reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th95:
  |[-1,0]|,|[1,0]| realize-max-dist-in C implies
  for Jc, Jd being compact with_the_max_arc Subset of TOP-REAL 2 st
  Jc is_an_arc_of |[-1,0]|,|[1,0]| & Jd is_an_arc_of |[-1,0]|,|[1,0]| &
  C = Jc \/ Jd & Jc /\ Jd = {|[-1,0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd &
  W-bound C = W-bound Jc & E-bound C = E-bound Jc
  for Ux being Subset of TOP-REAL 2 st Ux = Component_of Down ((1/2) *
  ((UMP (LSeg(LMP Jc,|[0,-3]|) /\ Jd)) + LMP Jc), C`)
  holds Ux is_inside_component_of C & for V being Subset of TOP-REAL 2 st
  V is_inside_component_of C holds V = Ux
proof
  set m = UMP C;
  set j = LMP C;
  assume
A1: a,b realize-max-dist-in C;
  let Jc, Jd being compact with_the_max_arc Subset of T2 such that
A2: Jc is_an_arc_of a,b and
A3: Jd is_an_arc_of a,b and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {a,b} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc;
  set l = LMP Jc;
  set LJ = LSeg(l,d) /\ Jd;
  set k = UMP LJ;
  set x = (1/2)*(k+l);
  set w = (W-bound C + E-bound C) / 2;
  let Ux be Subset of TOP-REAL 2 such that
A10: Ux = Component_of Down (x,C`);
A11: C c= R by A1,Th71;
A12: W-bound C = rl by A1,Th75;
A13: E-bound C = rp by A1,Th76;
A14: a in C by A1;
A15: b in C by A1;
A16: m in C by JORDAN21:30;
A17: l in Jc by JORDAN21:31;
A18: Jd c= C by A4,XBOOLE_1:7;
A19: Jc c= C by A4,XBOOLE_1:7;
  then
A20: l in C by A17;
A21: m`2 < c`2 by A1,Lm21,Th83,JORDAN21:30;
A22: l`1 = 0 by A8,A9,A12,A13,EUCLID:52;
A23: c`1 = w by A1,Lm87;
A24: m`1 = w by EUCLID:52;
A25: m <> a by A12,A13,Lm16,EUCLID:52;
A26: m <> b by A12,A13,Lm17,EUCLID:52;
A27: l <> a by A8,A9,A12,A13,Lm16,EUCLID:52;
A28: l <> b by A8,A9,A12,A13,Lm17,EUCLID:52;
  then consider Pml being Path of m,l such that
A29: rng Pml c= Jc and
A30: rng Pml misses {a,b} by A2,A6,A17,A25,A26,A27,Th44;
  set ml = rng Pml;
A31: ml c= C by A19,A29;
A32: j in C by A7,A18;
A33: LSeg(l,d) is vertical by A22,Lm22,SPPOL_1:16;
A34: d`2 <= j`2 by A1,A7,A18,Lm23,Th84;
A35: j`1 = 0 by A12,A13,EUCLID:52;
  l in Vertical_Line w by A12,A13,A22,JORDAN6:31;
  then
A36: l in C /\ Vertical_Line w by A17,A19,XBOOLE_0:def 4;
  then j`2 <= l`2 by JORDAN21:29;
  then j in LSeg(l,d) by A22,A34,A35,Lm22,GOBOARD7:7;
  then
A37: LJ is non empty by A7,XBOOLE_0:def 4;
A38: LJ is vertical by A33,Th4;
  then
A39: k in LJ by A37,JORDAN21:30;
  then
A40: k in LSeg(l,d) by XBOOLE_0:def 4;
A41: k in Jd by A39,XBOOLE_0:def 4;
  then
A42: k in C by A18;
A43: d in LSeg(l,d) by RLTOPSP1:68;
  then
A44: k`1 = 0 by A33,A40,Lm22;
  then
A45: k <> a by EUCLID:52;
A46: k <> b by A44,EUCLID:52;
A47: j <> a by A35,EUCLID:52;
  j <> b by A35,EUCLID:52;
  then consider Pkj being Path of k,j such that
A48: rng Pkj c= Jd and
A49: rng Pkj misses {a,b} by A3,A7,A41,A45,A46,A47,Th44;
  set kj = rng Pkj;
A50: kj c= C by A18,A48;
A51: x in LSeg(k,l) by RLTOPSP1:69;
A52: Component_of Down(x,C`) is a_component by CONNSP_1:40;
A53: the carrier of T2|C` = C` by PRE_TOPC:8;
A54: LSeg(l,k) is vertical by A22,A44,SPPOL_1:16;
A55: k in LSeg(l,k) by RLTOPSP1:68;
A56: l = |[l`1,l`2]| by EUCLID:53;
A57: k = |[k`1,k`2]| by EUCLID:53;
A58: d = |[d`1,d`2]| by EUCLID:53;
  d`2 <= l`2 by A1,A17,A19,Lm23,Th84;
  then
A59: k`2 <= l`2 by A22,A40,A56,A58,Lm22,JGRAPH_6:1;
A60: a <> k by A44,EUCLID:52;
  b <> k by A44,EUCLID:52;
  then not k in {a,b} by A60,TARSKI:def 2;
  then
A61: k <> l by A5,A17,A41,XBOOLE_0:def 4;
  then k`2 <> l`2 by A22,A44,TOPREAL3:