n) | R) by A12, PRE_TOPC:def 4;

      then

       A13: P11 is open by PRE_TOPC:def 2;

      P12 = (P22 /\ ( [#] (( TOP-REAL n) | R))) by XBOOLE_1: 28;

      then P12 in the topology of (( TOP-REAL n) | R) by A11, PRE_TOPC:def 4;

      then

       A14: P12 is open by PRE_TOPC:def 2;

      (( TOP-REAL n) | R9) is connected by A1, CONNSP_1:def 3;

      then P11 = ( {} (( TOP-REAL n) | R)) or P12 = ( {} (( TOP-REAL n) | R)) by A6, A13, A14, CONNSP_1: 11;

      hence thesis by A4, XBOOLE_1: 37;

    end;

    theorem :: JORDAN2C:79

    

     Th63: for R be Subset of ( TOP-REAL n), p,q be Point of ( TOP-REAL n) st R is connected & R is open & p in R & q in R & p <> q holds ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q

    proof

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

      assume that

       A1: R is connected & R is open & p in R and

       A2: q in R and

       A3: p <> q;

      set RR = { q2 : q2 = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q2 };

      RR c= the carrier of ( TOP-REAL n)

      proof

        let x be object;

        assume x in RR;

        then ex q2 st q2 = x & (q2 = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q2);

        hence thesis;

      end;

      then R c= RR by A1, Th62;

      then q in RR by A2;

      then ex q2 st q = q2 & (q2 = p or ex f be Function of I[01] , ( TOP-REAL n) st f is continuous & ( rng f) c= R & (f . 0 ) = p & (f . 1) = q2);

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:80

    

     Th64: for A be Subset of ( TOP-REAL n), a be Real st A = { q : |.q.| = a } holds (A ` ) is open & A is closed

    proof

      let A be Subset of ( TOP-REAL n), a be Real;

      assume

       A1: A = { q : |.q.| = a };

      reconsider a as Real;

      

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

      then

      reconsider P1 = (A ` ) as Subset of ( TopSpaceMetr ( Euclid n));

      for p be Point of ( Euclid n) st p in P1 holds ex r be Real st r > 0 & ( Ball (p,r)) c= P1

      proof

        let p be Point of ( Euclid n);

        reconsider q1 = p as Point of ( TOP-REAL n) by TOPREAL3: 8;

        assume p in P1;

        then not p in A by XBOOLE_0:def 5;

        then

         A3: |.q1.| <> a by A1;

        now

          per cases ;

            case

             A4: |.q1.| <= a;

            set r1 = ((a - |.q1.|) / 2);

             |.q1.| < a by A3, A4, XXREAL_0: 1;

            then

             A5: (a - |.q1.|) > 0 by XREAL_1: 50;

            ( Ball (p,r1)) c= P1

            proof

              let x be object;

              assume

               A6: x in ( Ball (p,r1));

              then

              reconsider p2 = x as Point of ( Euclid n);

              reconsider q2 = p2 as Point of ( TOP-REAL n) by TOPREAL3: 8;

              ( dist (p,p2)) < r1 by A6, METRIC_1: 11;

              then

               A7: |.(q2 - q1).| < r1 by JGRAPH_1: 28;

              now

                assume q2 in A;

                then

                 A8: ex q st q = q2 & |.q.| = a by A1;

                 |.(q2 - q1).| >= ( |.q2.| - |.q1.|) by TOPRNS_1: 32;

                then r1 > (r1 + r1) by A7, A8, XXREAL_0: 2;

                then (r1 - r1) > r1 by XREAL_1: 20;

                hence contradiction by A5;

              end;

              hence thesis by XBOOLE_0:def 5;

            end;

            hence thesis by A5, XREAL_1: 139;

          end;

            case

             A9: |.q1.| > a;

            set r1 = (( |.q1.| - a) / 2);

            

             A10: ( |.q1.| - a) > 0 by A9, XREAL_1: 50;

            ( Ball (p,r1)) c= P1

            proof

              let x be object;

              assume

               A11: x in ( Ball (p,r1));

              then

              reconsider p2 = x as Point of ( Euclid n);

              reconsider q2 = p2 as Point of ( TOP-REAL n) by TOPREAL3: 8;

              ( dist (p,p2)) < r1 by A11, METRIC_1: 11;

              then

               A12: |.(q1 - q2).| < r1 by JGRAPH_1: 28;

              now

                assume q2 in A;

                then

                 A13: ex q st q = q2 & |.q.| = a by A1;

                 |.(q1 - q2).| >= ( |.q1.| - |.q2.|) by TOPRNS_1: 32;

                then r1 > (r1 + r1) by A12, A13, XXREAL_0: 2;

                then (r1 - r1) > r1 by XREAL_1: 20;

                hence contradiction by A10;

              end;

              hence thesis by XBOOLE_0:def 5;

            end;

            hence thesis by A10, XREAL_1: 139;

          end;

        end;

        hence thesis;

      end;

      then P1 is open by TOPMETR: 15;

      hence (A ` ) is open by A2, PRE_TOPC: 30;

      hence thesis by TOPS_1: 3;

    end;

    theorem :: JORDAN2C:81

    

     Th65: for B be non empty Subset of ( TOP-REAL n) st B is open holds (( TOP-REAL n) | B) is locally_connected

    proof

      let B be non empty Subset of ( TOP-REAL n);

      

       A1: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      assume

       A2: B is open;

      for A be non empty Subset of (( TOP-REAL n) | B), C be Subset of (( TOP-REAL n) | B) st A is open & C is_a_component_of A holds C is open

      proof

        let A be non empty Subset of (( TOP-REAL n) | B), C be Subset of (( TOP-REAL n) | B);

        assume that

         A3: A is open and

         A4: C is_a_component_of A;

        consider C1 be Subset of ((( TOP-REAL n) | B) | A) such that

         A5: C1 = C and

         A6: C1 is a_component by A4, CONNSP_1:def 6;

        C1 c= ( [#] ((( TOP-REAL n) | B) | A));

        then

         A7: C1 c= A by PRE_TOPC:def 5;

        A c= the carrier of (( TOP-REAL n) | B);

        then A c= B by PRE_TOPC: 8;

        then C c= B by A5, A7;

        then

        reconsider C0 = C as Subset of ( TOP-REAL n) by XBOOLE_1: 1;

        reconsider CC = C0 as Subset of ( TopSpaceMetr ( Euclid n)) by A1;

        for p be Point of ( Euclid n) st p in C0 holds ex r be Real st r > 0 & ( Ball (p,r)) c= C0

        proof

          consider A0 be Subset of ( TOP-REAL n) such that

           A8: A0 is open and

           A9: (A0 /\ ( [#] (( TOP-REAL n) | B))) = A by A3, TOPS_2: 24;

          

           A10: (A0 /\ B) = A by A9, PRE_TOPC:def 5;

          

           A11: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

          then

          reconsider AA = (A0 /\ B) as Subset of ( TopSpaceMetr ( Euclid n));

          let p be Point of ( Euclid n);

          assume

           A12: p in C0;

          AA is open by A2, A8, A11, PRE_TOPC: 30;

          then

          consider r1 be Real such that

           A13: r1 > 0 and

           A14: ( Ball (p,r1)) c= AA by A5, A7, A12, A10, TOPMETR: 15;

          reconsider r1 as Real;

          

           A15: ( Ball (p,r1)) c= A by A9, A14, PRE_TOPC:def 5;

          then

          reconsider BL2 = ( Ball (p,r1)) as Subset of (( TOP-REAL n) | B) by XBOOLE_1: 1;

          ( Ball (p,r1)) c= ( [#] ((( TOP-REAL n) | B) | A)) by A15, PRE_TOPC:def 5;

          then

          reconsider BL = ( Ball (p,r1)) as Subset of ((( TOP-REAL n) | B) | A);

          reconsider BL as Subset of ((( TOP-REAL n) | B) | A);

          reconsider BL2 as Subset of (( TOP-REAL n) | B);

          reconsider BL1 = ( Ball (p,r1)) as Subset of ( TOP-REAL n) by TOPREAL3: 8;

          reconsider BL1 as Subset of ( TOP-REAL n);

          now

            p in BL by A13, GOBOARD6: 1;

            then (BL /\ C) <> ( {} ((( TOP-REAL n) | B) | A)) by A12, XBOOLE_0:def 4;

            then

             A16: BL meets C;

            BL1 is convex by Th55;

            then

             A17: BL2 is connected by CONNSP_1: 46;

            assume not ( Ball (p,r1)) c= C0;

            hence contradiction by A5, A6, A17, A16, CONNSP_1: 36, CONNSP_1: 46;

          end;

          hence thesis by A13;

        end;

        then CC is open by TOPMETR: 15;

        then

         A18: ( [#] (( TOP-REAL n) | B)) = B & C0 is open by A1, PRE_TOPC: 30, PRE_TOPC:def 5;

        C c= the carrier of (( TOP-REAL n) | B);

        then C c= B by PRE_TOPC: 8;

        then (C0 /\ B) = C by XBOOLE_1: 28;

        hence thesis by A18, TOPS_2: 24;

      end;

      hence thesis by CONNSP_2: 18;

    end;

    theorem :: JORDAN2C:82

    

     Th66: for B be non empty Subset of ( TOP-REAL n), A be Subset of ( TOP-REAL n), a be Real st A = { q : |.q.| = a } & (A ` ) = B holds (( TOP-REAL n) | B) is locally_connected

    proof

      let B be non empty Subset of ( TOP-REAL n), A be Subset of ( TOP-REAL n), a be Real;

      assume

       A1: A = { q : |.q.| = a } & (A ` ) = B;

      then (A ` ) is open by Th64;

      hence thesis by A1, Th65;

    end;

    theorem :: JORDAN2C:83

    

     Th67: for f be Function of ( TOP-REAL n), R^1 st (for q holds (f . q) = |.q.|) holds f is continuous

    proof

      let f be Function of ( TOP-REAL n), R^1 ;

      

       A1: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider f1 = f as Function of ( TopSpaceMetr ( Euclid n)), ( TopSpaceMetr RealSpace ) by TOPMETR:def 6;

      assume

       A2: for q holds (f . q) = |.q.|;

      now

        let r be Real, u be Element of ( Euclid n), u1 be Element of RealSpace ;

        assume that

         A3: r > 0 and

         A4: u1 = (f1 . u);

        set s1 = r;

        for w be Element of ( Euclid n), w1 be Element of RealSpace st w1 = (f1 . w) & ( dist (u,w)) < s1 holds ( dist (u1,w1)) < r

        proof

          let w be Element of ( Euclid n), w1 be Element of RealSpace ;

          assume that

           A5: w1 = (f1 . w) and

           A6: ( dist (u,w)) < s1;

          reconsider tu = u1, tw = w1 as Real;

          reconsider qw = w, qu = u as Point of ( TOP-REAL n) by TOPREAL3: 8;

          

           A7: ( dist (u1,w1)) = (the distance of RealSpace . (u1,w1))

          .= |.(tu - tw).| by METRIC_1:def 12;

          

           A8: tu = |.qu.| by A2, A4;

          w1 = |.qw.| by A2, A5;

          then ( dist (u,w)) = |.(qu - qw).| & ( dist (u1,w1)) <= |.(qu - qw).| by A7, A8, Th3, JGRAPH_1: 28;

          hence thesis by A6, XXREAL_0: 2;

        end;

        hence ex s be Real st s > 0 & for w be Element of ( Euclid n), w1 be Element of RealSpace st w1 = (f1 . w) & ( dist (u,w)) < s holds ( dist (u1,w1)) < r by A3;

      end;

      then f1 is continuous by UNIFORM1: 3;

      hence thesis by A1, PRE_TOPC: 32, TOPMETR:def 6;

    end;

    theorem :: JORDAN2C:84

    

     Th68: ex f be Function of ( TOP-REAL n), R^1 st (for q holds (f . q) = |.q.|) & f is continuous

    proof

      defpred P[ object, object] means (ex q st q = $1 & $2 = |.q.|);

      

       A1: for x be object st x in the carrier of ( TOP-REAL n) holds ex y be object st P[x, y]

      proof

        let x be object;

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

        then

        reconsider q3 = x as Point of ( TOP-REAL n);

        take |.q3.|;

        thus thesis;

      end;

      consider f1 be Function such that

       A2: ( dom f1) = the carrier of ( TOP-REAL n) & for x be object st x in the carrier of ( TOP-REAL n) holds P[x, (f1 . x)] from CLASSES1:sch 1( A1);

      ( rng f1) c= the carrier of R^1

      proof

        let z be object;

        assume z in ( rng f1);

        then

        consider xz be object such that

         A3: xz in ( dom f1) and

         A4: z = (f1 . xz) by FUNCT_1:def 3;

        consider q4 be Point of ( TOP-REAL n) such that

         A5: q4 = xz & (f1 . xz) = |.q4.| by A2, A3;

        z in REAL by A4, A5, XREAL_0:def 1;

        hence thesis by TOPMETR: 17;

      end;

      then

      reconsider f2 = f1 as Function of ( TOP-REAL n), R^1 by A2, FUNCT_2:def 1, RELSET_1: 4;

      

       A6: for q holds (f1 . q) = |.q.|

      proof

        let q;

        ex q2 st q2 = q & (f1 . q) = |.q2.| by A2;

        hence thesis;

      end;

      then f2 is continuous by Th67;

      hence thesis by A6;

    end;

    theorem :: JORDAN2C:85

    

     Th69: for g be Function of I[01] , ( TOP-REAL n) st g is continuous holds ex f be Function of I[01] , R^1 st (for t be Point of I[01] holds (f . t) = |.(g . t).|) & f is continuous

    proof

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

      consider h be Function of ( TOP-REAL n), R^1 such that

       A1: for q holds (h . q) = |.q.| and

       A2: h is continuous by Th68;

      set f1 = (h * g);

      

       A3: for t be Point of I[01] holds (f1 . t) = |.(g . t).|

      proof

        let t be Point of I[01] ;

        reconsider q = (g . t) as Point of ( TOP-REAL n);

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

        

        then (f1 . t) = (h . (g . t)) by FUNCT_1: 13

        .= |.q.| by A1;

        hence thesis;

      end;

      assume g is continuous;

      hence thesis by A2, A3;

    end;

    theorem :: JORDAN2C:86

    

     Th70: for g be Function of I[01] , ( TOP-REAL n), a be Real st g is continuous & |.(g /. 0 ).| <= a & a <= |.(g /. 1).| holds ex s be Point of I[01] st |.(g /. s).| = a

    proof

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

      

       A1: 0 in [. 0 , 1.] by XXREAL_1: 1;

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

      let g be Function of I[01] , ( TOP-REAL n), a be Real;

      assume that

       A2: g is continuous and

       A3: |.(g /. 0 ).| <= a & a <= |.(g /. 1).|;

      consider f be Function of I[01] , R^1 such that

       A4: for t be Point of I[01] holds (f . t) = |.(g . t).| and

       A5: f is continuous by A2, Th69;

      

       A6: (f . 0 ) = |.(g . o).| by A4

      .= |.(g /. 0 ).| by FUNCT_2:def 13;

      set c = |.(g /. 0 ).|, b = |.(g /. 1).|;

      

       A7: 1 in the carrier of I[01] by BORSUK_1: 40, XXREAL_1: 1;

      

       A8: (f . 1) = |.(g . I).| by A4

      .= |.(g /. 1).| by FUNCT_2:def 13;

      per cases by A3, XXREAL_0: 1;

        suppose c < a & a < b;

        then

        consider rc be Real such that

         A9: (f . rc) = a and

         A10: 0 < rc & rc < 1 by A5, A6, A8, TOPMETR: 20, TOPREAL5: 6;

        reconsider rc1 = rc as Point of I[01] by A10, BORSUK_1: 40, XXREAL_1: 1;

        

         A11: rc in the carrier of I[01] by A10, BORSUK_1: 40, XXREAL_1: 1;

         |.(g /. rc).| = |.(g . rc1).| by FUNCT_2:def 13

        .= a by A4, A9;

        hence thesis by A11;

      end;

        suppose c = a;

        hence thesis by A1, BORSUK_1: 40;

      end;

        suppose a = b;

        hence thesis by A7;

      end;

    end;

    theorem :: JORDAN2C:87

    

     Th71: q = <*r*> implies |.q.| = |.r.|

    proof

      assume

       A1: q = <*r*>;

      reconsider rr = r as Element of REAL by XREAL_0:def 1;

      reconsider xr = <*rr*> as Element of ( REAL 1);

      reconsider qk = ((q /. 1) ^2 ) as Element of REAL by XREAL_0:def 1;

      ( len xr) = 1 by FINSEQ_1: 39;

      then

       A2: (q /. 1) = (xr . 1) by A1, FINSEQ_4: 15;

      then ( len ( sqr xr)) = 1 & (( sqr xr) . 1) = ((q /. 1) ^2 ) by CARD_1:def 7, VALUED_1: 11;

      then

       A3: ( sqr xr) = <*qk*> by FINSEQ_1: 40;

      ( sqrt ((q /. 1) ^2 )) = |.(q /. 1).| by COMPLEX1: 72

      .= |.r.| by A2, FINSEQ_1: 40;

      then |.xr.| = |.rr.| by A3, FINSOP_1: 11;

      hence thesis by A1;

    end;

    theorem :: JORDAN2C:88

    for A be Subset of ( TOP-REAL n), a be Real st n >= 1 & a > 0 & A = { q : |.q.| = a } holds ( BDD A) is_inside_component_of A

    proof

      let A be Subset of ( TOP-REAL n), a be Real;

      { q where q be Point of ( TOP-REAL n) : |.q.| < a } c= the carrier of ( TOP-REAL n)

      proof

        let x be object;

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

        then ex q st q = x & |.q.| < a;

        hence thesis;

      end;

      then

      reconsider W = { q where q be Point of ( TOP-REAL n) : |.q.| < a } as Subset of ( Euclid n) by TOPREAL3: 8;

      reconsider P = W as Subset of ( TOP-REAL n) by TOPREAL3: 8;

      reconsider P as Subset of ( TOP-REAL n);

      

       A1: the carrier of (( TOP-REAL n) | (A ` )) = (A ` ) by PRE_TOPC: 8;

      then

      reconsider P1 = ( Component_of ( Down (P,(A ` )))) as Subset of ( TOP-REAL n) by XBOOLE_1: 1;

      assume

       A2: n >= 1 & a > 0 & A = { q : |.q.| = a };

      

       A3: P c= (A ` )

      proof

        let x be object;

        assume

         A4: x in P;

        then

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

        

         A5: ex q1 st q1 = q & |.q1.| < a by A4;

        now

          assume q in A;

          then ex q2 st q2 = q & |.q2.| = a by A2;

          hence contradiction by A5;

        end;

        hence thesis by XBOOLE_0:def 5;

      end;

      then

       A6: ( Down (P,(A ` ))) = P by XBOOLE_1: 28;

      P is convex by Th54;

      then (( TOP-REAL n) | P) is connected by CONNSP_1:def 3;

      then ((( TOP-REAL n) | (A ` )) | ( Down (P,(A ` )))) is connected by A3, A6, PRE_TOPC: 7;

      then

       A7: ( Down (P,(A ` ))) is connected by CONNSP_1:def 3;

       |.( 0. ( TOP-REAL n)).| = 0 by TOPRNS_1: 23;

      then

       A8: ( 0. ( TOP-REAL n)) in P by A2;

      then

      reconsider G = (A ` ) as non empty Subset of ( TOP-REAL n) by A3;

      

       A9: (( TOP-REAL n) | G) is non empty;

      

       A10: P c= ( Component_of ( Down (P,(A ` )))) by A6, A7, CONNSP_3: 13;

      

       A11: ( Down (P,(A ` ))) <> {} by A3, A8, XBOOLE_0:def 4;

      then

       A12: ( Component_of ( Down (P,(A ` )))) is a_component by A9, A7, CONNSP_3: 9;

      then

       A13: ( Component_of ( Down (P,(A ` )))) is connected by CONNSP_1:def 5;

      ( Component_of ( Down (P,(A ` )))) is bounded Subset of ( Euclid n)

      proof

        reconsider D2 = ( Component_of ( Down (P,(A ` )))) as Subset of ( TOP-REAL n) by A1, XBOOLE_1: 1;

        reconsider D = D2 as Subset of ( Euclid n) by TOPREAL3: 8;

        reconsider D as Subset of ( Euclid n);

        now

          reconsider B = (A ` ) as non empty Subset of ( TOP-REAL n) by A3, A8;

          set p = ( 0. ( TOP-REAL n));

          reconsider RR = (( TOP-REAL n) | B) as non empty TopSpace;

          assume not D2 is bounded;

          then

          consider q such that

           A14: q in D2 and

           A15: |.q.| >= a by Th21;

          

           A16: (A ` ) is open & D2 is connected by A2, A13, Th64, CONNSP_1: 23;

          D c= the carrier of (( TOP-REAL n) | (A ` ));

          then

           A17: D2 c= (A ` ) by PRE_TOPC: 8;

          then

           A18: D2 = ( Down (D2,(A ` ))) by XBOOLE_1: 28;

          RR is locally_connected by A2, Th66;

          then ( Component_of ( Down (P,(A ` )))) is open by A11, A7, CONNSP_2: 15, CONNSP_3: 9;

          then

          consider G be Subset of ( TOP-REAL n) such that

           A19: G is open and

           A20: ( Down (D2,(A ` ))) = (G /\ ( [#] (( TOP-REAL n) | (A ` )))) by A18, TOPS_2: 24;

          

           A21: (G /\ (A ` )) = D2 by A18, A20, PRE_TOPC:def 5;

          p <> q by A2, A15, TOPRNS_1: 23;

          then

          consider f1 be Function of I[01] , ( TOP-REAL n) such that

           A22: f1 is continuous and

           A23: ( rng f1) c= D2 and

           A24: (f1 . 0 ) = p and

           A25: (f1 . 1) = q by A8, A10, A14, A19, A21, A16, Th63;

          

           A26: |.(f1 /. 1).| >= a by A15, A25, BORSUK_1:def 15, FUNCT_2:def 13;

           |.p.| < a by A2, TOPRNS_1: 23;

          then |.(f1 /. 0 ).| < a by A24, BORSUK_1:def 14, FUNCT_2:def 13;

          then

          consider t0 be Point of I[01] such that

           A27: |.(f1 /. t0).| = a by A22, A26, Th70;

          reconsider q2 = (f1 . t0) as Point of ( TOP-REAL n);

          t0 in ( [#] I[01] );

          then t0 in ( dom f1) by FUNCT_2:def 1;

          then q2 in ( rng f1) by FUNCT_1:def 3;

          then

           A28: q2 in D2 by A23;

          q2 in A by A2, A27;

          then (A /\ (A ` )) <> ( {} the carrier of ( TOP-REAL n)) by A17, A28, XBOOLE_0:def 4;

          then A meets (A ` );

          hence contradiction by XBOOLE_1: 79;

        end;

        hence thesis by Th5;

      end;

      then

       A29: P1 is_inside_component_of A by A12, Th7;

      

       A30: P1 c= ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A })

      proof

        let x be object;

        assume

         A31: x in P1;

        P1 in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } by A29;

        hence thesis by A31, TARSKI:def 4;

      end;

      now

        per cases ;

          case

           A32: n >= 2;

          ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) c= P1

          proof

            reconsider E = (A ` ) as non empty Subset of ( TOP-REAL n) by A3, A8;

            let x be object;

            assume x in ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A });

            then

            consider y be set such that

             A33: x in y and

             A34: y in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } by TARSKI:def 4;

            consider B be Subset of ( TOP-REAL n) such that

             A35: B = y and

             A36: B is_inside_component_of A by A34;

            ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n) by A36, Th7;

            then

            reconsider p = x as Point of (( TOP-REAL n) | (A ` )) by A33, A35;

            

             A37: the carrier of (( TOP-REAL n) | (A ` )) = (A ` ) & p in the carrier of (( TOP-REAL n) | E) by PRE_TOPC: 8;

            then

            reconsider q2 = p as Point of ( TOP-REAL n);

             not p in A by A37, XBOOLE_0:def 5;

            then |.q2.| <> a by A2;

            then

             A38: |.q2.| < a or |.q2.| > a by XXREAL_0: 1;

            now

              per cases by A38;

                case

                 A39: p in { q : |.q.| > a };

                { q : |.q.| > a } c= (A ` )

                proof

                  let z be object;

                  assume z in { q : |.q.| > a };

                  then

                  consider q such that

                   A40: q = z and

                   A41: |.q.| > a;

                  now

                    assume q in A;

                    then ex q2 st q2 = q & |.q2.| = a by A2;

                    hence contradiction by A41;

                  end;

                  hence thesis by A40, XBOOLE_0:def 5;

                end;

                then

                reconsider Q = { q : |.q.| > a } as Subset of (( TOP-REAL n) | (A ` )) by PRE_TOPC: 8;

                reconsider Q as Subset of (( TOP-REAL n) | (A ` ));

                { q : |.q.| > a } c= the carrier of ( TOP-REAL n)

                proof

                  let z be object;

                  assume z in { q : |.q.| > a };

                  then ex q st q = z & |.q.| > a;

                  hence thesis;

                end;

                then

                reconsider P2 = { q : |.q.| > a } as Subset of ( TOP-REAL n);

                P2 is Subset of ( Euclid n) by TOPREAL3: 8;

                then

                reconsider W2 = { q : |.q.| > a } as Subset of ( Euclid n);

                P2 is connected by A32, Th38;

                then

                 A42: (( TOP-REAL n) | P2) is connected by CONNSP_1:def 3;

                

                 A43: not W2 is bounded by A32, Th46;

                 A44:

                now

                  assume W2 meets A;

                  then

                  consider z be object such that

                   A45: z in W2 & z in A by XBOOLE_0: 3;

                  (ex q st q = z & |.q.| > a) & ex q2 st q2 = z & |.q2.| = a by A2, A45;

                  hence contradiction;

                end;

                then (W2 /\ ((A ` ) ` )) = {} ;

                then (P2 \ (A ` )) = {} by SUBSET_1: 13;

                then

                 A46: W2 c= (A ` ) by XBOOLE_1: 37;

                then Q = ( Down (P2,(A ` ))) by XBOOLE_1: 28;

                then ( Up ( Component_of Q)) is_outside_component_of A by A32, A43, A44, Th38, Th48;

                then

                 A47: ( Component_of Q) c= ( UBD A) by Th14;

                (( TOP-REAL n) | P2) = ((( TOP-REAL n) | (A ` )) | Q) by A46, PRE_TOPC: 7;

                then Q is connected by A42, CONNSP_1:def 3;

                then Q c= ( Component_of Q) by CONNSP_3: 1;

                then

                 A48: p in ( Component_of Q) by A39;

                B c= ( BDD A) by A36, Th13;

                then p in (( BDD A) /\ ( UBD A)) by A33, A35, A47, A48, XBOOLE_0:def 4;

                then ( BDD A) meets ( UBD A);

                hence thesis by Th15;

              end;

                case

                 A49: p in { q1 : |.q1.| < a };

                ( Down (P,(A ` ))) c= ( Component_of ( Down (P,(A ` )))) by A7, CONNSP_3: 1;

                hence thesis by A6, A49;

              end;

            end;

            hence thesis;

          end;

          then P1 = ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) by A30;

          hence ex B be Subset of ( TOP-REAL n) st B is_inside_component_of A & B = ( BDD A) by A29;

        end;

          case n < 2;

          then n < (1 + 1);

          then

           A50: n <= 1 by NAT_1: 13;

          then

           A51: n = 1 by A2, XXREAL_0: 1;

          ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) c= P1

          proof

            reconsider E = (A ` ) as non empty Subset of ( TOP-REAL n) by A3, A8;

            let x be object;

            assume x in ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A });

            then

            consider y be set such that

             A52: x in y and

             A53: y in { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A } by TARSKI:def 4;

            consider B be Subset of ( TOP-REAL n) such that

             A54: B = y and

             A55: B is_inside_component_of A by A53;

            ex C be Subset of (( TOP-REAL n) | (A ` )) st C = B & C is a_component & C is bounded Subset of ( Euclid n) by A55, Th7;

            then

            reconsider p = x as Point of (( TOP-REAL n) | (A ` )) by A52, A54;

            

             A56: the carrier of (( TOP-REAL n) | (A ` )) = (A ` ) & p in the carrier of (( TOP-REAL n) | E) by PRE_TOPC: 8;

            then

            reconsider q2 = p as Point of ( TOP-REAL n);

             not p in A by A56, XBOOLE_0:def 5;

            then |.q2.| <> a by A2;

            then

             A57: |.q2.| < a or |.q2.| > a by XXREAL_0: 1;

            now

              per cases by A57;

                case p in { q : |.q.| > a };

                then

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

                 A58: q0 = p and

                 A59: |.q0.| > a;

                q0 is Element of ( REAL n) by EUCLID: 22;

                then

                consider r0 be Element of REAL such that

                 A60: q0 = <*r0*> by A51, FINSEQ_2: 97;

                

                 A61: |.q0.| = |.r0.| by A60, Th71;

                 A62:

                now

                  per cases ;

                    suppose r0 >= 0 ;

                    then r0 = |.r0.| by ABSVALUE:def 1;

                    hence p in { q : ex r st q = <*r*> & r > a } or p in { q1 : ex r1 st q1 = <*r1*> & r1 < ( - a) } by A58, A59, A60, A61;

                  end;

                    suppose r0 < 0 ;

                    then ( - r0) > a by A59, A61, ABSVALUE:def 1;

                    then ( - ( - r0)) < ( - a) by XREAL_1: 24;

                    hence p in { q : ex r st q = <*r*> & r > a } or p in { q1 : ex r1 st q1 = <*r1*> & r1 < ( - a) } by A58, A60;

                  end;

                end;

                now

                  per cases by A62;

                    suppose

                     A63: p in { q : ex r st q = <*r*> & r > a };

                    { q : ex r st q = <*r*> & r > a } c= (A ` )

                    proof

                      let z be object;

                      assume z in { q : ex r st q = <*r*> & r > a };

                      then

                      consider q such that

                       A64: q = z and

                       A65: ex r st q = <*r*> & r > a;

                      consider r such that

                       A66: q = <*r*> and

                       A67: r > a by A65;

                      reconsider rr = r as Element of REAL by XREAL_0:def 1;

                      n = 1 by A2, A50, XXREAL_0: 1;

                      then

                      reconsider xr = <*rr*> as Element of ( REAL n);

                      ( len xr) = 1 by FINSEQ_1: 39;

                      then

                       A68: (q /. 1) = (xr . 1) by A66, FINSEQ_4: 15;

                      then

                       A69: (( sqr xr) . 1) = ((q /. 1) ^2 ) by VALUED_1: 11;

                      

                       A70: ( sqrt ((q /. 1) ^2 )) = |.(q /. 1).| by COMPLEX1: 72

                      .= |.r.| by A68, FINSEQ_1: 40;

                      reconsider qk = ((q /. 1) ^2 ) as Element of REAL by XREAL_0:def 1;

                      ( len ( sqr xr)) = 1 by A51, CARD_1:def 7;

                      then ( sqr xr) = <*qk*> by A69, FINSEQ_1: 40;

                      

                      then

                       A71: |.q.| = |.r.| by A66, A70, FINSOP_1: 11

                      .= r by A2, A67, ABSVALUE:def 1;

                      now

                        assume q in A;

                        then ex q2 st q2 = q & |.q2.| = a by A2;

                        hence contradiction by A67, A71;

                      end;

                      hence thesis by A64, XBOOLE_0:def 5;

                    end;

                    then

                    reconsider Q = { q : ex r st q = <*r*> & r > a } as Subset of (( TOP-REAL n) | (A ` )) by PRE_TOPC: 8;

                    { q : ex r st q = <*r*> & r > a } c= the carrier of ( TOP-REAL n)

                    proof

                      let z be object;

                      assume z in { q : ex r st q = <*r*> & r > a };

                      then ex q st q = z & ex r st q = <*r*> & r > a;

                      hence thesis;

                    end;

                    then

                    reconsider P3 = { q : ex r st q = <*r*> & r > a } as Subset of ( TOP-REAL n);

                    reconsider W3 = P3 as Subset of ( Euclid n) by TOPREAL3: 8;

                    reconsider Q as Subset of (( TOP-REAL n) | (A ` ));

                    { q : |.q.| > a } c= the carrier of ( TOP-REAL n)

                    proof

                      let z be object;

                      assume z in { q : |.q.| > a };

                      then ex q st q = z & |.q.| > a;

                      hence thesis;

                    end;

                    then

                    reconsider P2 = { q : |.q.| > a } as Subset of ( TOP-REAL n);

                    P2 is Subset of ( Euclid n) by TOPREAL3: 8;

                    then

                    reconsider W2 = { q : |.q.| > a } as Subset of ( Euclid n);

                    

                     A72: W3 c= W2

                    proof

                      let z be object;

                      assume z in W3;

                      then

                      consider q such that

                       A73: q = z and

                       A74: ex r st q = <*r*> & r > a;

                      consider r such that

                       A75: q = <*r*> and

                       A76: r > a by A74;

                      

                       A77: r = |.r.| by A2, A76, ABSVALUE:def 1;

                      reconsider rr = r as Element of REAL by XREAL_0:def 1;

                      n = 1 by A2, A50, XXREAL_0: 1;

                      then

                      reconsider xr = <*rr*> as Element of ( REAL n);

                      ( len xr) = 1 by FINSEQ_1: 39;

                      then

                       A78: (q /. 1) = (xr . 1) by A75, FINSEQ_4: 15;

                      then

                       A79: (( sqr xr) . 1) = ((q /. 1) ^2 ) by VALUED_1: 11;

                      reconsider qk = ((q /. 1) ^2 ) as Element of REAL by XREAL_0:def 1;

                      ( len ( sqr xr)) = 1 by A51, CARD_1:def 7;

                      then

                       A80: ( sqr xr) = <*qk*> by A79, FINSEQ_1: 40;

                      ( sqrt ((q /. 1) ^2 )) = |.(q /. 1).| by COMPLEX1: 72

                      .= |.r.| by A78, FINSEQ_1: 40;

                      then |.xr.| = |.rr.| by A80, FINSOP_1: 11;

                      then |.q.| = |.r.| by A75;

                      hence thesis by A73, A76, A77;

                    end;

                     A81:

                    now

                      set z = the Element of (W2 /\ A);

                      assume

                       A82: not (W2 /\ A) = {} ;

                      then z in W2 by XBOOLE_0:def 4;

                      then

                       A83: ex q st q = z & |.q.| > a;

                      z in A by A82, XBOOLE_0:def 4;

                      then ex q2 st q2 = z & |.q2.| = a by A2;

                      hence contradiction by A83;

                    end;

                    then (W3 /\ A) = {} by A72, XBOOLE_1: 3, XBOOLE_1: 26;

                    then

                     A84: W3 misses A;

                    (W3 /\ ((A ` ) ` )) = {} by A81, A72, XBOOLE_1: 3, XBOOLE_1: 26;

                    then (W3 \ (A ` )) = {} by SUBSET_1: 13;

                    then

                     A85: W3 c= (A ` ) by XBOOLE_1: 37;

                    then

                     A86: (( TOP-REAL n) | P3) = ((( TOP-REAL n) | (A ` )) | Q) by PRE_TOPC: 7;

                    

                     A87: P3 is convex by A51, Th42;

                    then (( TOP-REAL n) | P3) is connected by CONNSP_1:def 3;

                    then Q is connected by A86, CONNSP_1:def 3;

                    then Q c= ( Component_of Q) by CONNSP_3: 1;

                    then

                     A88: p in ( Component_of Q) by A63;

                    

                     A89: Q = ( Down (P3,(A ` ))) by A85, XBOOLE_1: 28;

                     not W3 is bounded by A51, Th44;

                    then ( Up ( Component_of Q)) is_outside_component_of A by A87, A84, A89, Th48;

                    then

                     A90: ( Component_of Q) c= ( UBD A) by Th14;

                    B c= ( BDD A) by A55, Th13;

                    then (( BDD A) /\ ( UBD A)) <> {} by A52, A54, A90, A88, XBOOLE_0:def 4;

                    then ( BDD A) meets ( UBD A);

                    hence thesis by Th15;

                  end;

                    suppose

                     A91: p in { q1 : ex r1 st q1 = <*r1*> & r1 < ( - a) };

                    { q : ex r st q = <*r*> & r < ( - a) } c= (A ` )

                    proof

                      let z be object;

                      assume z in { q : ex r st q = <*r*> & r < ( - a) };

                      then

                      consider q such that

                       A92: q = z and

                       A93: ex r st q = <*r*> & r < ( - a);

                      consider r such that

                       A94: q = <*r*> and

                       A95: r < ( - a) by A93;

                      

                       A96: r < ( - 0 ) by A2, A95;

                      reconsider rr = r as Element of REAL by XREAL_0:def 1;

                      n = 1 by A2, A50, XXREAL_0: 1;

                      then

                      reconsider xr = <*rr*> as Element of ( REAL n);

                      ( len xr) = 1 by FINSEQ_1: 39;

                      then

                       A97: (q /. 1) = (xr . 1) by A94, FINSEQ_4: 15;

                      then

                       A98: (( sqr xr) . 1) = ((q /. 1) ^2 ) by VALUED_1: 11;

                      reconsider qk = ((q /. 1) ^2 ) as Element of REAL by XREAL_0:def 1;

                      ( len ( sqr xr)) = 1 by A51, CARD_1:def 7;

                      then

                       A99: ( sqr xr) = <*qk*> by A98, FINSEQ_1: 40;

                      ( sqrt ((q /. 1) ^2 )) = |.(q /. 1).| by COMPLEX1: 72

                      .= |.r.| by A97, FINSEQ_1: 40;

                      

                      then

                       A100: |.q.| = |.r.| by A94, A99, FINSOP_1: 11

                      .= ( - r) by A96, ABSVALUE:def 1;

                      now

                        assume q in A;

                        then ex q2 st q2 = q & |.q2.| = a by A2;

                        hence contradiction by A95, A100;

                      end;

                      hence thesis by A92, XBOOLE_0:def 5;

                    end;

                    then

                    reconsider Q = { q : ex r st q = <*r*> & r < ( - a) } as Subset of (( TOP-REAL n) | (A ` )) by PRE_TOPC: 8;

                    { q : ex r st q = <*r*> & r < ( - a) } c= the carrier of ( TOP-REAL n)

                    proof

                      let z be object;

                      assume z in { q : ex r st q = <*r*> & r < ( - a) };

                      then ex q st q = z & ex r st q = <*r*> & r < ( - a);

                      hence thesis;

                    end;

                    then

                    reconsider P3 = { q : ex r st q = <*r*> & r < ( - a) } as Subset of ( TOP-REAL n);

                    reconsider W3 = P3 as Subset of ( Euclid n) by TOPREAL3: 8;

                    reconsider Q as Subset of (( TOP-REAL n) | (A ` ));

                    { q : |.q.| > a } c= the carrier of ( TOP-REAL n)

                    proof

                      let z be object;

                      assume z in { q : |.q.| > a };

                      then ex q st q = z & |.q.| > a;

                      hence thesis;

                    end;

                    then

                    reconsider P2 = { q : |.q.| > a } as Subset of ( TOP-REAL n);

                    P2 is Subset of ( Euclid n) by TOPREAL3: 8;

                    then

                    reconsider W2 = { q : |.q.| > a } as Subset of ( Euclid n);

                    

                     A101: W3 c= W2

                    proof

                      let z be object;

                      assume z in W3;

                      then

                      consider q such that

                       A102: q = z and

                       A103: ex r st q = <*r*> & r < ( - a);

                      consider r such that

                       A104: q = <*r*> and

                       A105: r < ( - a) by A103;

                      

                       A106: r < ( - 0 ) & ( - r) > ( - ( - a)) by A2, A105, XREAL_1: 24;

                      reconsider rr = r as Element of REAL by XREAL_0:def 1;

                      n = 1 by A2, A50, XXREAL_0: 1;

                      then

                      reconsider xr = <*rr*> as Element of ( REAL n);

                      ( len xr) = 1 by FINSEQ_1: 39;

                      then

                       A107: (q /. 1) = (xr . 1) by A104, FINSEQ_4: 15;

                      then

                       A108: (( sqr xr) . 1) = ((q /. 1) ^2 ) by VALUED_1: 11;

                      reconsider qk = ((q /. 1) ^2 ) as Element of REAL by XREAL_0:def 1;

                      ( len ( sqr xr)) = 1 by A51, CARD_1:def 7;

                      then

                       A109: ( sqr xr) = <*qk*> by A108, FINSEQ_1: 40;

                      ( sqrt ((q /. 1) ^2 )) = |.(q /. 1).| by COMPLEX1: 72

                      .= |.r.| by A107, FINSEQ_1: 40;

                      then |.q.| = |.r.| by A104, A109, FINSOP_1: 11;

                      then |.q.| > a by A106, ABSVALUE:def 1;

                      hence thesis by A102;

                    end;

                     A110:

                    now

                      set z = the Element of (W2 /\ A);

                      assume

                       A111: not (W2 /\ A) = {} ;

                      then z in W2 by XBOOLE_0:def 4;

                      then

                       A112: ex q st q = z & |.q.| > a;

                      z in A by A111, XBOOLE_0:def 4;

                      then ex q2 st q2 = z & |.q2.| = a by A2;

                      hence contradiction by A112;

                    end;

                    then (W3 /\ A) = {} by A101, XBOOLE_1: 3, XBOOLE_1: 26;

                    then

                     A113: W3 misses A;

                    (W3 /\ ((A ` ) ` )) = {} by A110, A101, XBOOLE_1: 3, XBOOLE_1: 26;

                    then (W3 \ (A ` )) = {} by SUBSET_1: 13;

                    then

                     A114: W3 c= (A ` ) by XBOOLE_1: 37;

                    then

                     A115: (( TOP-REAL n) | P3) = ((( TOP-REAL n) | (A ` )) | Q) by PRE_TOPC: 7;

                    

                     A116: P3 is convex by A51, Th43;

                    then (( TOP-REAL n) | P3) is connected by CONNSP_1:def 3;

                    then Q is connected by A115, CONNSP_1:def 3;

                    then Q c= ( Component_of Q) by CONNSP_3: 1;

                    then

                     A117: p in ( Component_of Q) by A91;

                    

                     A118: Q = ( Down (P3,(A ` ))) by A114, XBOOLE_1: 28;

                    ( Up ( Component_of Q)) is_outside_component_of A by A116, A113, A118, Th48, A51, Th45;

                    then

                     A119: ( Component_of Q) c= ( UBD A) by Th14;

                    B c= ( BDD A) by A55, Th13;

                    then p in (( BDD A) /\ ( UBD A)) by A52, A54, A119, A117, XBOOLE_0:def 4;

                    then ( BDD A) meets ( UBD A);

                    hence thesis by Th15;

                  end;

                end;

                hence thesis;

              end;

                case

                 A120: p in { q1 : |.q1.| < a };

                ( Down (P,(A ` ))) c= ( Component_of ( Down (P,(A ` )))) by A7, CONNSP_3: 1;

                hence thesis by A6, A120;

              end;

            end;

            hence thesis;

          end;

          then P1 = ( union { B where B be Subset of ( TOP-REAL n) : B is_inside_component_of A }) by A30;

          hence ex B be Subset of ( TOP-REAL n) st B is_inside_component_of A & B = ( BDD A) by A29;

        end;

      end;

      hence thesis;

    end;

    begin

    reserve D for non vertical non horizontal non empty compact Subset of ( TOP-REAL 2);

    theorem :: JORDAN2C:89

    

     Th73: ( len ( GoB ( SpStSeq D))) = 2 & ( width ( GoB ( SpStSeq D))) = 2 & (( SpStSeq D) /. 1) = (( GoB ( SpStSeq D)) * (1,2)) & (( SpStSeq D) /. 2) = (( GoB ( SpStSeq D)) * (2,2)) & (( SpStSeq D) /. 3) = (( GoB ( SpStSeq D)) * (2,1)) & (( SpStSeq D) /. 4) = (( GoB ( SpStSeq D)) * (1,1)) & (( SpStSeq D) /. 5) = (( GoB ( SpStSeq D)) * (1,2))

    proof

      set f = ( SpStSeq D);

      

       A1: ( S-bound ( L~ f)) < ( N-bound ( L~ f)) by SPRECT_1: 32;

      

       A2: ( len f) = 5 by SPRECT_1: 82;

      then

       A3: (f /. 5) = (f /. 1) by FINSEQ_6:def 1;

      4 in ( Seg ( len f)) by A2, FINSEQ_1: 1;

      then

       A4: 4 in ( dom f) by FINSEQ_1:def 3;

      then 4 in ( dom ( X_axis f)) by SPRECT_2: 15;

      then (f /. 4) = ( W-min ( L~ f)) & (( X_axis f) . 4) = ((f /. 4) `1 ) by GOBOARD1:def 1, SPRECT_1: 86;

      then

       A5: (( X_axis f) . 4) = ( W-bound ( L~ f)) by EUCLID: 52;

      

       A6: (f /. 3) = ( S-max ( L~ f)) by SPRECT_1: 85;

      3 in ( Seg ( len f)) by A2, FINSEQ_1: 1;

      then

       A7: 3 in ( dom f) by FINSEQ_1:def 3;

      then 3 in ( dom ( X_axis f)) by SPRECT_2: 15;

      then (f /. 3) = ( E-min ( L~ f)) & (( X_axis f) . 3) = ((f /. 3) `1 ) by GOBOARD1:def 1, SPRECT_1: 85;

      then

       A8: (( X_axis f) . 3) = ( E-bound ( L~ f)) by EUCLID: 52;

      

       A9: (f /. (1 + 1)) = ( N-max ( L~ f)) by SPRECT_1: 84;

      3 in ( dom ( Y_axis f)) by A7, SPRECT_2: 16;

      then (( Y_axis f) . 3) = ((f /. 3) `2 ) by GOBOARD1:def 2;

      then

       A10: (( Y_axis f) . 3) = ( S-bound ( L~ f)) by A6, EUCLID: 52;

      

       A11: (f /. 1) = ( N-min ( L~ f)) by SPRECT_1: 83;

      1 in ( Seg ( len f)) by A2, FINSEQ_1: 1;

      then

       A12: 1 in ( dom f) by FINSEQ_1:def 3;

      then 1 in ( dom ( Y_axis f)) by SPRECT_2: 16;

      then (( Y_axis f) . 1) = ((f /. 1) `2 ) by GOBOARD1:def 2;

      then

       A13: (( Y_axis f) . 1) = ( N-bound ( L~ f)) by A11, EUCLID: 52;

      

       A14: (f /. 4) = ( S-min ( L~ f)) by SPRECT_1: 86;

      2 in ( Seg ( len f)) by A2, FINSEQ_1: 1;

      then

       A15: 2 in ( dom f) by FINSEQ_1:def 3;

      then 2 in ( dom ( X_axis f)) by SPRECT_2: 15;

      then (f /. (1 + 1)) = ( E-max ( L~ f)) & (( X_axis f) . 2) = ((f /. 2) `1 ) by GOBOARD1:def 1, SPRECT_1: 84;

      then

       A16: (( X_axis f) . 2) = ( E-bound ( L~ f)) by EUCLID: 52;

      4 in ( dom ( Y_axis f)) by A4, SPRECT_2: 16;

      then (( Y_axis f) . 4) = ((f /. 4) `2 ) by GOBOARD1:def 2;

      then

       A17: (( Y_axis f) . 4) = ( S-bound ( L~ f)) by A14, EUCLID: 52;

      2 in ( dom ( Y_axis f)) by A15, SPRECT_2: 16;

      then (( Y_axis f) . 2) = ((f /. 2) `2 ) by GOBOARD1:def 2;

      then

       A18: (( Y_axis f) . 2) = ( N-bound ( L~ f)) by A9, EUCLID: 52;

      

       A19: {( S-bound ( L~ f)), ( N-bound ( L~ f))} c= ( rng ( Y_axis f))

      proof

        let z be object;

        assume

         A20: z in {( S-bound ( L~ f)), ( N-bound ( L~ f))};

        now

          per cases by A20, TARSKI:def 2;

            case

             A21: z = ( S-bound ( L~ f));

            4 in ( dom ( Y_axis f)) by A4, SPRECT_2: 16;

            hence thesis by A17, A21, FUNCT_1:def 3;

          end;

            case

             A22: z = ( N-bound ( L~ f));

            2 in ( dom ( Y_axis f)) by A15, SPRECT_2: 16;

            hence thesis by A18, A22, FUNCT_1:def 3;

          end;

        end;

        hence thesis;

      end;

      

       A23: (f /. 1) = ( W-max ( L~ f)) by SPRECT_1: 83;

      1 in ( dom ( X_axis f)) by A12, SPRECT_2: 15;

      then (( X_axis f) . 1) = ((f /. 1) `1 ) by GOBOARD1:def 1;

      then

       A24: (( X_axis f) . 1) = ( W-bound ( L~ f)) by A23, EUCLID: 52;

      

       A25: {( W-bound ( L~ f)), ( E-bound ( L~ f))} c= ( rng ( X_axis f))

      proof

        let z be object;

        assume

         A26: z in {( W-bound ( L~ f)), ( E-bound ( L~ f))};

        now

          per cases by A26, TARSKI:def 2;

            case

             A27: z = ( W-bound ( L~ f));

            1 in ( dom ( X_axis f)) by A12, SPRECT_2: 15;

            hence thesis by A24, A27, FUNCT_1:def 3;

          end;

            case

             A28: z = ( E-bound ( L~ f));

            2 in ( dom ( X_axis f)) by A15, SPRECT_2: 15;

            hence thesis by A16, A28, FUNCT_1:def 3;

          end;

        end;

        hence thesis;

      end;

      

       A29: ( GoB f) = ( GoB (( Incr ( X_axis f)),( Incr ( Y_axis f)))) by GOBOARD2:def 2;

      5 in ( Seg ( len f)) by A2, FINSEQ_1: 1;

      then

       A30: 5 in ( dom f) by FINSEQ_1:def 3;

      then 5 in ( dom ( X_axis f)) by SPRECT_2: 15;

      then (( X_axis f) . 5) = ((f /. 5) `1 ) by GOBOARD1:def 1;

      then

       A31: (( X_axis f) . 5) = ( W-bound ( L~ f)) by A23, A3, EUCLID: 52;

      ( rng ( X_axis f)) c= {( W-bound ( L~ f)), ( E-bound ( L~ f))}

      proof

        let z be object;

        assume z in ( rng ( X_axis f));

        then

        consider u be object such that

         A32: u in ( dom ( X_axis f)) and

         A33: z = (( X_axis f) . u) by FUNCT_1:def 3;

        reconsider mu = u as Element of NAT by A32;

        u in ( dom f) by A32, SPRECT_2: 15;

        then u in ( Seg ( len f)) by FINSEQ_1:def 3;

        then 1 <= mu & mu <= 5 by A2, FINSEQ_1: 1;

        then

         A34: mu = (1 + 0 ) or ... or mu = (1 + 4) by NAT_1: 62;

        per cases by A34;

          suppose mu = 1;

          hence thesis by A24, A33, TARSKI:def 2;

        end;

          suppose mu = 2;

          hence thesis by A16, A33, TARSKI:def 2;

        end;

          suppose mu = 3;

          hence thesis by A8, A33, TARSKI:def 2;

        end;

          suppose mu = 4;

          hence thesis by A5, A33, TARSKI:def 2;

        end;

          suppose mu = 5;

          hence thesis by A31, A33, TARSKI:def 2;

        end;

      end;

      then

       A35: ( rng ( X_axis f)) = {( W-bound ( L~ f)), ( E-bound ( L~ f))} by A25;

      then

       A36: ( rng ( Incr ( X_axis f))) = {( W-bound ( L~ f)), ( E-bound ( L~ f))} by SEQ_4:def 21;

      5 in ( dom ( Y_axis f)) by A30, SPRECT_2: 16;

      then (( Y_axis f) . 5) = ((f /. 5) `2 ) by GOBOARD1:def 2;

      then

       A37: (( Y_axis f) . 5) = ( N-bound ( L~ f)) by A11, A3, EUCLID: 52;

      ( rng ( Y_axis f)) c= {( S-bound ( L~ f)), ( N-bound ( L~ f))}

      proof

        let z be object;

        assume z in ( rng ( Y_axis f));

        then

        consider u be object such that

         A38: u in ( dom ( Y_axis f)) and

         A39: z = (( Y_axis f) . u) by FUNCT_1:def 3;

        reconsider mu = u as Element of NAT by A38;

        u in ( dom f) by A38, SPRECT_2: 16;

        then u in ( Seg ( len f)) by FINSEQ_1:def 3;

        then 1 <= mu & mu <= 5 by A2, FINSEQ_1: 1;

        then

         A40: mu = (1 + 0 ) or ... or mu = (1 + 4) by NAT_1: 62;

        per cases by A40;

          suppose mu = 1;

          hence thesis by A13, A39, TARSKI:def 2;

        end;

          suppose mu = 2;

          hence thesis by A18, A39, TARSKI:def 2;

        end;

          suppose mu = 3;

          hence thesis by A10, A39, TARSKI:def 2;

        end;

          suppose mu = 4;

          hence thesis by A17, A39, TARSKI:def 2;

        end;

          suppose mu = 5;

          hence thesis by A37, A39, TARSKI:def 2;

        end;

      end;

      then

       A41: ( rng ( Y_axis f)) = {( S-bound ( L~ f)), ( N-bound ( L~ f))} by A19;

      then ( card ( rng ( Y_axis f))) = 2 by A1, CARD_2: 57;

      then

       A42: ( len ( Incr ( Y_axis f))) = 2 by SEQ_4:def 21;

      

       A43: ( W-bound ( L~ f)) < ( E-bound ( L~ f)) by SPRECT_1: 31;

      then

       A44: ( card ( rng ( X_axis f))) = 2 by A35, CARD_2: 57;

      then

       A45: ( len ( Incr ( X_axis f))) = 2 by SEQ_4:def 21;

      

       A46: ( len ( GoB f)) = ( card ( rng ( X_axis f))) by GOBOARD2: 13

      .= (1 + 1) by A43, A35, CARD_2: 57;

      then

       A47: 1 in ( Seg ( len ( GoB f))) by FINSEQ_1: 1;

      

       A48: ( width ( GoB f)) = ( card ( rng ( Y_axis f))) by GOBOARD2: 13

      .= (1 + 1) by A1, A41, CARD_2: 57;

      for p be FinSequence of the carrier of ( TOP-REAL 2) st p in ( rng ( GoB f)) holds ( len p) = 2

      proof

        ( len ( GoB (( Incr ( X_axis f)),( Incr ( Y_axis f))))) = ( len ( Incr ( X_axis f))) by GOBOARD2:def 1

        .= 2 by A44, SEQ_4:def 21;

        then

        consider s1 be FinSequence such that

         A49: s1 in ( rng ( GoB (( Incr ( X_axis f)),( Incr ( Y_axis f))))) and

         A50: ( len s1) = ( width ( GoB (( Incr ( X_axis f)),( Incr ( Y_axis f))))) by MATRIX_0:def 3;

        let p be FinSequence of the carrier of ( TOP-REAL 2);

        consider n be Nat such that

         A51: for x st x in ( rng ( GoB f)) holds ex s be FinSequence st s = x & ( len s) = n by MATRIX_0:def 1;

        assume p in ( rng ( GoB f));

        then

         A52: ex s2 be FinSequence st s2 = p & ( len s2) = n by A51;

        s1 in ( rng ( GoB f)) by A49, GOBOARD2:def 2;

        then ex s be FinSequence st s = s1 & ( len s) = n by A51;

        hence thesis by A48, A50, A52, GOBOARD2:def 2;

      end;

      then

       A53: ( GoB f) is Matrix of 2, 2, the carrier of ( TOP-REAL 2) by A46, MATRIX_0:def 2;

      

       A54: 1 in ( Seg ( width ( GoB f))) by A48, FINSEQ_1: 1;

      then [1, 1] in [:( Seg ( len ( GoB f))), ( Seg ( width ( GoB f))):] by A47, ZFMISC_1: 87;

      then

       A55: [1, 1] in ( Indices ( GoB f)) by A46, A48, A53, MATRIX_0: 24;

      

       A56: ( width ( GoB f)) in ( Seg ( width ( GoB f))) by A48, FINSEQ_1: 1;

      then [1, ( width ( GoB f))] in [:( Seg ( len ( GoB f))), ( Seg ( width ( GoB f))):] by A47, ZFMISC_1: 87;

      then

       A57: [1, ( width ( GoB f))] in ( Indices ( GoB f)) by A46, A48, A53, MATRIX_0: 24;

      

       A58: ( len ( GoB f)) in ( Seg ( len ( GoB f))) by A46, FINSEQ_1: 1;

      then [( len ( GoB f)), 1] in [:( Seg ( len ( GoB f))), ( Seg ( width ( GoB f))):] by A54, ZFMISC_1: 87;

      then

       A59: [( len ( GoB f)), 1] in ( Indices ( GoB f)) by A46, A48, A53, MATRIX_0: 24;

      (( S-max ( L~ f)) `1 ) = (( SE-corner D) `1 ) by SPRECT_1: 81

      .= ( E-bound D) by EUCLID: 52

      .= ( E-bound ( L~ f)) by SPRECT_1: 61

      .= (( Incr ( X_axis f)) . 2) by A43, A36, A45, Th1;

      then (( S-max ( L~ f)) `1 ) = ( |[(( Incr ( X_axis f)) . 2), (( Incr ( Y_axis f)) . 1)]| `1 ) by EUCLID: 52;

      then

       A60: (( S-max ( L~ f)) `1 ) = ((( GoB f) * (( len ( GoB f)),1)) `1 ) by A29, A46, A59, GOBOARD2:def 1;

      (( S-min ( L~ f)) `1 ) = (( SW-corner D) `1 ) by SPRECT_1: 80

      .= ( W-bound D) by EUCLID: 52

      .= ( W-bound ( L~ f)) by SPRECT_1: 58

      .= (( Incr ( X_axis f)) . 1) by A43, A36, A45, Th1;

      then (( S-min ( L~ f)) `1 ) = ( |[(( Incr ( X_axis f)) . 1), (( Incr ( Y_axis f)) . 1)]| `1 ) by EUCLID: 52;

      then

       A61: (( S-min ( L~ f)) `1 ) = ((( GoB f) * (1,1)) `1 ) by A29, A55, GOBOARD2:def 1;

       [( len ( GoB f)), ( width ( GoB f))] in [:( Seg ( len ( GoB f))), ( Seg ( width ( GoB f))):] by A58, A56, ZFMISC_1: 87;

      then

       A62: [( len ( GoB f)), ( width ( GoB f))] in ( Indices ( GoB f)) by A46, A48, A53, MATRIX_0: 24;

      ( W-bound ( L~ f)) = (( Incr ( X_axis f)) . 1) by A43, A36, A45, Th1;

      then (( W-max ( L~ f)) `1 ) = (( Incr ( X_axis f)) . 1) by EUCLID: 52;

      then (( W-max ( L~ f)) `1 ) = ( |[(( Incr ( X_axis f)) . 1), (( Incr ( Y_axis f)) . (1 + 1))]| `1 ) by EUCLID: 52;

      then

       A63: (( W-max ( L~ f)) `1 ) = ((( GoB f) * (1,( width ( GoB f)))) `1 ) by A29, A48, A57, GOBOARD2:def 1;

      

       A64: (f /. 3) = |[((f /. 3) `1 ), ((f /. 3) `2 )]| & (f /. 4) = |[((f /. 4) `1 ), ((f /. 4) `2 )]| by EUCLID: 53;

      

       A65: (f /. 1) = |[((f /. 1) `1 ), ((f /. 1) `2 )]| & (f /. (1 + 1)) = |[((f /. (1 + 1)) `1 ), ((f /. (1 + 1)) `2 )]| by EUCLID: 53;

      

       A66: ( rng ( Incr ( Y_axis f))) = {( S-bound ( L~ f)), ( N-bound ( L~ f))} by A41, SEQ_4:def 21;

      then

       A67: ( N-bound ( L~ f)) = (( Incr ( Y_axis f)) . 2) by A1, A42, Th1;

      then (( N-min ( L~ f)) `2 ) = (( Incr ( Y_axis f)) . 2) by EUCLID: 52;

      then (( N-min ( L~ f)) `2 ) = ( |[(( Incr ( X_axis f)) . 1), (( Incr ( Y_axis f)) . 2)]| `2 ) by EUCLID: 52;

      then

       A68: (( N-min ( L~ f)) `2 ) = ((( GoB f) * (1,( width ( GoB f)))) `2 ) by A29, A48, A57, GOBOARD2:def 1;

      

       A69: ( S-bound ( L~ f)) = (( Incr ( Y_axis f)) . 1) by A1, A66, A42, Th1;

      then (( S-min ( L~ f)) `2 ) = (( Incr ( Y_axis f)) . 1) by EUCLID: 52;

      then (( S-min ( L~ f)) `2 ) = ( |[(( Incr ( X_axis f)) . 1), (( Incr ( Y_axis f)) . 1)]| `2 ) by EUCLID: 52;

      then

       A70: (( S-min ( L~ f)) `2 ) = ((( GoB f) * (1,1)) `2 ) by A29, A55, GOBOARD2:def 1;

      (( N-max ( L~ f)) `2 ) = ( N-bound ( L~ f)) by EUCLID: 52;

      then (( N-max ( L~ f)) `2 ) = ( |[(( Incr ( X_axis f)) . 2), (( Incr ( Y_axis f)) . 2)]| `2 ) by A67, EUCLID: 52;

      then

       A71: (( N-max ( L~ f)) `2 ) = ((( GoB f) * (( len ( GoB f)),( width ( GoB f)))) `2 ) by A29, A46, A48, A62, GOBOARD2:def 1;

      (( S-max ( L~ f)) `2 ) = (( Incr ( Y_axis f)) . 1) by A69, EUCLID: 52;

      then (( S-max ( L~ f)) `2 ) = ( |[(( Incr ( X_axis f)) . 2), (( Incr ( Y_axis f)) . 1)]| `2 ) by EUCLID: 52;

      then

       A72: (( S-max ( L~ f)) `2 ) = ((( GoB f) * (( len ( GoB f)),1)) `2 ) by A29, A46, A59, GOBOARD2:def 1;

      (( N-max ( L~ f)) `1 ) = (( NE-corner D) `1 ) by SPRECT_1: 77

      .= ( E-bound D) by EUCLID: 52

      .= ( E-bound ( L~ f)) by SPRECT_1: 61

      .= (( Incr ( X_axis f)) . 2) by A43, A36, A45, Th1;

      then (( N-max ( L~ f)) `1 ) = ( |[(( Incr ( X_axis f)) . (1 + 1)), (( Incr ( Y_axis f)) . (1 + 1))]| `1 ) by EUCLID: 52;

      then (( N-max ( L~ f)) `1 ) = ((( GoB f) * (( len ( GoB f)),( width ( GoB f)))) `1 ) by A29, A46, A48, A62, GOBOARD2:def 1;

      hence thesis by A11, A23, A9, A6, A14, A3, A46, A48, A63, A60, A61, A68, A71, A72, A70, A65, A64, EUCLID: 53;

    end;

    theorem :: JORDAN2C:90

    

     Th74: ( LeftComp ( SpStSeq D)) is non bounded

    proof

      set f = ( SpStSeq D);

      set q3 = the Element of ( LeftComp f);

      reconsider q4 = q3 as Point of ( TOP-REAL 2);

      set r1 = |.((1 / 2) * ((f /. 1) + (f /. 2))).|;

      reconsider f1 = f as non constant standard special_circular_sequence;

      

       A1: ( W-bound ( L~ f1)) < ( E-bound ( L~ f1)) by SPRECT_1: 31;

      

       A2: (( N-min ( L~ f1)) `2 ) = ( N-bound ( L~ f1)) by EUCLID: 52;

      

      then ((( N-min ( L~ f1)) `2 ) + (( N-max ( L~ f1)) `2 )) = (( N-bound ( L~ f1)) + ( N-bound ( L~ f1))) by EUCLID: 52

      .= (2 * ( N-bound ( L~ f)));

      then

       A3: ((1 / 2) * ((( N-min ( L~ f)) `2 ) + (( N-max ( L~ f)) `2 ))) = ( N-bound ( L~ f));

      

       A4: ( len f1) = 5 by SPRECT_1: 82;

      then 5 in ( Seg ( len f1)) by FINSEQ_1: 1;

      then

       A5: 5 in ( dom f1) by FINSEQ_1:def 3;

      then 5 in ( dom ( Y_axis f1)) by SPRECT_2: 16;

      then

       A6: (( Y_axis f1) . 5) = ((f1 /. 5) `2 ) by GOBOARD1:def 2;

      4 in ( Seg ( len f1)) by A4, FINSEQ_1: 1;

      then

       A7: 4 in ( dom f1) by FINSEQ_1:def 3;

      then 4 in ( dom ( Y_axis f1)) by SPRECT_2: 16;

      then (f1 /. 4) = ( S-min ( L~ f1)) & (( Y_axis f1) . 4) = ((f1 /. 4) `2 ) by GOBOARD1:def 2, SPRECT_1: 86;

      then

       A8: (( Y_axis f1) . 4) = ( S-bound ( L~ f1)) by EUCLID: 52;

      3 in ( Seg ( len f1)) by A4, FINSEQ_1: 1;

      then

       A9: 3 in ( dom f1) by FINSEQ_1:def 3;

      then 3 in ( dom ( Y_axis f1)) by SPRECT_2: 16;

      then (f1 /. 3) = ( S-max ( L~ f1)) & (( Y_axis f1) . 3) = ((f1 /. 3) `2 ) by GOBOARD1:def 2, SPRECT_1: 85;

      then

       A10: (( Y_axis f1) . 3) = ( S-bound ( L~ f1)) by EUCLID: 52;

      3 in ( dom ( X_axis f1)) by A9, SPRECT_2: 15;

      then (f1 /. 3) = ( E-min ( L~ f1)) & (( X_axis f1) . 3) = ((f1 /. 3) `1 ) by GOBOARD1:def 1, SPRECT_1: 85;

      then

       A11: (( X_axis f1) . 3) = ( E-bound ( L~ f1)) by EUCLID: 52;

      5 in ( dom ( X_axis f1)) by A5, SPRECT_2: 15;

      then

       A12: (( X_axis f1) . 5) = ((f1 /. 5) `1 ) by GOBOARD1:def 1;

      assume ( LeftComp f) is bounded;

      then

      consider r be Real such that

       A13: for q be Point of ( TOP-REAL 2) st q in ( LeftComp f) holds |.q.| < r by Th21;

      set q1 = ( |[ 0 , ((r1 + r) + 1)]| + ((1 / 2) * ((f /. 1) + (f /. 2))));

      

       A14: (f1 /. 1) = ( N-min ( L~ f1)) by SPRECT_1: 83;

      4 in ( dom ( X_axis f1)) by A7, SPRECT_2: 15;

      then (f1 /. 4) = ( W-min ( L~ f1)) & (( X_axis f1) . 4) = ((f1 /. 4) `1 ) by GOBOARD1:def 1, SPRECT_1: 86;

      then

       A15: (( X_axis f1) . 4) = ( W-bound ( L~ f1)) by EUCLID: 52;

      

       A16: ( GoB f1) = ( GoB (( Incr ( X_axis f1)),( Incr ( Y_axis f1)))) by GOBOARD2:def 2;

      

       A17: (f1 /. 2) = ( E-max ( L~ f1)) by SPRECT_1: 84;

      2 in ( Seg ( len f1)) by A4, FINSEQ_1: 1;

      then

       A18: 2 in ( dom f1) by FINSEQ_1:def 3;

      then

       A19: 2 in ( dom ( X_axis f1)) by SPRECT_2: 15;

      then (( X_axis f1) . 2) = ((f1 /. 2) `1 ) by GOBOARD1:def 1;

      then

       A20: (( X_axis f1) . 2) = ( E-bound ( L~ f1)) by A17, EUCLID: 52;

      

       A21: 1 in ( Seg ( len f1)) by A4, FINSEQ_1: 1;

      then

       A22: 1 in ( dom f1) by FINSEQ_1:def 3;

      then 1 in ( dom ( Y_axis f1)) by SPRECT_2: 16;

      then (( Y_axis f1) . 1) = ((f1 /. 1) `2 ) by GOBOARD1:def 2;

      then

       A23: (( Y_axis f1) . 1) = ( N-bound ( L~ f1)) by A14, EUCLID: 52;

      (( X_axis f1) . 2) = ((f1 /. 2) `1 ) by A19, GOBOARD1:def 1;

      then

       A24: ((f1 /. 2) `1 ) in ( rng ( X_axis f1)) by A19, FUNCT_1:def 3;

      ( len ( X_axis f1)) = ( len f1) by GOBOARD1:def 1;

      then

       A25: ( dom ( X_axis f1)) = ( Seg ( len f1)) by FINSEQ_1:def 3;

      then (( X_axis f1) . 1) = ((f1 /. 1) `1 ) by A21, GOBOARD1:def 1;

      then

       A26: ((f1 /. 1) `1 ) in ( rng ( X_axis f1)) by A21, A25, FUNCT_1:def 3;

       {((f1 /. 1) `1 ), ((f1 /. 2) `1 )} c= ( rng ( X_axis f1)) by A26, A24, TARSKI:def 2;

      then ( {((f1 /. 1) `1 )} \/ {((f1 /. 2) `1 )}) c= ( rng ( X_axis f1)) by ENUMSET1: 1;

      then

       A27: ( card ( {((f1 /. 1) `1 )} \/ {((f1 /. 2) `1 )})) <= ( card ( rng ( X_axis f1))) by NAT_1: 43;

      

       A28: (f1 /. (1 + 1)) = ( N-max ( L~ f1)) by SPRECT_1: 84;

      then ((f1 /. 1) `1 ) < ((f1 /. 2) `1 ) by A14, SPRECT_2: 51;

      then not ((f1 /. 2) `1 ) in {((f1 /. 1) `1 )} by TARSKI:def 1;

      

      then

       A29: ( card ( {((f1 /. 1) `1 )} \/ {((f1 /. 2) `1 )})) = (( card {((f1 /. 1) `1 )}) + 1) by CARD_2: 41

      .= (1 + 1) by CARD_1: 30

      .= 2;

      

       A30: 1 <> (( len ( GoB f1)) + 1) by A27, GOBOARD2: 13, XREAL_1: 29;

      2 in ( dom ( Y_axis f1)) by A18, SPRECT_2: 16;

      then (( Y_axis f1) . 2) = ((f1 /. 2) `2 ) by GOBOARD1:def 2;

      then

       A31: (( Y_axis f1) . 2) = ( N-bound ( L~ f1)) by A28, EUCLID: 52;

      (f1 /. 5) = (f1 /. 1) by A4, FINSEQ_6:def 1;

      then

       A32: (( Y_axis f1) . 5) = ( N-bound ( L~ f1)) by A14, A6, EUCLID: 52;

      

       A33: ( rng ( Y_axis f1)) c= {( S-bound ( L~ f1)), ( N-bound ( L~ f1))}

      proof

        let z be object;

        assume z in ( rng ( Y_axis f1));

        then

        consider u be object such that

         A34: u in ( dom ( Y_axis f1)) and

         A35: z = (( Y_axis f1) . u) by FUNCT_1:def 3;

        reconsider mu = u as Element of NAT by A34;

        u in ( dom f1) by A34, SPRECT_2: 16;

        then u in ( Seg ( len f1)) by FINSEQ_1:def 3;

        then 1 <= mu & mu <= 5 by A4, FINSEQ_1: 1;

        then

         A36: mu = (1 + 0 ) or ... or mu = (1 + 4) by NAT_1: 62;

        per cases by A36;

          suppose mu = 1;

          hence thesis by A23, A35, TARSKI:def 2;

        end;

          suppose mu = 2;

          hence thesis by A31, A35, TARSKI:def 2;

        end;

          suppose mu = 3;

          hence thesis by A10, A35, TARSKI:def 2;

        end;

          suppose mu = 4;

          hence thesis by A8, A35, TARSKI:def 2;

        end;

          suppose mu = 5;

          hence thesis by A32, A35, TARSKI:def 2;

        end;

      end;

       {( S-bound ( L~ f1)), ( N-bound ( L~ f1))} c= ( rng ( Y_axis f1))

      proof

        let z be object;

        assume

         A37: z in {( S-bound ( L~ f1)), ( N-bound ( L~ f1))};

        per cases by A37, TARSKI:def 2;

          suppose

           A38: z = ( S-bound ( L~ f1));

          4 in ( dom ( Y_axis f1)) by A7, SPRECT_2: 16;

          hence thesis by A8, A38, FUNCT_1:def 3;

        end;

          suppose

           A39: z = ( N-bound ( L~ f1));

          2 in ( dom ( Y_axis f1)) by A18, SPRECT_2: 16;

          hence thesis by A31, A39, FUNCT_1:def 3;

        end;

      end;

      then

       A40: ( S-bound ( L~ f1)) < ( N-bound ( L~ f1)) & ( rng ( Y_axis f1)) = {( S-bound ( L~ f1)), ( N-bound ( L~ f1))} by A33, SPRECT_1: 32;

      

       A41: ( width ( GoB f1)) = ( card ( rng ( Y_axis f1))) by GOBOARD2: 13

      .= (1 + 1) by A40, CARD_2: 57;

      then

       A42: ( width ( GoB f1)) in ( Seg ( width ( GoB f1))) by FINSEQ_1: 1;

      (f1 /. (1 + 1)) = ( E-max ( L~ f1)) by SPRECT_1: 84;

      then

       A43: (f /. 2) = |[(( E-max ( L~ f)) `1 ), (( N-max ( L~ f)) `2 )]| by A28, EUCLID: 53;

      

       A44: (f1 /. 1) = ( W-max ( L~ f1)) by SPRECT_1: 83;

      then (f /. 1) = |[(( W-max ( L~ f)) `1 ), (( N-min ( L~ f)) `2 )]| by A14, EUCLID: 53;

      then ((f /. 1) + (f /. 2)) = |[((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 )), ((( N-min ( L~ f)) `2 ) + (( N-max ( L~ f)) `2 ))]| by A43, EUCLID: 56;

      then ((1 / 2) * ((f /. 1) + (f /. 2))) = |[((1 / 2) * ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 ))), ( N-bound ( L~ f))]| by A3, EUCLID: 58;

      

      then

       A45: q1 = |[( 0 + ((1 / 2) * ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 )))), (((r1 + r) + 1) + ( N-bound ( L~ f)))]| by EUCLID: 56

      .= |[((1 / 2) * ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 ))), (((r1 + r) + 1) + ( N-bound ( L~ f)))]|;

      (( W-max ( L~ f)) `1 ) = ( W-bound ( L~ f)) by EUCLID: 52;

      then

       A46: (( W-max ( L~ f)) `1 ) < (( E-max ( L~ f)) `1 ) by A1, EUCLID: 52;

      

       A47: (f1 /. 1) = ( W-max ( L~ f1)) by SPRECT_1: 83;

      then

       A48: ((( GoB f1) * (1,1)) `1 ) <= (( W-max ( L~ f)) `1 ) by A4, A41, JORDAN5D: 5;

      then ((( GoB f1) * (1,1)) `1 ) < (( E-max ( L~ f)) `1 ) by A46, XXREAL_0: 2;

      then (((( GoB f1) * (1,1)) `1 ) + ((( GoB f1) * (1,1)) `1 )) < ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 )) by A48, XREAL_1: 8;

      then

       A49: ((1 / 2) * (2 * ((( GoB f1) * (1,1)) `1 ))) < ((1 / 2) * ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 ))) by XREAL_1: 68;

      1 in ( dom ( X_axis f1)) by A22, SPRECT_2: 15;

      then (( X_axis f1) . 1) = ((f1 /. 1) `1 ) by GOBOARD1:def 1;

      then

       A50: (( X_axis f1) . 1) = ( W-bound ( L~ f1)) by A47, EUCLID: 52;

      (f1 /. 5) = ( W-max ( L~ f1)) by A4, A44, FINSEQ_6:def 1;

      then

       A51: (( X_axis f1) . 5) = ( W-bound ( L~ f1)) by A12, EUCLID: 52;

      

       A52: ( rng ( X_axis f1)) c= {( W-bound ( L~ f1)), ( E-bound ( L~ f1))}

      proof

        let z be object;

        assume z in ( rng ( X_axis f1));

        then

        consider u be object such that

         A53: u in ( dom ( X_axis f1)) and

         A54: z = (( X_axis f1) . u) by FUNCT_1:def 3;

        reconsider mu = u as Element of NAT by A53;

        u in ( dom f1) by A53, SPRECT_2: 15;

        then u in ( Seg ( len f1)) by FINSEQ_1:def 3;

        then 1 <= mu & mu <= 5 by A4, FINSEQ_1: 1;

        then

         A55: mu = (1 + 0 ) or ... or mu = (1 + 4) by NAT_1: 62;

        per cases by A55;

          suppose mu = 1;

          hence thesis by A50, A54, TARSKI:def 2;

        end;

          suppose mu = 2;

          hence thesis by A20, A54, TARSKI:def 2;

        end;

          suppose mu = 3;

          hence thesis by A11, A54, TARSKI:def 2;

        end;

          suppose mu = 4;

          hence thesis by A15, A54, TARSKI:def 2;

        end;

          suppose mu = 5;

          hence thesis by A51, A54, TARSKI:def 2;

        end;

      end;

       {( W-bound ( L~ f1)), ( E-bound ( L~ f1))} c= ( rng ( X_axis f1))

      proof

        let z be object;

        assume

         A56: z in {( W-bound ( L~ f1)), ( E-bound ( L~ f1))};

        per cases by A56, TARSKI:def 2;

          suppose

           A57: z = ( W-bound ( L~ f1));

          1 in ( dom ( X_axis f1)) by A22, SPRECT_2: 15;

          hence thesis by A50, A57, FUNCT_1:def 3;

        end;

          suppose

           A58: z = ( E-bound ( L~ f1));

          2 in ( dom ( X_axis f1)) by A18, SPRECT_2: 15;

          hence thesis by A20, A58, FUNCT_1:def 3;

        end;

      end;

      then

       A59: ( rng ( X_axis f1)) = {( W-bound ( L~ f1)), ( E-bound ( L~ f1))} by A52;

      

       A60: ( len ( GoB f1)) = ( card ( rng ( X_axis f1))) by GOBOARD2: 13

      .= (1 + 1) by A1, A59, CARD_2: 57;

      then

       A61: ((( GoB f1) * ((1 + 1),1)) `1 ) >= (( E-max ( L~ f)) `1 ) by A4, A17, A41, JORDAN5D: 5;

      then (( W-max ( L~ f)) `1 ) < ((( GoB f1) * ((1 + 1),1)) `1 ) by A46, XXREAL_0: 2;

      then ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 )) < (((( GoB f1) * ((1 + 1),1)) `1 ) + ((( GoB f1) * ((1 + 1),1)) `1 )) by A61, XREAL_1: 8;

      then

       A62: ((1 / 2) * ((( W-max ( L~ f)) `1 ) + (( E-max ( L~ f)) `1 ))) < ((1 / 2) * (2 * ((( GoB f1) * ((1 + 1),1)) `1 ))) by XREAL_1: 68;

      

       A63: ( card ( rng ( X_axis f1))) = 2 by A1, A59, CARD_2: 57;

      for p be FinSequence of the carrier of ( TOP-REAL 2) st p in ( rng ( GoB f1)) holds ( len p) = 2

      proof

        ( len ( GoB (( Incr ( X_axis f1)),( Incr ( Y_axis f1))))) = ( len ( Incr ( X_axis f1))) by GOBOARD2:def 1

        .= 2 by A63, SEQ_4:def 21;

        then

        consider s1 be FinSequence such that

         A64: s1 in ( rng ( GoB (( Incr ( X_axis f1)),( Incr ( Y_axis f1))))) and

         A65: ( len s1) = ( width ( GoB (( Incr ( X_axis f1)),( Incr ( Y_axis f1))))) by MATRIX_0:def 3;

        let p be FinSequence of the carrier of ( TOP-REAL 2);

        consider n be Nat such that

         A66: for x st x in ( rng ( GoB f1)) holds ex s be FinSequence st s = x & ( len s) = n by MATRIX_0:def 1;

        assume p in ( rng ( GoB f1));

        then

         A67: ex s2 be FinSequence st s2 = p & ( len s2) = n by A66;

        ex s be FinSequence st s = s1 & ( len s) = n by A16, A64, A66;

        hence thesis by A41, A65, A67, GOBOARD2:def 2;

      end;

      then

       A68: ( GoB f1) is Matrix of 2, 2, the carrier of ( TOP-REAL 2) by A60, MATRIX_0:def 2;

      ( len ( GoB f1)) in ( Seg ( len ( GoB f1))) by A60, FINSEQ_1: 1;

      then [( len ( GoB f1)), ( width ( GoB f1))] in [:( Seg ( len ( GoB f1))), ( Seg ( width ( GoB f1))):] by A42, ZFMISC_1: 87;

      then

       A69: [( len ( GoB f1)), ( width ( GoB f1))] in ( Indices ( GoB f1)) by A60, A41, A68, MATRIX_0: 24;

      1 in ( Seg ( len ( GoB f1))) by A60, FINSEQ_1: 1;

      then [1, ( width ( GoB f1))] in [:( Seg ( len ( GoB f1))), ( Seg ( width ( GoB f1))):] by A42, ZFMISC_1: 87;

      then

       A70: [1, ( width ( GoB f1))] in ( Indices ( GoB f1)) by A60, A41, A68, MATRIX_0: 24;

      ( card ( rng ( X_axis f1))) > 1 by A27, A29, XXREAL_0: 2;

      then

       A71: 1 < ( len ( GoB f1)) by GOBOARD2: 13;

      

       A72: (f1 /. 1) = (( GoB f1) * (1,( width ( GoB f1)))) by A41, Th73;

      set p = |[ 0 , ((r1 + r) + 1)]|;

      

       A73: (p `1 ) = 0 & (p `2 ) = ((r1 + r) + 1) by EUCLID: 52;

      

       A74: |.q1.| >= ( |. |[ 0 , ((r1 + r) + 1)]|.| - r1) & r < (r + 1) by TOPRNS_1: 31, XREAL_1: 29;

      

       A75: ( Int ( left_cell (f1,1))) c= ( LeftComp f) by GOBOARD9:def 1;

      

       A76: ( width ( GoB f1)) <> (( width ( GoB f1)) + 1);

      (f1 /. (1 + 1)) = (( GoB f1) * (( len ( GoB f1)),( width ( GoB f1)))) by A60, A41, Th73;

      then ( left_cell (f1,1)) = ( cell (( GoB f1),1,( width ( GoB f1)))) by A4, A70, A69, A72, A30, A76, GOBOARD5:def 7;

      then

       A77: ( Int ( left_cell (f1,1))) = { |[r2, s]| : ((( GoB f1) * (1,1)) `1 ) < r2 & r2 < ((( GoB f1) * ((1 + 1),1)) `1 ) & ((( GoB f1) * (1,( width ( GoB f1)))) `2 ) < s } by A71, GOBOARD6: 25;

      

       A78: |.q4.| < r by A13;

      

       A79: |. |[ 0 , ((r1 + r) + 1)]|.| = ( sqrt (((p `1 ) ^2 ) + ((p `2 ) ^2 ))) by JGRAPH_1: 30

      .= ((r1 + r) + 1) by A78, A73, SQUARE_1: 22;

      ((( GoB f1) * (1,( width ( GoB f1)))) `2 ) < (( N-bound ( L~ f)) + ((r1 + r) + 1)) by A78, A14, A72, A2, XREAL_1: 29;

      then q1 in ( Int ( left_cell (f1,1))) by A77, A45, A49, A62;

      hence contradiction by A13, A79, A74, A75, XXREAL_0: 2;

    end;

    theorem :: JORDAN2C:91

    

     Th75: ( LeftComp ( SpStSeq D)) c= ( UBD ( L~ ( SpStSeq D)))

    proof

      set f = ( SpStSeq D);

      set A = ( L~ ( SpStSeq D));

      ( LeftComp f) is_a_component_of (A ` ) & not ( LeftComp f) is bounded by Th74, GOBOARD9:def 1;

      then

       A1: ( LeftComp f) is_outside_component_of A;

      ( LeftComp f) c= ( union { B where B be Subset of ( TOP-REAL 2) : B is_outside_component_of A })

      proof

        let x be object;

        assume

         A2: x in ( LeftComp f);

        ( LeftComp f) in { B where B be Subset of ( TOP-REAL 2) : B is_outside_component_of A } by A1;

        hence thesis by A2, TARSKI:def 4;

      end;

      hence thesis;

    end;

    theorem :: JORDAN2C:92

    

     Th76: for G be TopSpace, A,B,C be Subset of G st A is a_component & B is a_component & C is connected & A meets C & B meets C holds A = B

    proof

      let G be TopSpace, A,B,C be Subset of G;

      assume that

       A1: A is a_component and

       A2: B is a_component and

       A3: C is connected and

       A4: A meets C and

       A5: B meets C;

      

       A6: (C /\ A) = ( {} G) or C c= A by A1, A3, A4, CONNSP_1: 36;

      

       A7: C misses B or C c= B by A2, A3, CONNSP_1: 36;

      per cases by A1, A2, CONNSP_1: 1, CONNSP_1: 34;

        suppose A = B;

        hence thesis;

      end;

        suppose A misses B;

        then

         A8: (A /\ B) = {} ;

        C c= (A /\ B) by A4, A5, A6, A7, XBOOLE_1: 19;

        then C = {} by A8;

        then (C /\ A) = {} ;

        hence thesis by A4;

      end;

    end;

    theorem :: JORDAN2C:93

    

     Th77: for B be Subset of ( TOP-REAL 2) st B is_a_component_of (( L~ ( SpStSeq D)) ` ) & not B is bounded holds B = ( LeftComp ( SpStSeq D))

    proof

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

      set f = ( SpStSeq D);

      assume that

       A1: B is_a_component_of (( L~ f) ` ) and

       A2: not B is bounded;

      

       A3: ex B1 be Subset of (( TOP-REAL 2) | (( L~ f) ` )) st B1 = B & B1 is a_component by A1, CONNSP_1:def 6;

      consider r1 be Real such that

       A4: for q be Point of ( TOP-REAL 2) st q in ( L~ f) holds |.q.| < r1 by Th21;

      consider q4 be Point of ( TOP-REAL 2) such that

       A5: q4 in B and

       A6: |.q4.| >= r1 by A2, Th21;

       A7:

      now

        assume q4 in { q where q be Point of ( TOP-REAL 2) : |.q.| < r1 };

        then ex q be Point of ( TOP-REAL 2) st q = q4 & |.q.| < r1;

        hence contradiction by A6;

      end;

      reconsider P = (( REAL 2) \ { q where q be Point of ( TOP-REAL 2) : |.q.| < r1 }) as Subset of ( TOP-REAL 2) by EUCLID: 22;

      P c= (the carrier of ( TOP-REAL 2) \ ( L~ f))

      proof

        let z be object;

        assume

         A8: z in P;

        now

          assume

           A9: z in ( L~ f);

          then

          reconsider q3 = z as Point of ( TOP-REAL 2);

          

           A10: not q3 in { q where q be Point of ( TOP-REAL 2) : |.q.| < r1 } by A8, XBOOLE_0:def 5;

           |.q3.| < r1 by A4, A9;

          hence contradiction by A10;

        end;

        hence thesis by A8, XBOOLE_0:def 5;

      end;

      then

       A11: (P /\ (the carrier of ( TOP-REAL 2) \ ( L~ f))) = P by XBOOLE_1: 28;

      then

       A12: ( Down (P,(( L~ f) ` ))) is connected by Th40, CONNSP_1: 46;

       not ( LeftComp f) is bounded by Th74;

      then

      consider q3 be Point of ( TOP-REAL 2) such that

       A13: q3 in ( LeftComp f) and

       A14: |.q3.| >= r1 by Th21;

       A15:

      now

        assume q3 in { q where q be Point of ( TOP-REAL 2) : |.q.| < r1 };

        then ex q be Point of ( TOP-REAL 2) st q = q3 & |.q.| < r1;

        hence contradiction by A14;

      end;

      q4 in the carrier of ( TOP-REAL 2);

      then q4 in ( REAL 2) by EUCLID: 22;

      then q4 in P by A7, XBOOLE_0:def 5;

      then

       A16: B meets P by A5, XBOOLE_0: 3;

      ( LeftComp f) is_a_component_of (( L~ f) ` ) by GOBOARD9:def 1;

      then

      consider L1 be Subset of (( TOP-REAL 2) | (( L~ f) ` )) such that

       A17: L1 = ( LeftComp f) and

       A18: L1 is a_component by CONNSP_1:def 6;

      q3 in the carrier of ( TOP-REAL 2);

      then q3 in ( REAL 2) by EUCLID: 22;

      then q3 in P by A15, XBOOLE_0:def 5;

      then L1 meets P by A17, A13, XBOOLE_0: 3;

      hence thesis by A3, A17, A18, A11, A12, A16, Th76;

    end;

    theorem :: JORDAN2C:94

    

     Th78: ( RightComp ( SpStSeq D)) c= ( BDD ( L~ ( SpStSeq D))) & ( RightComp ( SpStSeq D)) is bounded

    proof

      set f = ( SpStSeq D);

      set A = ( L~ ( SpStSeq D));

      

       A1: ( RightComp f) is_a_component_of (A ` ) by GOBOARD9:def 2;

       A2:

      now

        

         A3: ( LeftComp f) misses ( RightComp f) by SPRECT_1: 88;

        assume not ( RightComp f) is bounded;

        hence contradiction by A1, A3, Th77;

      end;

      then

       A4: ( RightComp f) is_inside_component_of A by A1;

      ( RightComp f) c= ( union { B where B be Subset of ( TOP-REAL 2) : B is_inside_component_of A })

      proof

        let x be object;

        assume

         A5: x in ( RightComp f);

        ( RightComp f) in { B where B be Subset of ( TOP-REAL 2) : B is_inside_component_of A } by A4;

        hence thesis by A5, TARSKI:def 4;

      end;

      hence ( RightComp f) c= ( BDD ( L~ ( SpStSeq D)));

      thus thesis by A2;

    end;

    theorem :: JORDAN2C:95

    

     Th79: ( LeftComp ( SpStSeq D)) = ( UBD ( L~ ( SpStSeq D))) & ( RightComp ( SpStSeq D)) = ( BDD ( L~ ( SpStSeq D)))

    proof

      set f = ( SpStSeq D);

      

       A1: (( L~ f) ` ) = (( LeftComp f) \/ ( RightComp f)) by GOBRD12: 10;

      

       A2: ( LeftComp f) c= ( UBD ( L~ ( SpStSeq D))) by Th75;

      

       A3: ( RightComp f) c= ( BDD ( L~ ( SpStSeq D))) by Th78;

       A4:

      now

        assume not ( LeftComp f) = ( UBD ( L~ ( SpStSeq D)));

        then not ( UBD ( L~ ( SpStSeq D))) c= ( LeftComp f) by A2;

        then

        consider z be object such that

         A5: z in ( UBD ( L~ ( SpStSeq D))) and

         A6: not z in ( LeftComp f);

        ( UBD ( L~ f)) c= (( L~ f) ` ) by Th17;

        then z in ( LeftComp f) or z in ( RightComp f) by A1, A5, XBOOLE_0:def 3;

        then ( BDD ( L~ f)) meets ( UBD ( L~ f)) by A3, A5, A6, XBOOLE_0: 3;

        hence contradiction by Th15;

      end;

      now

        assume not ( RightComp f) = ( BDD ( L~ ( SpStSeq D)));

        then not ( BDD ( L~ ( SpStSeq D))) c= ( RightComp f) by A3;

        then

        consider z be object such that

         A7: z in ( BDD ( L~ ( SpStSeq D))) and

         A8: not z in ( RightComp f);

        ( BDD ( L~ f)) c= (( L~ f) ` ) by Th16;

        then z in ( LeftComp f) or z in ( RightComp f) by A1, A7, XBOOLE_0:def 3;

        then ( BDD ( L~ f)) meets ( UBD ( L~ f)) by A2, A7, A8, XBOOLE_0: 3;

        hence contradiction by Th15;

      end;

      hence thesis by A4;

    end;

    theorem :: JORDAN2C:96

    

     Th80: ( UBD ( L~ ( SpStSeq D))) <> {} & ( UBD ( L~ ( SpStSeq D))) is_outside_component_of ( L~ ( SpStSeq D)) & ( BDD ( L~ ( SpStSeq D))) <> {} & ( BDD ( L~ ( SpStSeq D))) is_inside_component_of ( L~ ( SpStSeq D))

    proof

      set f = ( SpStSeq D);

      

       A1: ( UBD ( L~ ( SpStSeq D))) = ( LeftComp ( SpStSeq D)) by Th79;

      hence ( UBD ( L~ ( SpStSeq D))) <> {} ;

      ( LeftComp f) is_a_component_of (( L~ f) ` ) & not ( LeftComp f) is bounded by Th74, GOBOARD9:def 1;

      hence ( UBD ( L~ ( SpStSeq D))) is_outside_component_of ( L~ ( SpStSeq D)) by A1;

      

       A2: ( BDD ( L~ ( SpStSeq D))) = ( RightComp ( SpStSeq D)) by Th79;

      hence ( BDD ( L~ ( SpStSeq D))) <> {} ;

      ( RightComp ( SpStSeq D)) is_a_component_of (( L~ f) ` ) & ( RightComp ( SpStSeq D)) is bounded by Th78, GOBOARD9:def 2;

      hence thesis by A2;

    end;

    begin

    theorem :: JORDAN2C:97

    

     Th81: for G be non empty TopSpace, A be Subset of G st (A ` ) <> {} holds A is boundary iff for x be set, V be Subset of G st x in A & x in V & V is open holds ex B be Subset of G st B is_a_component_of (A ` ) & V meets B

    proof

      let G be non empty TopSpace, A be Subset of G;

      assume

       A1: (A ` ) <> {} ;

      hereby

        reconsider A1 = (A ` ) as non empty Subset of G by A1;

        reconsider A2 = (A ` ) as Subset of G;

        assume A is boundary;

        then (A ` ) is dense by TOPS_1:def 4;

        then

         A2: ( Cl (A ` )) = ( [#] G) by TOPS_1:def 3;

        let x be set, V be Subset of G;

        assume that x in A and

         A3: x in V & V is open;

        A2 meets V by A3, A2, PRE_TOPC:def 7;

        then

        consider z be object such that

         A4: z in (A ` ) and

         A5: z in V by XBOOLE_0: 3;

        reconsider p = z as Point of (G | (A ` )) by A4, PRE_TOPC: 8;

        ( Component_of p) c= the carrier of (G | (A ` ));

        then ( Component_of p) c= (A ` ) by PRE_TOPC: 8;

        then

        reconsider B0 = ( Component_of p) as Subset of G by XBOOLE_1: 1;

        

         A6: (G | A1) is non empty;

        then p in ( Component_of p) by CONNSP_1: 38;

        then p in (V /\ B0) by A5, XBOOLE_0:def 4;

        then

         A7: V meets B0;

        ( Component_of p) is a_component by A6, CONNSP_1: 40;

        then B0 is_a_component_of (A ` ) by CONNSP_1:def 6;

        hence ex B be Subset of G st B is_a_component_of (A ` ) & V meets B by A7;

      end;

      assume

       A8: for x be set, V be Subset of G st x in A & x in V & V is open holds ex B be Subset of G st B is_a_component_of (A ` ) & V meets B;

      the carrier of G c= ( Cl (A ` ))

      proof

        let z be object;

        assume

         A9: z in the carrier of G;

        per cases ;

          suppose

           A10: z in A;

          for G1 be Subset of G st G1 is open holds z in G1 implies (A ` ) meets G1

          proof

            let G1 be Subset of G;

            assume

             A11: G1 is open;

            assume z in G1;

            then

            consider B be Subset of G such that

             A12: B is_a_component_of (A ` ) and

             A13: G1 meets B by A8, A10, A11;

            

             A14: (G1 /\ B) <> {} by A13;

            consider B1 be Subset of (G | (A ` )) such that

             A15: B1 = B and B1 is a_component by A12, CONNSP_1:def 6;

            B1 c= the carrier of (G | (A ` ));

            then B1 c= (A ` ) by PRE_TOPC: 8;

            then ((A ` ) /\ G1) <> ( {} G) by A15, A14, XBOOLE_1: 3, XBOOLE_1: 26;

            hence thesis;

          end;

          hence thesis by A9, PRE_TOPC:def 7;

        end;

          suppose

           A16: not z in A;

          

           A17: (A ` ) c= ( Cl (A ` )) by PRE_TOPC: 18;

          z in (the carrier of G \ A) by A9, A16, XBOOLE_0:def 5;

          hence thesis by A17;

        end;

      end;

      then ( Cl (A ` )) = ( [#] G);

      then (A ` ) is dense by TOPS_1:def 3;

      hence thesis by TOPS_1:def 4;

    end;

    theorem :: JORDAN2C:98

    

     Th82: for A be Subset of ( TOP-REAL 2) st (A ` ) <> {} holds A is boundary & A is Jordan iff ex A1,A2 be Subset of ( TOP-REAL 2) st (A ` ) = (A1 \/ A2) & A1 misses A2 & (( Cl A1) \ A1) = (( Cl A2) \ A2) & A = (( Cl A1) \ A1) & for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component

    proof

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

      assume

       A1: (A ` ) <> {} ;

      hereby

        assume that

         A2: A is boundary and

         A3: A is Jordan;

        consider A1,A2 be Subset of ( TOP-REAL 2) such that

         A4: (A ` ) = (A1 \/ A2) and

         A5: A1 misses A2 and

         A6: (( Cl A1) \ A1) = (( Cl A2) \ A2) and

         A7: for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component by A3, JORDAN1:def 2;

        A = ((A1 \/ A2) ` ) by A4;

        then

         A8: A = ((A1 ` ) /\ (A2 ` )) by XBOOLE_1: 53;

        A2 c= (A ` ) by A4, XBOOLE_1: 7;

        then

        reconsider D2 = A2 as Subset of (( TOP-REAL 2) | (A ` )) by PRE_TOPC: 8;

        A1 c= (A ` ) by A4, XBOOLE_1: 7;

        then

        reconsider D1 = A1 as Subset of (( TOP-REAL 2) | (A ` )) by PRE_TOPC: 8;

        D2 = A2;

        then

         A9: D1 is a_component by A7;

        

         A10: A c= (( Cl A1) \ A1)

        proof

          let z be object;

          assume

           A11: z in A;

          for G be Subset of ( TOP-REAL 2) st G is open holds z in G implies (A1 \/ A2) meets G

          proof

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

            assume

             A12: G is open;

            hereby

              assume z in G;

              then

              consider B be Subset of ( TOP-REAL 2) such that

               A13: B is_a_component_of (A ` ) and

               A14: G meets B by A1, A2, A11, A12, Th81;

              consider B1 be Subset of (( TOP-REAL 2) | (A ` )) such that

               A15: B1 = B and

               A16: B1 is a_component by A13, CONNSP_1:def 6;

               A17:

              now

                per cases by A9, A16, CONNSP_1: 34;

                  case B1 = D1;

                  hence B1 c= (A1 \/ A2) by XBOOLE_1: 7;

                end;

                  case (B1,D1) are_separated ;

                  then

                   A18: ( Cl B1) misses D1 or B1 misses ( Cl D1) by CONNSP_1:def 1;

                  B1 is closed & D1 is closed by A9, A16, CONNSP_1: 33;

                  then B1 misses D1 by A18, PRE_TOPC: 22;

                  then

                   A19: (B1 /\ D1) = {} ;

                  B1 c= the carrier of (( TOP-REAL 2) | (A ` ));

                  then B1 c= (A ` ) by PRE_TOPC: 8;

                  

                  then B1 = (B1 /\ (A ` )) by XBOOLE_1: 28

                  .= ((B1 /\ A1) \/ (B1 /\ A2)) by A4, XBOOLE_1: 23

                  .= (B1 /\ A2) by A19;

                  then

                   A20: B1 c= A2 by XBOOLE_1: 17;

                  A2 c= (A1 \/ A2) by XBOOLE_1: 7;

                  hence B1 c= (A1 \/ A2) by A20;

                end;

              end;

              (G /\ B) <> {} by A14;

              then ((A1 \/ A2) /\ G) <> {} by A15, A17, XBOOLE_1: 3, XBOOLE_1: 26;

              hence (A1 \/ A2) meets G;

            end;

          end;

          then z in ( Cl (A1 \/ A2)) by A11, PRE_TOPC:def 7;

          then z in (( Cl A1) \/ ( Cl A2)) by PRE_TOPC: 20;

          then

           A21: z in ( Cl A1) or z in ( Cl A2) by XBOOLE_0:def 3;

           not z in (A ` ) by A11, XBOOLE_0:def 5;

          then ( not z in A1) & not z in A2 by A4, XBOOLE_0:def 3;

          hence thesis by A6, A21, XBOOLE_0:def 5;

        end;

        (( Cl A1) \ A1) c= (A1 ` ) & (( Cl A2) \ A2) c= (A2 ` ) by XBOOLE_1: 33;

        then (( Cl A1) \ A1) c= A by A6, A8, XBOOLE_1: 19;

        then A = (( Cl A1) \ A1) by A10;

        hence ex A1,A2 be Subset of ( TOP-REAL 2) st (A ` ) = (A1 \/ A2) & A1 misses A2 & (( Cl A1) \ A1) = (( Cl A2) \ A2) & A = (( Cl A1) \ A1) & for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component by A4, A5, A6, A7;

      end;

      hereby

        assume ex A1,A2 be Subset of ( TOP-REAL 2) st (A ` ) = (A1 \/ A2) & A1 misses A2 & (( Cl A1) \ A1) = (( Cl A2) \ A2) & A = (( Cl A1) \ A1) & for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component;

        then

        consider A1,A2 be Subset of ( TOP-REAL 2) such that

         A22: (A ` ) = (A1 \/ A2) and

         A23: A1 misses A2 & (( Cl A1) \ A1) = (( Cl A2) \ A2) and

         A24: A = (( Cl A1) \ A1) and

         A25: for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component;

        for x be set, V be Subset of ( TOP-REAL 2) st x in A & x in V & V is open holds ex B be Subset of ( TOP-REAL 2) st B is_a_component_of (A ` ) & V meets B

        proof

          A2 c= (A ` ) by A22, XBOOLE_1: 7;

          then

          reconsider D2 = A2 as Subset of (( TOP-REAL 2) | (A ` )) by PRE_TOPC: 8;

          A1 c= (A ` ) by A22, XBOOLE_1: 7;

          then

          reconsider D1 = A1 as Subset of (( TOP-REAL 2) | (A ` )) by PRE_TOPC: 8;

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

          assume that

           A26: x in A and

           A27: x in V & V is open;

          D2 = A2;

          then D1 is a_component by A25;

          then

           A28: A1 is_a_component_of (A ` ) by CONNSP_1:def 6;

          x in ( Cl A1) by A24, A26, XBOOLE_0:def 5;

          then A1 meets V by A27, PRE_TOPC:def 7;

          hence thesis by A28;

        end;

        hence A is boundary & A is Jordan by A1, A22, A23, A25, Th81, JORDAN1:def 2;

      end;

    end;

    theorem :: JORDAN2C:99

    

     Th83: for p be Point of ( TOP-REAL n), P be Subset of ( TOP-REAL n) st n >= 1 & P = {p} holds P is boundary

    proof

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

      assume that

       A1: n >= 1 and

       A2: P = {p};

      the carrier of ( TOP-REAL n) c= ( Cl (P ` ))

      proof

        let z be object;

        assume

         A3: z in the carrier of ( TOP-REAL n);

        per cases ;

          suppose

           A4: z = p;

          reconsider ez = z as Point of ( Euclid n) by A3, TOPREAL3: 8;

          for G1 be Subset of ( TOP-REAL n) st G1 is open holds z in G1 implies (P ` ) meets G1

          proof

            let G1 be Subset of ( TOP-REAL n);

            assume

             A5: G1 is open;

            thus z in G1 implies (P ` ) meets G1

            proof

              

               A6: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

              then

              reconsider GG = G1 as Subset of ( TopSpaceMetr ( Euclid n));

              assume

               A7: z in G1;

              GG is open by A5, A6, PRE_TOPC: 30;

              then

              consider r be Real such that

               A8: r > 0 and

               A9: ( Ball (ez,r)) c= GG by A7, TOPMETR: 15;

              reconsider r as Real;

              set p2 = (p - (((r / 2) / ( sqrt n)) * ( 1.REAL n)));

              reconsider ep2 = p2 as Point of ( Euclid n) by TOPREAL3: 8;

              

               A10: 0 < ( sqrt n) by A1, SQUARE_1: 25;

              

               A11: |.(p - p2).| = |.((p - p) + (((r / 2) / ( sqrt n)) * ( 1.REAL n))).| by RLVECT_1: 29

              .= |.(((((r / 2) / ( sqrt n)) * ( 1.REAL n)) + p) - p).| by RLVECT_1:def 3

              .= |.(((r / 2) / ( sqrt n)) * ( 1.REAL n)).| by RLVECT_4: 1

              .= ( |.((r / 2) / ( sqrt n)).| * |.( 1.REAL n).|) by TOPRNS_1: 7

              .= ( |.((r / 2) / ( sqrt n)).| * ( sqrt n)) by EUCLID: 73

              .= (( |.(r / 2).| / |.( sqrt n).|) * ( sqrt n)) by COMPLEX1: 67

              .= (( |.(r / 2).| / ( sqrt n)) * ( sqrt n)) by A10, ABSVALUE:def 1

              .= |.(r / 2).| by A10, XCMPLX_1: 87

              .= (r / 2) by A8, ABSVALUE:def 1;

              (r / 2) > 0 by A8, XREAL_1: 139;

              then p <> p2 by A11, TOPRNS_1: 28;

              then not p2 in P by A2, TARSKI:def 1;

              then

               A12: p2 in (P ` ) by XBOOLE_0:def 5;

              (r / 2) < r by A8, XREAL_1: 216;

              then ( dist (ez,ep2)) < r by A4, A11, JGRAPH_1: 28;

              then p2 in ( Ball (ez,r)) by METRIC_1: 11;

              hence thesis by A9, A12, XBOOLE_0: 3;

            end;

          end;

          hence thesis by A3, PRE_TOPC:def 7;

        end;

          suppose z <> p;

          then not z in P by A2, TARSKI:def 1;

          then

           A13: z in (P ` ) by A3, XBOOLE_0:def 5;

          (P ` ) c= ( Cl (P ` )) by PRE_TOPC: 18;

          hence thesis by A13;

        end;

      end;

      then ( Cl (P ` )) = ( [#] ( TOP-REAL n));

      then (P ` ) is dense by TOPS_1:def 3;

      hence thesis by TOPS_1:def 4;

    end;

    theorem :: JORDAN2C:100

    

     Th84: for p,q be Point of ( TOP-REAL 2), r st (p `1 ) = (q `2 ) & ( - (p `2 )) = (q `1 ) & p = (r * q) holds (p `1 ) = 0 & (p `2 ) = 0 & p = ( 0. ( TOP-REAL 2))

    proof

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

      

       A1: (1 + (r * r)) > ( 0 + 0 ) by XREAL_1: 8, XREAL_1: 63;

      assume (p `1 ) = (q `2 ) & ( - (p `2 )) = (q `1 ) & p = (r * q);

      then

       A2: p = |[(r * ( - (p `2 ))), (r * (p `1 ))]| by EUCLID: 57;

      then (p `2 ) = (r * (p `1 )) by EUCLID: 52;

      

      then (p `1 ) = ( - (r * (r * (p `1 )))) by A2, EUCLID: 52

      .= ( - ((r * r) * (p `1 )));

      then ((1 + (r * r)) * (p `1 )) = 0 ;

      hence

       A3: (p `1 ) = 0 by A1, XCMPLX_1: 6;

      (p `1 ) = (r * ( - (p `2 ))) by A2, EUCLID: 52;

      then (p `2 ) = ( - ((r * r) * (p `2 ))) by A2, EUCLID: 52;

      then ((1 + (r * r)) * (p `2 )) = 0 ;

      hence (p `2 ) = 0 by A1, XCMPLX_1: 6;

      hence thesis by A3, EUCLID: 53, EUCLID: 54;

    end;

    theorem :: JORDAN2C:101

    

     Th85: for q1,q2 be Point of ( TOP-REAL 2) holds ( LSeg (q1,q2)) is boundary

    proof

      let q1,q2 be Point of ( TOP-REAL 2);

      per cases ;

        suppose q1 = q2;

        then ( LSeg (q1,q2)) = {q1} by RLTOPSP1: 70;

        hence thesis by Th83;

      end;

        suppose

         A1: q1 <> q2;

        set P = ( LSeg (q1,q2));

        the carrier of ( TOP-REAL 2) c= ( Cl (P ` ))

        proof

          let z be object;

          assume

           A2: z in the carrier of ( TOP-REAL 2);

          per cases ;

            suppose

             A3: z in P;

            reconsider ez = z as Point of ( Euclid 2) by A2, TOPREAL3: 8;

            set p1 = (q1 - q2);

            consider s be Real such that

             A4: z = (((1 - s) * q1) + (s * q2)) and 0 <= s and s <= 1 by A3;

            set p = (((1 - s) * q1) + (s * q2));

             A5:

            now

              assume |.p1.| = 0 ;

              then p1 = ( 0. ( TOP-REAL 2)) by TOPRNS_1: 24;

              hence contradiction by A1, RLVECT_1: 21;

            end;

            for G1 be Subset of ( TOP-REAL 2) st G1 is open holds z in G1 implies (P ` ) meets G1

            proof

              let G1 be Subset of ( TOP-REAL 2);

              assume

               A6: G1 is open;

              thus z in G1 implies (P ` ) meets G1

              proof

                

                 A7: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

                then

                reconsider GG = G1 as Subset of ( TopSpaceMetr ( Euclid 2));

                assume

                 A8: z in G1;

                GG is open by A6, A7, PRE_TOPC: 30;

                then

                consider r be Real such that

                 A9: r > 0 and

                 A10: ( Ball (ez,r)) c= G1 by A8, TOPMETR: 15;

                reconsider r as Real;

                

                 A11: (r / 2) < r by A9, XREAL_1: 216;

                set p2 = ((((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|) + p);

                now

                  assume p2 in P;

                  then

                  consider s2 be Real such that

                   A12: p2 = (((1 - s2) * q1) + (s2 * q2)) and 0 <= s2 and s2 <= 1;

                   A13:

                  now

                    assume (s - s2) = 0 ;

                    then (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|) = (p - p) by A12, RLVECT_4: 1;

                    then

                     A14: (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 5;

                    

                     A15: ((r / 2) / |.p1.|) = ((r * (2 " )) * ( |.p1.| " )) by XCMPLX_0:def 9

                    .= (r * ((2 " ) * ( |.p1.| " )));

                    ((2 " ) * ( |.p1.| " )) <> 0 by A5;

                    then |[( - (p1 `2 )), (p1 `1 )]| = ( 0. ( TOP-REAL 2)) by A9, A14, A15, RLVECT_1: 11, XCMPLX_1: 6;

                    then

                     A16: (( 0. ( TOP-REAL 2)) `1 ) = ( - (p1 `2 )) & (( 0. ( TOP-REAL 2)) `2 ) = (p1 `1 ) by EUCLID: 52;

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

                    hence contradiction by A1, A16, EUCLID: 53, EUCLID: 54, RLVECT_1: 21;

                  end;

                  

                   A17: (p2 - p) = (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|) by RLVECT_4: 1;

                  (p2 - p) = (((((1 - s2) * q1) + (s2 * q2)) - ((1 - s) * q1)) - (s * q2)) by A12, RLVECT_1: 27

                  .= (((((1 - s2) * q1) - ((1 - s) * q1)) + (s2 * q2)) - (s * q2)) by RLVECT_1:def 3

                  .= (((((1 - s2) - (1 - s)) * q1) + (s2 * q2)) - (s * q2)) by RLVECT_1: 35

                  .= (((s - s2) * q1) + ((s2 * q2) - (s * q2))) by RLVECT_1:def 3

                  .= (((s - s2) * q1) + ((s2 - s) * q2)) by RLVECT_1: 35

                  .= (((s - s2) * q1) + (( - (s - s2)) * q2))

                  .= (((s - s2) * q1) - ((s - s2) * q2)) by RLVECT_1: 79

                  .= ((s - s2) * p1) by RLVECT_1: 34;

                  then (((1 / (s - s2)) * (s - s2)) * p1) = ((1 / (s - s2)) * (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)) by A17, RLVECT_1:def 7;

                  then (1 * p1) = ((1 / (s - s2)) * (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)) by A13, XCMPLX_1: 106;

                  then p1 = ((1 / (s - s2)) * (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)) by RLVECT_1:def 8;

                  then

                   A18: p1 = (((1 / (s - s2)) * ((r / 2) / |.p1.|)) * |[( - (p1 `2 )), (p1 `1 )]|) by RLVECT_1:def 7;

                  (p1 `1 ) = ( |[( - (p1 `2 )), (p1 `1 )]| `2 ) & ( - (p1 `2 )) = ( |[( - (p1 `2 )), (p1 `1 )]| `1 ) by EUCLID: 52;

                  then p1 = ( 0. ( TOP-REAL 2)) by A18, Th84;

                  hence contradiction by A1, RLVECT_1: 21;

                end;

                then

                 A19: p2 in (the carrier of ( TOP-REAL 2) \ P) by XBOOLE_0:def 5;

                reconsider ep2 = p2 as Point of ( Euclid 2) by TOPREAL3: 8;

                

                 A20: ((p + ( - (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|))) - p) = ( - (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)) by RLVECT_4: 1;

                

                 A21: ( |[( - (p1 `2 )), (p1 `1 )]| `1 ) = ( - (p1 `2 )) & ( |[( - (p1 `2 )), (p1 `1 )]| `2 ) = (p1 `1 ) by EUCLID: 52;

                 |.(p - p2).| = |.((p - (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)) - p).| by RLVECT_1: 27

                .= |.( - (((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|)).| by A20

                .= |.(((r / 2) / |.p1.|) * |[( - (p1 `2 )), (p1 `1 )]|).| by TOPRNS_1: 26

                .= ( |.((r / 2) / |.p1.|).| * |. |[( - (p1 `2 )), (p1 `1 )]|.|) by TOPRNS_1: 7

                .= ( |.((r / 2) / |.p1.|).| * ( sqrt ((( - (p1 `2 )) ^2 ) + ((p1 `1 ) ^2 )))) by A21, JGRAPH_1: 30

                .= ( |.((r / 2) / |.p1.|).| * ( sqrt (((p1 `1 ) ^2 ) + ((p1 `2 ) ^2 ))))

                .= ( |.((r / 2) / |.p1.|).| * |.p1.|) by JGRAPH_1: 30

                .= (( |.(r / 2).| / |. |.p1.|.|) * |.p1.|) by COMPLEX1: 67

                .= (( |.(r / 2).| / |.p1.|) * |.p1.|) by ABSVALUE:def 1

                .= |.(r / 2).| by A5, XCMPLX_1: 87

                .= (r / 2) by A9, ABSVALUE:def 1;

                then ( dist (ez,ep2)) < r by A4, A11, JGRAPH_1: 28;

                then p2 in ( Ball (ez,r)) by METRIC_1: 11;

                hence thesis by A10, A19, XBOOLE_0: 3;

              end;

            end;

            hence thesis by A2, PRE_TOPC:def 7;

          end;

            suppose

             A22: not z in P;

            

             A23: (P ` ) c= ( Cl (P ` )) by PRE_TOPC: 18;

            z in (the carrier of ( TOP-REAL 2) \ P) by A2, A22, XBOOLE_0:def 5;

            hence thesis by A23;

          end;

        end;

        then ( Cl (P ` )) = ( [#] ( TOP-REAL 2));

        then (P ` ) is dense by TOPS_1:def 3;

        hence thesis by TOPS_1:def 4;

      end;

    end;

    registration

      let q1,q2 be Point of ( TOP-REAL 2);

      cluster ( LSeg (q1,q2)) -> boundary;

      coherence by Th85;

    end

    theorem :: JORDAN2C:102

    

     Th86: for f be FinSequence of ( TOP-REAL 2) holds ( L~ f) is boundary

    proof

      let f be FinSequence of ( TOP-REAL 2);

      

       A1: ( L~ f) = ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= ( len f) }) by TOPREAL1:def 4;

      defpred P[ Nat] means for R1 be Subset of ( TOP-REAL 2) st R1 = ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= $1 }) holds R1 is boundary;

      

       A2: for k be Nat st P[k] holds P[(k + 1)]

      proof

        let k be Nat;

         A3:

        now

          per cases ;

            case 1 <= k & (k + 1) <= ( len f);

            then ( LSeg (f,k)) = ( LSeg ((f /. k),(f /. (k + 1)))) by TOPREAL1:def 3;

            hence ( LSeg (f,k)) is boundary;

          end;

            case not (1 <= k & (k + 1) <= ( len f));

            then ( LSeg (f,k)) = {} by TOPREAL1:def 3;

            hence ( LSeg (f,k)) is boundary;

          end;

        end;

        ( union { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= k }) c= the carrier of ( TOP-REAL 2)

        proof

          let z be object;

          assume z in ( union { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= k });

          then

          consider x be set such that

           A4: z in x & x in { ( LSeg (f,i)) : 1 <= i & (i + 1) <= k } by TARSKI:def 4;

          ex i st x = ( LSeg (f,i)) & 1 <= i & (i + 1) <= k by A4;

          hence thesis by A4;

        end;

        then

        reconsider R3 = ( union { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= k }) as Subset of ( TOP-REAL 2);

        assume for R1 be Subset of ( TOP-REAL 2) st R1 = ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= k }) holds R1 is boundary;

        then

         A5: R3 is boundary;

        thus for R2 be Subset of ( TOP-REAL 2) st R2 = ( union { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= (k + 1) }) holds R2 is boundary

        proof

          let R2 be Subset of ( TOP-REAL 2);

          assume

           A6: R2 = ( union { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= (k + 1) });

          

           A7: (R3 \/ ( LSeg (f,k))) c= R2

          proof

            let z be object;

            assume

             A8: z in (R3 \/ ( LSeg (f,k)));

            per cases by A8, XBOOLE_0:def 3;

              suppose z in R3;

              then

              consider x be set such that

               A9: z in x & x in { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= k } by TARSKI:def 4;

              consider i2 such that

               A10: x = ( LSeg (f,i2)) & 1 <= i2 and

               A11: (i2 + 1) <= k by A9;

              (i2 + 1) < (k + 1) by A11, NAT_1: 13;

              then x in { ( LSeg (f,j)) : 1 <= j & (j + 1) <= (k + 1) } by A10;

              hence thesis by A6, A9, TARSKI:def 4;

            end;

              suppose

               A12: z in ( LSeg (f,k));

              now

                per cases ;

                  suppose 1 <= k;

                  then ( LSeg (f,k)) in { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= (k + 1) };

                  hence thesis by A6, A12, TARSKI:def 4;

                end;

                  suppose k < 1;

                  hence thesis by A12, TOPREAL1:def 3;

                end;

              end;

              hence thesis;

            end;

          end;

          R2 c= (R3 \/ ( LSeg (f,k)))

          proof

            let z be object;

            assume z in R2;

            then

            consider x be set such that

             A13: z in x & x in { ( LSeg (f,i2)) : 1 <= i2 & (i2 + 1) <= (k + 1) } by A6, TARSKI:def 4;

            consider i2 such that

             A14: x = ( LSeg (f,i2)) and

             A15: 1 <= i2 and

             A16: (i2 + 1) <= (k + 1) by A13;

            now

              per cases ;

                case (i2 + 1) <= k;

                then x in { ( LSeg (f,j)) : 1 <= j & (j + 1) <= k } by A14, A15;

                hence z in R3 or z in ( LSeg (f,k)) by A13, TARSKI:def 4;

              end;

                case (i2 + 1) > k;

                then (k + 1) <= (i2 + 1) by NAT_1: 13;

                then (i2 + 1) = (k + 1) by A16, XXREAL_0: 1;

                hence z in R3 or z in ( LSeg (f,k)) by A13, A14;

              end;

            end;

            hence thesis by XBOOLE_0:def 3;

          end;

          then R2 = (R3 \/ ( LSeg (f,k))) by A7;

          hence thesis by A5, A3, TOPS_1: 49;

        end;

      end;

      ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= 0 }) c= {}

      proof

        let z be object;

        assume z in ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= 0 });

        then

        consider x be set such that

         A17: z in x & x in { ( LSeg (f,i)) : 1 <= i & (i + 1) <= 0 } by TARSKI:def 4;

        ex i st x = ( LSeg (f,i)) & 1 <= i & (i + 1) <= 0 by A17;

        hence thesis;

      end;

      then

       A18: P[ 0 ];

      for j holds P[j] from NAT_1:sch 2( A18, A2);

      hence thesis by A1;

    end;

    registration

      let f be FinSequence of ( TOP-REAL 2);

      cluster ( L~ f) -> boundary;

      coherence by Th86;

    end

    theorem :: JORDAN2C:103

    

     Th87: for ep be Point of ( Euclid n), p,q be Point of ( TOP-REAL n) st p = ep & q in ( Ball (ep,r)) holds |.(p - q).| < r & |.(q - p).| < r

    proof

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

      assume that

       A1: p = ep and

       A2: q in ( Ball (ep,r));

      reconsider eq = q as Point of ( Euclid n) by TOPREAL3: 8;

      ( dist (ep,eq)) < r by A2, METRIC_1: 11;

      hence thesis by A1, JGRAPH_1: 28;

    end;

    theorem :: JORDAN2C:104

    for a be Real, p be Point of ( TOP-REAL 2) st a > 0 & p in ( L~ ( SpStSeq D)) holds ex q be Point of ( TOP-REAL 2) st q in ( UBD ( L~ ( SpStSeq D))) & |.(p - q).| < a

    proof

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

      assume that

       A1: a > 0 and

       A2: p in ( L~ ( SpStSeq D));

      set q1 = the Element of ( UBD ( L~ ( SpStSeq D)));

      set A = ( L~ ( SpStSeq D));

      (A ` ) <> {} by SPRECT_1:def 3;

      then

      consider A1,A2 be Subset of ( TOP-REAL 2) such that

       A3: (A ` ) = (A1 \/ A2) and A1 misses A2 and

       A4: (( Cl A1) \ A1) = (( Cl A2) \ A2) and

       A5: A = (( Cl A1) \ A1) and

       A6: for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component by Th82;

      

       A7: ( Down (A2,(A ` ))) = A2 by A3, XBOOLE_1: 21;

      ( UBD A) is_outside_component_of A by Th53;

      then ( UBD ( L~ ( SpStSeq D))) is_a_component_of (A ` );

      then

      consider B1 be Subset of (( TOP-REAL 2) | (A ` )) such that

       A8: B1 = ( UBD ( L~ ( SpStSeq D))) and

       A9: B1 is a_component by CONNSP_1:def 6;

      B1 c= ( [#] (( TOP-REAL 2) | (A ` )));

      then

       A10: ( UBD ( L~ ( SpStSeq D))) c= (A1 \/ A2) by A3, A8, PRE_TOPC:def 5;

      

       A11: ( Down (A1,(A ` ))) = A1 by A3, XBOOLE_1: 21;

      then

       A12: ( Down (A1,(A ` ))) is a_component by A6, A7;

      

       A13: ( Down (A2,(A ` ))) is a_component by A6, A11, A7;

      

       A14: ( UBD ( L~ ( SpStSeq D))) <> {} by Th80;

      then

       A15: q1 in ( UBD ( L~ ( SpStSeq D)));

      per cases by A10, A15, XBOOLE_0:def 3;

        suppose q1 in A1;

        then (B1 /\ ( Down (A1,(A ` )))) <> ( {} (( TOP-REAL 2) | (A ` ))) by A11, A8, A14, XBOOLE_0:def 4;

        then B1 meets ( Down (A1,(A ` )));

        then B1 = ( Down (A1,(A ` ))) by A12, A9, CONNSP_1: 35;

        then

         A16: p in ( Cl ( UBD ( L~ ( SpStSeq D)))) by A2, A5, A11, A8, XBOOLE_0:def 5;

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

        reconsider G2 = ( Ball (ep,a)) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

        the distance of ( Euclid 2) is Reflexive by METRIC_1:def 6;

        then ( dist (ep,ep)) = 0 ;

        then

         A17: p in ( Ball (ep,a)) by A1, METRIC_1: 11;

        G2 is open by GOBOARD6: 3;

        then ( UBD ( L~ ( SpStSeq D))) meets G2 by A16, A17, PRE_TOPC:def 7;

        then

        consider t2 be object such that

         A18: t2 in ( UBD ( L~ ( SpStSeq D))) and

         A19: t2 in G2 by XBOOLE_0: 3;

        reconsider qt2 = t2 as Point of ( TOP-REAL 2) by A18;

         |.(p - qt2).| < a by A19, Th87;

        hence thesis by A18;

      end;

        suppose q1 in A2;

        then (B1 /\ ( Down (A2,(A ` )))) <> ( {} (( TOP-REAL 2) | (A ` ))) by A7, A8, A14, XBOOLE_0:def 4;

        then B1 meets ( Down (A2,(A ` )));

        then B1 = ( Down (A2,(A ` ))) by A13, A9, CONNSP_1: 35;

        then

         A20: p in ( Cl ( UBD ( L~ ( SpStSeq D)))) by A2, A4, A5, A7, A8, XBOOLE_0:def 5;

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

        reconsider G2 = ( Ball (ep,a)) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

        the distance of ( Euclid 2) is Reflexive by METRIC_1:def 6;

        then ( dist (ep,ep)) = 0 ;

        then

         A21: p in ( Ball (ep,a)) by A1, METRIC_1: 11;

        G2 is open by GOBOARD6: 3;

        then ( UBD ( L~ ( SpStSeq D))) meets G2 by A20, A21, PRE_TOPC:def 7;

        then

        consider t2 be object such that

         A22: t2 in ( UBD ( L~ ( SpStSeq D))) and

         A23: t2 in G2 by XBOOLE_0: 3;

        reconsider qt2 = t2 as Point of ( TOP-REAL 2) by A22;

         |.(p - qt2).| < a by A23, Th87;

        hence thesis by A22;

      end;

    end;

    theorem :: JORDAN2C:105

    ( REAL 0 ) = {( 0. ( TOP-REAL 0 ))} by EUCLID: 77;

    theorem :: JORDAN2C:106

    

     Th90: for A be Subset of ( TOP-REAL n) st A is bounded holds ( BDD A) is bounded

    proof

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

      assume A is bounded;

      then

      consider r be Real such that

       A1: for q be Point of ( TOP-REAL n) st q in A holds |.q.| < r by Th21;

      per cases ;

        suppose

         A2: n >= 1;

        set a = r;

        reconsider P = (( REAL n) \ { q : |.q.| < a }) as Subset of ( TOP-REAL n) by EUCLID: 22;

        

         A3: P c= (A ` )

        proof

          let z be object;

          assume

           A4: z in P;

          then

          reconsider q0 = z as Point of ( TOP-REAL n);

           not z in { q : |.q.| < a } by A4, XBOOLE_0:def 5;

          then |.q0.| >= a;

          then not q0 in A by A1;

          hence thesis by XBOOLE_0:def 5;

        end;

        then

         A5: ( Down (P,(A ` ))) = P by XBOOLE_1: 28;

        now

          per cases ;

            suppose n >= 2;

            then

             A6: P is connected by Th40;

            now

              assume not ( BDD A) is bounded;

              then

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

               A7: q in ( BDD A) and

               A8: not |.q.| < r by Th21;

              consider y be set such that

               A9: q in y and

               A10: y in { B3 where B3 be Subset of ( TOP-REAL n) : B3 is_inside_component_of A } by A7, TARSKI:def 4;

              consider B3 be Subset of ( TOP-REAL n) such that

               A11: y = B3 and

               A12: B3 is_inside_component_of A by A10;

              q in the carrier of ( TOP-REAL n);

              then

               A13: q in ( REAL n) by EUCLID: 22;

              B3 is_a_component_of (A ` ) by A12;

              then

              consider B4 be Subset of (( TOP-REAL n) | (A ` )) such that

               A14: B4 = B3 and

               A15: B4 is a_component by CONNSP_1:def 6;

              for q2 be Point of ( TOP-REAL n) st q2 = q holds |.q2.| >= a by A8;

              then not q in { q2 where q2 be Point of ( TOP-REAL n) : |.q2.| < a };

              then q in P by A13, XBOOLE_0:def 5;

              then (P /\ B4) <> ( {} (( TOP-REAL n) | (A ` ))) by A9, A11, A14, XBOOLE_0:def 4;

              then P meets B4;

              then

               A16: P c= B4 by A5, A6, A15, CONNSP_1: 36, CONNSP_1: 46;

              B3 is bounded by A12;

              hence contradiction by A2, A14, A16, Th41, RLTOPSP1: 42;

            end;

            hence thesis;

          end;

            suppose

             A17: n < 2;

            { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 >= a } c= the carrier of ( TOP-REAL n)

            proof

              let z be object;

              assume z in { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 >= a };

              then ex q be Point of ( TOP-REAL n) st q = z & for r2 be Real st q = |[r2]| holds r2 >= a;

              hence thesis;

            end;

            then

            reconsider P2 = { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 >= a } as Subset of ( TOP-REAL n);

            { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 <= ( - a) } c= the carrier of ( TOP-REAL n)

            proof

              let z be object;

              assume z in { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 <= ( - a) };

              then ex q be Point of ( TOP-REAL n) st q = z & for r2 be Real st q = |[r2]| holds r2 <= ( - a);

              hence thesis;

            end;

            then

            reconsider P1 = { q where q be Point of ( TOP-REAL n) : for r2 be Real st q = |[r2]| holds r2 <= ( - a) } as Subset of ( TOP-REAL n);

            n < (1 + 1) by A17;

            then n <= 1 by NAT_1: 13;

            then

             A18: n = 1 by A2, XXREAL_0: 1;

            

             A19: P c= (P1 \/ P2)

            proof

              let z be object;

              assume

               A20: z in P;

              then

              reconsider q0 = z as Point of ( TOP-REAL n);

              consider r3 be Real such that

               A21: q0 = <*r3*> by A18, JORDAN2B: 20;

               not z in { q : |.q.| < a } by A20, XBOOLE_0:def 5;

              then |.q0.| >= a;

              then

               A22: |.r3.| >= a by A21, Th4;

              per cases by A22, SEQ_2: 1;

                suppose ( - a) >= r3;

                then for r2 be Real st q0 = |[r2]| holds r2 <= ( - a) by A21, JORDAN2B: 23;

                then q0 in P1;

                hence thesis by XBOOLE_0:def 3;

              end;

                suppose r3 >= a;

                then for r2 be Real st q0 = |[r2]| holds r2 >= a by A21, JORDAN2B: 23;

                then q0 in P2;

                hence thesis by XBOOLE_0:def 3;

              end;

            end;

            (P1 \/ P2) c= P

            proof

              let z be object;

              assume

               A23: z in (P1 \/ P2);

              per cases by A23, XBOOLE_0:def 3;

                suppose

                 A24: z in P1;

                then

                 A25: ex q be Point of ( TOP-REAL n) st q = z & for r2 be Real st q = |[r2]| holds r2 <= ( - a);

                for q2 be Point of ( TOP-REAL n) st q2 = z holds |.q2.| >= a

                proof

                  let q2 be Point of ( TOP-REAL n);

                  consider r3 be Real such that

                   A26: q2 = <*r3*> by A18, JORDAN2B: 20;

                  assume

                   A27: q2 = z;

                  then

                   A28: r3 <= ( - a) by A25, A26;

                  now

                    per cases ;

                      case a < 0 ;

                      hence |.r3.| >= a by COMPLEX1: 46;

                    end;

                      case a >= 0 ;

                      then ( - a) <= ( - 0 );

                      then |.r3.| = ( - r3) by A28, ABSVALUE: 30;

                      hence |.r3.| >= a by A25, A27, A26, XREAL_1: 25;

                    end;

                  end;

                  hence thesis by A26, Th4;

                end;

                then

                 A29: not z in { q2 where q2 be Point of ( TOP-REAL n) : |.q2.| < a };

                z in the carrier of ( TOP-REAL n) by A24;

                then z in ( REAL n) by EUCLID: 22;

                hence thesis by A29, XBOOLE_0:def 5;

              end;

                suppose

                 A30: z in P2;

                then

                 A31: ex q be Point of ( TOP-REAL n) st q = z & for r2 be Real st q = |[r2]| holds r2 >= a;

                for q2 be Point of ( TOP-REAL n) st q2 = z holds |.q2.| >= a

                proof

                  let q2 be Point of ( TOP-REAL n);

                  consider r3 be Real such that

                   A32: q2 = <*r3*> by A18, JORDAN2B: 20;

                  assume q2 = z;

                  then

                   A33: r3 >= a by A31, A32;

                  now

                    per cases ;

                      suppose a < 0 ;

                      hence |.r3.| >= a by COMPLEX1: 46;

                    end;

                      suppose a >= 0 ;

                      hence |.r3.| >= a by A33, ABSVALUE:def 1;

                    end;

                  end;

                  hence thesis by A32, Th4;

                end;

                then

                 A34: not z in { q2 where q2 be Point of ( TOP-REAL n) : |.q2.| < a };

                z in the carrier of ( TOP-REAL n) by A30;

                then z in ( REAL n) by EUCLID: 22;

                hence thesis by A34, XBOOLE_0:def 5;

              end;

            end;

            then

             A35: P = (P1 \/ P2) by A19;

            then P2 c= P by XBOOLE_1: 7;

            then

             A36: ( Down (P2,(A ` ))) = P2 by A3, XBOOLE_1: 1, XBOOLE_1: 28;

            for w1,w2 be Point of ( TOP-REAL n) st w1 in P2 & w2 in P2 holds ( LSeg (w1,w2)) c= P2

            proof

              let w1,w2 be Point of ( TOP-REAL n);

              assume that

               A37: w1 in P2 and

               A38: w2 in P2;

              

               A39: ex q2 be Point of ( TOP-REAL n) st q2 = w2 & for r2 be Real st q2 = |[r2]| holds r2 >= a by A38;

              consider r3 be Real such that

               A40: w1 = <*r3*> by A18, JORDAN2B: 20;

              consider r4 be Real such that

               A41: w2 = <*r4*> by A18, JORDAN2B: 20;

              

               A42: ex q1 be Point of ( TOP-REAL n) st q1 = w1 & for r2 be Real st q1 = |[r2]| holds r2 >= a by A37;

              thus ( LSeg (w1,w2)) c= P2

              proof

                let z be object;

                assume z in ( LSeg (w1,w2));

                then

                consider r2 such that

                 A43: z = (((1 - r2) * w1) + (r2 * w2)) and

                 A44: 0 <= r2 and

                 A45: r2 <= 1;

                reconsider q4 = z as Point of ( TOP-REAL n) by A43;

                ((1 - r2) * w1) = |[((1 - r2) * r3)]| & (r2 * w2) = |[(r2 * r4)]| by A18, A40, A41, JORDAN2B: 21;

                then

                 A46: z = |[(((1 - r2) * r3) + (r2 * r4))]| by A18, A43, JORDAN2B: 22;

                for r5 be Real st q4 = |[r5]| holds r5 >= a

                proof

                  let r5 be Real;

                  assume q4 = |[r5]|;

                  then

                   A47: r5 = (((1 - r2) * r3) + (r2 * r4)) by A46, JORDAN2B: 23;

                  (1 - r2) >= 0 by A45, XREAL_1: 48;

                  then

                   A48: ((1 - r2) * r3) >= ((1 - r2) * a) by A42, A40, XREAL_1: 64;

                  (r2 * r4) >= (r2 * a) & (((1 - r2) * a) + (r2 * a)) = a by A39, A41, A44, XREAL_1: 64;

                  hence thesis by A47, A48, XREAL_1: 7;

                end;

                hence thesis;

              end;

            end;

            then P2 is convex by JORDAN1:def 1;

            then

             A49: ( Down (P2,(A ` ))) is connected by A36, CONNSP_1: 46;

            P1 c= P by A35, XBOOLE_1: 7;

            then

             A50: ( Down (P1,(A ` ))) = P1 by A3, XBOOLE_1: 1, XBOOLE_1: 28;

             A51:

            now

              assume P2 is bounded;

              then

              consider r be Real such that

               A52: for q be Point of ( TOP-REAL n) st q in P2 holds |.q.| < r by Th21;

               0 <= |.r.| & 0 <= |.a.| by COMPLEX1: 46;

              then

               A53: |.( |.r.| + |.a.|).| = ( |.r.| + |.a.|) by ABSVALUE:def 1;

              set p3 = |[( |.r.| + |.a.|)]|;

              

               A54: |.r.| <= ( |.r.| + |.a.|) by COMPLEX1: 46, XREAL_1: 31;

              for r5 be Real st p3 = |[r5]| holds r5 >= a

              proof

                let r5 be Real;

                assume p3 = |[r5]|;

                then

                 A55: r5 = ( |.r.| + |.a.|) by JORDAN2B: 23;

                a <= |.a.| & |.a.| <= ( |.a.| + |.r.|) by ABSVALUE: 4, COMPLEX1: 46, XREAL_1: 31;

                hence thesis by A55, XXREAL_0: 2;

              end;

              then

               A56: p3 in P2 by A18;

               |.p3.| = |.( |.r.| + |.a.|).| & r <= |.r.| by Th4, ABSVALUE: 4;

              hence contradiction by A52, A56, A53, A54, XXREAL_0: 2;

            end;

             A57:

            now

              assume P1 is bounded;

              then

              consider r be Real such that

               A58: for q be Point of ( TOP-REAL n) st q in P1 holds |.q.| < r by Th21;

               0 <= |.r.| & 0 <= |.a.| by COMPLEX1: 46;

              then

               A59: |.( |.r.| + |.a.|).| = ( |.r.| + |.a.|) by ABSVALUE:def 1;

              set p3 = |[( - ( |.r.| + |.a.|))]|;

              

               A60: r <= |.r.| & |.r.| <= ( |.r.| + |.a.|) by ABSVALUE: 4, COMPLEX1: 46, XREAL_1: 31;

              for r5 be Real st p3 = |[r5]| holds r5 <= ( - a)

              proof

                let r5 be Real;

                a <= |.a.| by ABSVALUE: 4;

                then

                 A61: ( - |.a.|) <= ( - a) by XREAL_1: 24;

                 |.a.| <= ( |.a.| + |.r.|) by COMPLEX1: 46, XREAL_1: 31;

                then

                 A62: ( - |.a.|) >= ( - ( |.a.| + |.r.|)) by XREAL_1: 24;

                assume p3 = |[r5]|;

                then r5 = ( - ( |.r.| + |.a.|)) by JORDAN2B: 23;

                hence thesis by A61, A62, XXREAL_0: 2;

              end;

              then

               A63: p3 in P1 by A18;

               |.p3.| = |.( - ( |.r.| + |.a.|)).| by Th4

              .= |.( |.r.| + |.a.|).| by COMPLEX1: 52;

              hence contradiction by A58, A63, A59, A60, XXREAL_0: 2;

            end;

            for w1,w2 be Point of ( TOP-REAL n) st w1 in P1 & w2 in P1 holds ( LSeg (w1,w2)) c= P1

            proof

              let w1,w2 be Point of ( TOP-REAL n);

              assume that

               A64: w1 in P1 and

               A65: w2 in P1;

              consider r4 be Real such that

               A66: w2 = <*r4*> by A18, JORDAN2B: 20;

              ex q2 be Point of ( TOP-REAL n) st q2 = w2 & for r2 be Real st q2 = |[r2]| holds r2 <= ( - a) by A65;

              then

               A67: r4 <= ( - a) by A66;

              consider r3 be Real such that

               A68: w1 = <*r3*> by A18, JORDAN2B: 20;

              ex q1 be Point of ( TOP-REAL n) st q1 = w1 & for r2 be Real st q1 = |[r2]| holds r2 <= ( - a) by A64;

              then

               A69: r3 <= ( - a) by A68;

              thus ( LSeg (w1,w2)) c= P1

              proof

                let z be object;

                assume z in ( LSeg (w1,w2));

                then

                consider r2 such that

                 A70: z = (((1 - r2) * w1) + (r2 * w2)) and

                 A71: 0 <= r2 and

                 A72: r2 <= 1;

                reconsider q4 = z as Point of ( TOP-REAL n) by A70;

                

                 A73: (r2 * w2) = |[(r2 * r4)]| by A18, A66, JORDAN2B: 21;

                ((1 - r2) * w1) = ((1 - r2) * |[r3]|) by A68

                .= |[((1 - r2) * r3)]| by JORDAN2B: 21;

                then

                 A74: z = |[(((1 - r2) * r3) + (r2 * r4))]| by A18, A70, A73, JORDAN2B: 22;

                for r5 be Real st q4 = |[r5]| holds r5 <= ( - a)

                proof

                  let r5 be Real;

                  assume q4 = |[r5]|;

                  then

                   A75: r5 = (((1 - r2) * r3) + (r2 * r4)) by A74, JORDAN2B: 23;

                  (1 - r2) >= 0 by A72, XREAL_1: 48;

                  then

                   A76: ((1 - r2) * r3) <= ((1 - r2) * ( - a)) by A69, XREAL_1: 64;

                  (r2 * r4) <= (r2 * ( - a)) & (((1 - r2) * ( - a)) + (r2 * ( - a))) = ( - a) by A67, A71, XREAL_1: 64;

                  hence thesis by A75, A76, XREAL_1: 7;

                end;

                hence thesis;

              end;

            end;

            then P1 is convex by JORDAN1:def 1;

            then

             A77: ( Down (P1,(A ` ))) is connected by A50, CONNSP_1: 46;

            now

              assume not ( BDD A) is bounded;

              then

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

               A78: q in ( BDD A) and

               A79: not |.q.| < r by Th21;

              consider y be set such that

               A80: q in y and

               A81: y in { B3 where B3 be Subset of ( TOP-REAL n) : B3 is_inside_component_of A } by A78, TARSKI:def 4;

              consider B3 be Subset of ( TOP-REAL n) such that

               A82: y = B3 and

               A83: B3 is_inside_component_of A by A81;

              q in the carrier of ( TOP-REAL n);

              then

               A84: q in ( REAL n) by EUCLID: 22;

              for q2 be Point of ( TOP-REAL n) st q2 = q holds |.q2.| >= a by A79;

              then not q in { q2 where q2 be Point of ( TOP-REAL n) : |.q2.| < a };

              then

               A85: q in P by A84, XBOOLE_0:def 5;

              B3 is_a_component_of (A ` ) by A83;

              then

              consider B4 be Subset of (( TOP-REAL n) | (A ` )) such that

               A86: B4 = B3 and

               A87: B4 is a_component by CONNSP_1:def 6;

              per cases by A19, A85, XBOOLE_0:def 3;

                suppose q in P1;

                then (P1 /\ B4) <> ( {} (( TOP-REAL n) | (A ` ))) by A80, A82, A86, XBOOLE_0:def 4;

                then

                 A88: P1 meets B4;

                B3 is bounded by A83;

                hence contradiction by A50, A57, A77, A86, A87, A88, CONNSP_1: 36, RLTOPSP1: 42;

              end;

                suppose q in P2;

                then (P2 /\ B4) <> ( {} (( TOP-REAL n) | (A ` ))) by A80, A82, A86, XBOOLE_0:def 4;

                then

                 A89: P2 meets B4;

                B3 is bounded by A83;

                hence contradiction by A36, A51, A49, A86, A87, A89, CONNSP_1: 36, RLTOPSP1: 42;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

        suppose n < 1;

        then n < ( 0 + 1);

        then

         A90: n = 0 by NAT_1: 13;

        for q2 be Point of ( TOP-REAL n) holds |.q2.| < 1

        proof

          let q2 be Point of ( TOP-REAL n);

          q2 = ( 0. ( TOP-REAL n)) by A90, EUCLID: 77;

          hence thesis by TOPRNS_1: 23;

        end;

        then for q2 be Point of ( TOP-REAL n) st q2 in ( [#] ( TOP-REAL n)) holds |.q2.| < 1;

        then ( [#] ( TOP-REAL n)) is bounded by Th21;

        hence thesis by RLTOPSP1: 42;

      end;

    end;

    theorem :: JORDAN2C:107

    

     Th91: for G be non empty TopSpace, A,B,C,D be Subset of G st B is a_component & C is a_component & (A \/ B) = the carrier of G & C misses A holds C = B

    proof

      let G be non empty TopSpace, A,B,C,D be Subset of G;

      assume that

       A1: B is a_component and

       A2: C is a_component and

       A3: (A \/ B) = the carrier of G and

       A4: C misses A;

      now

        (C /\ the carrier of G) = C by XBOOLE_1: 28;

        then

         A5: ((C /\ A) \/ (C /\ B)) = C by A3, XBOOLE_1: 23;

        assume C misses B;

        then

         A6: (C /\ B) = {} ;

        C <> ( {} G) by A2, CONNSP_1: 32;

        hence contradiction by A4, A6, A5;

      end;

      hence thesis by A1, A2, CONNSP_1: 35;

    end;

    theorem :: JORDAN2C:108

    

     Th92: for A be Subset of ( TOP-REAL 2) st A is bounded & A is Jordan holds ( BDD A) is_inside_component_of A

    proof

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

      assume that

       A1: A is bounded and

       A2: A is Jordan;

      reconsider D = (A ` ) as non empty Subset of ( TOP-REAL 2) by A2, JORDAN1:def 2;

      consider A1,A2 be Subset of ( TOP-REAL 2) such that

       A3: (A ` ) = (A1 \/ A2) and

       A4: A1 misses A2 and (( Cl A1) \ A1) = (( Cl A2) \ A2) and

       A5: for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component by A2, JORDAN1:def 2;

      

       A6: ( UBD A) is_outside_component_of A by A1, Th53;

      then ( UBD A) is_a_component_of (A ` );

      then

      consider B1 be Subset of (( TOP-REAL 2) | (A ` )) such that

       A7: B1 = ( UBD A) and

       A8: B1 is a_component by CONNSP_1:def 6;

      

       A9: ( Down (A1,(A ` ))) = A1 by A3, XBOOLE_1: 21;

      

       A10: ( Down (A2,(A ` ))) = A2 by A3, XBOOLE_1: 21;

      then

       A11: ( Down (A2,(A ` ))) is a_component by A5, A9;

      then

       A12: A2 is_a_component_of (A ` ) by A10, CONNSP_1:def 6;

      

       A13: (( TOP-REAL 2) | D) is non empty;

      

       A14: ( Down (A1,(A ` ))) is a_component by A5, A9, A10;

      then

       A15: A1 is_a_component_of (A ` ) by A9, CONNSP_1:def 6;

      per cases by A9, A14, A8, CONNSP_1: 35;

        suppose

         A16: B1 = A1;

         A17:

        now

          assume not ( BDD A) c= A2;

          then

          consider x be object such that

           A18: x in ( BDD A) and

           A19: not x in A2;

          consider y be set such that

           A20: x in y and

           A21: y in { B3 where B3 be Subset of ( TOP-REAL 2) : B3 is_inside_component_of A } by A18, TARSKI:def 4;

          consider B3 be Subset of ( TOP-REAL 2) such that

           A22: y = B3 and

           A23: B3 is_inside_component_of A by A21;

          

           A24: B3 is_a_component_of (A ` ) by A23;

          then

          consider B4 be Subset of (( TOP-REAL 2) | (A ` )) such that

           A25: B4 = B3 and

           A26: B4 is a_component by CONNSP_1:def 6;

          

           A27: B3 <> ( {} (( TOP-REAL 2) | (A ` ))) by A13, A25, A26, CONNSP_1: 32;

          B4 <> ( Down (A1,(A ` ))) by A9, A7, A16, A23, A25, A6;

          then

           A28: B3 misses A1 by A9, A14, A25, A26, CONNSP_1: 35;

          B4 = ( Down (A2,(A ` ))) or B4 misses ( Down (A2,(A ` ))) by A11, A26, CONNSP_1: 35;

          then

           A29: B4 = ( Down (A2,(A ` ))) or (B4 /\ ( Down (A2,(A ` )))) = ( {} (( TOP-REAL 2) | (A ` )));

          B3 = (B3 /\ (A1 \/ A2)) by A3, A24, SPRECT_1: 5, XBOOLE_1: 28

          .= ((B3 /\ A1) \/ (B3 /\ A2)) by XBOOLE_1: 23

          .= {} by A10, A19, A20, A22, A25, A29, A28;

          hence contradiction by A27;

        end;

        now

          assume not A2 is bounded;

          then A2 is_outside_component_of A by A12;

          then (A2 /\ ( UBD A)) <> {} by Th14, XBOOLE_1: 28;

          hence contradiction by A4, A7, A16;

        end;

        then

         A30: A2 is_inside_component_of A by A12;

        then A2 c= ( BDD A) by Th13;

        hence thesis by A30, A17, XBOOLE_0:def 10;

      end;

        suppose

         A31: B1 misses ( Down (A1,(A ` )));

        set E1 = ( Down (A1,(A ` ))), E2 = ( Down (A2,(A ` )));

        (E1 \/ E2) = ( [#] (( TOP-REAL 2) | (A ` ))) by A3, A9, A10, PRE_TOPC:def 5;

        then

         A32: ( UBD A) = A2 by A10, A11, A13, A7, A8, A31, Th91;

        

         A33: (( BDD A) \/ ( UBD A)) = (A ` ) by Th18;

        

         A34: ( BDD A) misses ( UBD A) by Th15;

        

         A35: ( BDD A) c= A1

        proof

          let z be object;

          assume z in ( BDD A);

          then z in (A ` ) & not z in ( UBD A) by A34, A33, XBOOLE_0: 3, XBOOLE_0:def 3;

          hence thesis by A3, A32, XBOOLE_0:def 3;

        end;

        

         A36: ( BDD A) is bounded by A1, Th90;

        A1 c= ( BDD A)

        proof

          let z be object;

          assume z in A1;

          then z in (A ` ) & not z in ( UBD A) by A3, A4, A32, XBOOLE_0: 3, XBOOLE_0:def 3;

          hence thesis by A33, XBOOLE_0:def 3;

        end;

        then ( BDD A) = A1 by A35;

        hence thesis by A15, A36;

      end;

    end;

    theorem :: JORDAN2C:109

    for a be Real, p be Point of ( TOP-REAL 2) st a > 0 & p in ( L~ ( SpStSeq D)) holds ex q be Point of ( TOP-REAL 2) st q in ( BDD ( L~ ( SpStSeq D))) & |.(p - q).| < a

    proof

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

      assume that

       A1: a > 0 and

       A2: p in ( L~ ( SpStSeq D));

      set q1 = the Element of ( BDD ( L~ ( SpStSeq D)));

      set A = ( L~ ( SpStSeq D));

      (A ` ) <> {} by SPRECT_1:def 3;

      then

      consider A1,A2 be Subset of ( TOP-REAL 2) such that

       A3: (A ` ) = (A1 \/ A2) and A1 misses A2 and

       A4: (( Cl A1) \ A1) = (( Cl A2) \ A2) and

       A5: A = (( Cl A1) \ A1) and

       A6: for C1,C2 be Subset of (( TOP-REAL 2) | (A ` )) st C1 = A1 & C2 = A2 holds C1 is a_component & C2 is a_component by Th82;

      

       A7: ( Down (A2,(A ` ))) = A2 by A3, XBOOLE_1: 21;

      ( BDD A) is_inside_component_of A by Th92;

      then ( BDD ( L~ ( SpStSeq D))) is_a_component_of (A ` );

      then

      consider B1 be Subset of (( TOP-REAL 2) | (A ` )) such that

       A8: B1 = ( BDD ( L~ ( SpStSeq D))) and

       A9: B1 is a_component by CONNSP_1:def 6;

      B1 c= the carrier of (( TOP-REAL 2) | (A ` ));

      then

       A10: ( BDD ( L~ ( SpStSeq D))) c= (A1 \/ A2) by A3, A8, PRE_TOPC: 8;

      

       A11: ( Down (A1,(A ` ))) = A1 by A3, XBOOLE_1: 21;

      then

       A12: ( Down (A1,(A ` ))) is a_component by A6, A7;

      

       A13: ( Down (A2,(A ` ))) is a_component by A6, A11, A7;

      

       A14: ( BDD ( L~ ( SpStSeq D))) <> {} by Th80;

      then

       A15: q1 in ( BDD ( L~ ( SpStSeq D)));

      per cases by A10, A15, XBOOLE_0:def 3;

        suppose q1 in A1;

        then (B1 /\ ( Down (A1,(A ` )))) <> ( {} (( TOP-REAL 2) | (A ` ))) by A11, A8, A14, XBOOLE_0:def 4;

        then B1 meets ( Down (A1,(A ` )));

        then B1 = ( Down (A1,(A ` ))) by A12, A9, CONNSP_1: 35;

        then

         A16: p in ( Cl ( BDD ( L~ ( SpStSeq D)))) by A2, A5, A11, A8, XBOOLE_0:def 5;

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

        reconsider G2 = ( Ball (ep,a)) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

        the distance of ( Euclid 2) is Reflexive by METRIC_1:def 6;

        then ( dist (ep,ep)) = 0 ;

        then

         A17: p in ( Ball (ep,a)) by A1, METRIC_1: 11;

        G2 is open by GOBOARD6: 3;

        then ( BDD ( L~ ( SpStSeq D))) meets G2 by A16, A17, PRE_TOPC:def 7;

        then

        consider t2 be object such that

         A18: t2 in ( BDD ( L~ ( SpStSeq D))) and

         A19: t2 in G2 by XBOOLE_0: 3;

        reconsider qt2 = t2 as Point of ( TOP-REAL 2) by A18;

         |.(p - qt2).| < a by A19, Th87;

        hence thesis by A18;

      end;

        suppose q1 in A2;

        then (B1 /\ ( Down (A2,(A ` )))) <> ( {} (( TOP-REAL 2) | (A ` ))) by A7, A8, A14, XBOOLE_0:def 4;

        then B1 meets ( Down (A2,(A ` )));

        then B1 = ( Down (A2,(A ` ))) by A13, A9, CONNSP_1: 35;

        then

         A20: p in ( Cl ( BDD ( L~ ( SpStSeq D)))) by A2, A4, A5, A7, A8, XBOOLE_0:def 5;

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

        reconsider G2 = ( Ball (ep,a)) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

        the distance of ( Euclid 2) is Reflexive by METRIC_1:def 6;

        then ( dist (ep,ep)) = 0 ;

        then

         A21: p in ( Ball (ep,a)) by A1, METRIC_1: 11;

        G2 is open by GOBOARD6: 3;

        then ( BDD ( L~ ( SpStSeq D))) meets G2 by A20, A21, PRE_TOPC:def 7;

        then

        consider t2 be object such that

         A22: t2 in ( BDD ( L~ ( SpStSeq D))) and

         A23: t2 in G2 by XBOOLE_0: 3;

        reconsider qt2 = t2 as Point of ( TOP-REAL 2) by A22;

         |.(p - qt2).| < a by A23, Th87;

        hence thesis by A22;

      end;

    end;

    begin

    reserve f for clockwise_oriented non constant standard special_circular_sequence;

    theorem :: JORDAN2C:110

    for p be Point of ( TOP-REAL 2) st (f /. 1) = ( N-min ( L~ f)) & (p `1 ) < ( W-bound ( L~ f)) holds p in ( LeftComp f)

    proof

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

      assume that

       A1: (f /. 1) = ( N-min ( L~ f)) and

       A2: (p `1 ) < ( W-bound ( L~ f));

      set g = ( SpStSeq ( L~ f));

      

       A3: ( LeftComp g) c= ( LeftComp f) by A1, SPRECT_3: 41;

      ( W-bound ( L~ g)) = ( W-bound ( L~ f)) by SPRECT_1: 58;

      then p in ( LeftComp g) by A2, SPRECT_3: 40;

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:111

    for p be Point of ( TOP-REAL 2) st (f /. 1) = ( N-min ( L~ f)) & (p `1 ) > ( E-bound ( L~ f)) holds p in ( LeftComp f)

    proof

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

      assume that

       A1: (f /. 1) = ( N-min ( L~ f)) and

       A2: (p `1 ) > ( E-bound ( L~ f));

      set g = ( SpStSeq ( L~ f));

      

       A3: ( LeftComp g) c= ( LeftComp f) by A1, SPRECT_3: 41;

      ( E-bound ( L~ g)) = ( E-bound ( L~ f)) by SPRECT_1: 61;

      then p in ( LeftComp g) by A2, SPRECT_3: 40;

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:112

    for p be Point of ( TOP-REAL 2) st (f /. 1) = ( N-min ( L~ f)) & (p `2 ) < ( S-bound ( L~ f)) holds p in ( LeftComp f)

    proof

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

      assume that

       A1: (f /. 1) = ( N-min ( L~ f)) and

       A2: (p `2 ) < ( S-bound ( L~ f));

      set g = ( SpStSeq ( L~ f));

      

       A3: ( LeftComp g) c= ( LeftComp f) by A1, SPRECT_3: 41;

      ( S-bound ( L~ g)) = ( S-bound ( L~ f)) by SPRECT_1: 59;

      then p in ( LeftComp g) by A2, SPRECT_3: 40;

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:113

    for p be Point of ( TOP-REAL 2) st (f /. 1) = ( N-min ( L~ f)) & (p `2 ) > ( N-bound ( L~ f)) holds p in ( LeftComp f)

    proof

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

      assume that

       A1: (f /. 1) = ( N-min ( L~ f)) and

       A2: (p `2 ) > ( N-bound ( L~ f));

      set g = ( SpStSeq ( L~ f));

      

       A3: ( LeftComp g) c= ( LeftComp f) by A1, SPRECT_3: 41;

      ( N-bound ( L~ g)) = ( N-bound ( L~ f)) by SPRECT_1: 60;

      then p in ( LeftComp g) by A2, SPRECT_3: 40;

      hence thesis by A3;

    end;

    theorem :: JORDAN2C:114

    for T be TopSpace, A be Subset of T, B be Subset of T st B is_a_component_of A holds B is connected

    proof

      let T be TopSpace, A be Subset of T, B be Subset of T;

      assume B is_a_component_of A;

      then

      consider C be Subset of (T | A) such that

       A1: C = B and

       A2: C is a_component by CONNSP_1:def 6;

      C is connected by A2, CONNSP_1:def 5;

      hence thesis by A1, CONNSP_1: 23;

    end;

    theorem :: JORDAN2C:115

    for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) st B is_inside_component_of A holds B is connected

    proof

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

      assume B is_inside_component_of A;

      then

      consider C be Subset of (( TOP-REAL n) | (A ` )) such that

       A1: C = B and

       A2: C is a_component and C is bounded Subset of ( Euclid n) by Th7;

      C is connected by A2, CONNSP_1:def 5;

      hence thesis by A1, CONNSP_1: 23;

    end;

    theorem :: JORDAN2C:116

    

     Th100: for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) st B is_outside_component_of A holds B is connected

    proof

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

      assume B is_outside_component_of A;

      then

      consider C be Subset of (( TOP-REAL n) | (A ` )) such that

       A1: C = B and

       A2: C is a_component and not C is bounded Subset of ( Euclid n) by Th8;

      C is connected by A2, CONNSP_1:def 5;

      hence thesis by A1, CONNSP_1: 23;

    end;

    theorem :: JORDAN2C:117

    for A be Subset of ( TOP-REAL n), B be Subset of ( TOP-REAL n) st B is_a_component_of (A ` ) holds A misses B by SPRECT_1: 5, SUBSET_1: 23;

    theorem :: JORDAN2C:118

    P is_outside_component_of Q & R is_inside_component_of Q implies P misses R

    proof

      assume

       A1: P is_outside_component_of Q;

      assume

       A2: R is_inside_component_of Q;

      ( BDD Q) misses ( UBD Q) by Th15;

      then P misses ( BDD Q) by A1, Th14, XBOOLE_1: 63;

      hence thesis by A2, Th13, XBOOLE_1: 63;

    end;

    theorem :: JORDAN2C:119

    2 <= n implies for A,B,P be Subset of ( TOP-REAL n) st P is bounded & A is_outside_component_of P & B is_outside_component_of P holds A = B

    proof

      assume

       A1: 2 <= n;

      let A,B,P be Subset of ( TOP-REAL n) such that

       A2: P is bounded and

       A3: A is_outside_component_of P and

       A4: B is_outside_component_of P;

      

       A5: B is_a_component_of (P ` ) by A4;

      ( UBD P) is_outside_component_of P by A1, A2, Th53;

      then

       A6: ( UBD P) is_a_component_of (P ` );

      

       A7: (P ` ) is non empty by A1, A2, Th51, XXREAL_0: 2;

      

       A8: B <> {} by A4;

      

       A9: B c= ( UBD P) by A4, Th14;

      

       A10: A c= ( UBD P) by A3, Th14;

      

       A11: A is_a_component_of (P ` ) by A3;

      then A <> {} by A7, SPRECT_1: 4;

      then A = ( UBD P) by A11, A6, A10, GOBOARD9: 1, XBOOLE_1: 69;

      hence thesis by A5, A8, A6, A9, GOBOARD9: 1, XBOOLE_1: 69;

    end;

    registration

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

      cluster ( BDD C) -> open;

      coherence

      proof

        set F = { B where B be Subset of ( TOP-REAL 2) : B is_inside_component_of C };

        F c= ( bool the carrier of ( TOP-REAL 2))

        proof

          let f be object;

          assume f in F;

          then ex B be Subset of ( TOP-REAL 2) st f = B & B is_inside_component_of C;

          hence thesis;

        end;

        then

        reconsider F as Subset-Family of ( TOP-REAL 2);

        F is open

        proof

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

          assume P in F;

          then

          consider B be Subset of ( TOP-REAL 2) such that

           A1: P = B and

           A2: B is_inside_component_of C;

          B is_a_component_of (C ` ) by A2;

          hence thesis by A1, SPRECT_3: 8;

        end;

        hence thesis by TOPS_2: 19;

      end;

      cluster ( UBD C) -> open;

      coherence

      proof

        set F = { B where B be Subset of ( TOP-REAL 2) : B is_outside_component_of C };

        F c= ( bool the carrier of ( TOP-REAL 2))

        proof

          let f be object;

          assume f in F;

          then ex B be Subset of ( TOP-REAL 2) st f = B & B is_outside_component_of C;

          hence thesis;

        end;

        then

        reconsider F as Subset-Family of ( TOP-REAL 2);

        F is open

        proof

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

          assume P in F;

          then

          consider B be Subset of ( TOP-REAL 2) such that

           A3: P = B and

           A4: B is_outside_component_of C;

          B is_a_component_of (C ` ) by A4;

          hence thesis by A3, SPRECT_3: 8;

        end;

        hence thesis by TOPS_2: 19;

      end;

    end

    registration

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

      cluster ( UBD C) -> connected;

      coherence by Th53, Th100;

    end

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

    theorem :: JORDAN2C:120

    

     Th104: ( west_halfline p) is non bounded

    proof

      set Wp = ( west_halfline p);

      set p11 = (p `1 ), p12 = (p `2 );

      assume Wp is bounded;

      then

      reconsider C = Wp as bounded Subset of ( Euclid 2) by Th5;

      consider r be Real such that

       A1: 0 < r and

       A2: for x,y be Point of ( Euclid 2) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

      set EX1 = ((p `1 ) - (2 * r));

      reconsider p1 = p, EX = |[((p `1 ) - (2 * r)), (p `2 )]| as Point of ( Euclid 2) by EUCLID: 67;

      ( 0 + (p `1 )) <= ((2 * r) + (p `1 )) by A1, XREAL_1: 6;

      then ((p `1 ) - (2 * r)) <= (p `1 ) by XREAL_1: 20;

      then

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

      then

       A4: p1 in Wp by TOPREAL1:def 13;

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

      then

       A5: EX in Wp by A3, TOPREAL1:def 13;

      p = |[p11, p12]| by EUCLID: 53;

      

      then ( dist (p1,EX)) = ( sqrt (((p11 - EX1) ^2 ) + ((p12 - (p `2 )) ^2 ))) by GOBOARD6: 6

      .= (2 * r) by A1, SQUARE_1: 22;

      hence thesis by A1, A2, A5, A4, XREAL_1: 155;

    end;

    theorem :: JORDAN2C:121

    

     Th105: ( east_halfline p) is non bounded

    proof

      set Wp = ( east_halfline p);

      set p11 = (p `1 ), p12 = (p `2 );

      assume Wp is bounded;

      then

      reconsider C = Wp as bounded Subset of ( Euclid 2) by Th5;

      consider r be Real such that

       A1: 0 < r and

       A2: for x,y be Point of ( Euclid 2) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

      set EX1 = ((p `1 ) + (2 * r)), EX2 = (p `2 );

      reconsider p1 = p, EX = |[((p `1 ) + (2 * r)), (p `2 )]| as Point of ( Euclid 2) by EUCLID: 67;

      ( 0 + (p `1 )) <= ((2 * r) + (p `1 )) by A1, XREAL_1: 6;

      then

       A3: ( |[EX1, (p `2 )]| `1 ) >= (p `1 ) by EUCLID: 52;

      then

       A4: p1 in Wp by TOPREAL1:def 11;

      ( |[EX1, (p `2 )]| `2 ) = (p `2 ) by EUCLID: 52;

      then

       A5: EX in Wp by A3, TOPREAL1:def 11;

      p = |[p11, p12]| by EUCLID: 53;

      

      then ( dist (p1,EX)) = ( sqrt (((p11 - EX1) ^2 ) + ((p12 - EX2) ^2 ))) by GOBOARD6: 6

      .= ( sqrt (((EX1 - p11) ^2 ) + 0 ))

      .= (2 * r) by A1, SQUARE_1: 22;

      hence thesis by A1, A2, A5, A4, XREAL_1: 155;

    end;

    theorem :: JORDAN2C:122

    

     Th106: ( north_halfline p) is non bounded

    proof

      set Wp = ( north_halfline p);

      set p11 = (p `1 ), p12 = (p `2 );

      assume Wp is bounded;

      then

      reconsider C = Wp as bounded Subset of ( Euclid 2) by Th5;

      consider r be Real such that

       A1: 0 < r and

       A2: for x,y be Point of ( Euclid 2) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

      set EX2 = ((p `2 ) + (2 * r)), EX1 = (p `1 );

      reconsider p1 = p, EX = |[(p `1 ), ((p `2 ) + (2 * r))]| as Point of ( Euclid 2) by EUCLID: 67;

      

       A3: ( |[(p `1 ), EX2]| `1 ) = (p `1 ) by EUCLID: 52;

      then

       A4: p1 in Wp by TOPREAL1:def 10;

      ( 0 + (p `2 )) <= ((2 * r) + (p `2 )) by A1, XREAL_1: 6;

      then ( |[(p `1 ), EX2]| `2 ) >= (p `2 ) by EUCLID: 52;

      then

       A5: EX in Wp by A3, TOPREAL1:def 10;

      p = |[p11, p12]| by EUCLID: 53;

      

      then ( dist (p1,EX)) = ( sqrt (((p11 - EX1) ^2 ) + ((p12 - EX2) ^2 ))) by GOBOARD6: 6

      .= ( sqrt (((EX2 - p12) ^2 ) + 0 ))

      .= (2 * r) by A1, SQUARE_1: 22;

      hence thesis by A1, A2, A5, A4, XREAL_1: 155;

    end;

    theorem :: JORDAN2C:123

    

     Th107: ( south_halfline p) is non bounded

    proof

      set Wp = ( south_halfline p);

      set p11 = (p `1 ), p12 = (p `2 );

      assume Wp is bounded;

      then

      reconsider C = Wp as bounded Subset of ( Euclid 2) by Th5;

      consider r be Real such that

       A1: 0 < r and

       A2: for x,y be Point of ( Euclid 2) st x in C & y in C holds ( dist (x,y)) <= r by TBSP_1:def 7;

      set EX2 = ((p `2 ) - (2 * r)), EX1 = (p `1 );

      reconsider p1 = p, EX = |[(p `1 ), ((p `2 ) - (2 * r))]| as Point of ( Euclid 2) by EUCLID: 67;

      p = |[p11, p12]| by EUCLID: 53;

      

      then

       A3: ( dist (p1,EX)) = ( sqrt (((p11 - EX1) ^2 ) + ((p12 - EX2) ^2 ))) by GOBOARD6: 6

      .= (2 * r) by A1, SQUARE_1: 22;

      

       A4: ( |[(p `1 ), EX2]| `1 ) = (p `1 ) by EUCLID: 52;

      then

       A5: p1 in Wp by TOPREAL1:def 12;

      ( 0 + (p `2 )) <= ((2 * r) + (p `2 )) by A1, XREAL_1: 6;

      then ((p `2 ) - (2 * r)) <= (p `2 ) by XREAL_1: 20;

      then ( |[(p `1 ), EX2]| `2 ) <= (p `2 ) by EUCLID: 52;

      then EX in Wp by A4, TOPREAL1:def 12;

      hence thesis by A1, A2, A5, A3, XREAL_1: 155;

    end;

    registration

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

      cluster ( UBD C) -> non empty;

      coherence

      proof

        

         A1: ( UBD C) is_outside_component_of C by Th53;

        thus thesis by A1;

      end;

    end

    theorem :: JORDAN2C:124

    

     Th108: for C be compact Subset of ( TOP-REAL 2) holds ( UBD C) is_a_component_of (C ` )

    proof

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

      ( UBD C) is_outside_component_of C by Th53;

      hence thesis;

    end;

    theorem :: JORDAN2C:125

    

     Th109: for C be compact Subset of ( TOP-REAL 2) holds for WH be connected Subset of ( TOP-REAL 2) st WH is non bounded & WH misses C holds WH c= ( UBD C)

    proof

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

      let WH be connected Subset of ( TOP-REAL 2);

      assume that

       A1: WH is non bounded and

       A2: WH misses C;

      

       A3: WH meets ( UBD C)

      proof

        (( BDD C) \/ ( UBD C)) = (C ` ) & ( [#] the carrier of ( TOP-REAL 2)) = (C \/ (C ` )) by Th18, SUBSET_1: 10;

        then

         A4: WH c= (( UBD C) \/ ( BDD C)) by A2, XBOOLE_1: 73;

        assume

         A5: WH misses ( UBD C);

        ( BDD C) is bounded by Th90;

        hence thesis by A1, A5, A4, RLTOPSP1: 42, XBOOLE_1: 73;

      end;

      WH c= (C ` ) by A2, SUBSET_1: 23;

      hence thesis by A3, Th108, GOBOARD9: 4;

    end;

    theorem :: JORDAN2C:126

    for C be compact Subset of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st ( west_halfline p) misses C holds ( west_halfline p) c= ( UBD C)

    proof

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

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

      set WH = ( west_halfline p);

      assume

       A1: WH misses C;

      WH is non bounded non empty connected by Th104;

      hence thesis by A1, Th109;

    end;

    theorem :: JORDAN2C:127

    for C be compact Subset of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st ( east_halfline p) misses C holds ( east_halfline p) c= ( UBD C)

    proof

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

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

      set WH = ( east_halfline p);

      assume

       A1: WH misses C;

      WH is non bounded non empty connected by Th105;

      hence thesis by A1, Th109;

    end;

    theorem :: JORDAN2C:128

    for C be compact Subset of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st ( south_halfline p) misses C holds ( south_halfline p) c= ( UBD C)

    proof

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

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

      set WH = ( south_halfline p);

      assume

       A1: WH misses C;

      WH is non bounded non empty connected by Th107;

      hence thesis by A1, Th109;

    end;

    theorem :: JORDAN2C:129

    for C be compact Subset of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st ( north_halfline p) misses C holds ( north_halfline p) c= ( UBD C)

    proof

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

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

      set WH = ( north_halfline p);

      assume

       A1: WH misses C;

      WH is non bounded non empty connected by Th106;

      hence thesis by A1, Th109;

    end;

    theorem :: JORDAN2C:130

    for n be Nat, r be Real st r > 0 holds for x,y,z be Element of ( Euclid n) st x = ( 0* n) holds for p be Element of ( TOP-REAL n) st p = y & (r * p) = z holds (r * ( dist (x,y))) = ( dist (x,z)) by Lm1;

    theorem :: JORDAN2C:131

    for n be Nat, r,s be Real st r > 0 holds for x be Element of ( Euclid n) st x = ( 0* n) holds for A be Subset of ( TOP-REAL n) st A = ( Ball (x,s)) holds (r * A) = ( Ball (x,(r * s))) by Lm2;

    theorem :: JORDAN2C:132

    for n be Nat, r,s,t be Real st 0 < s & s <= t holds for x be Element of ( Euclid n) st x = ( 0* n) holds for BA be Subset of ( TOP-REAL n) st BA = ( Ball (x,r)) holds (s * BA) c= (t * BA) by Lm3;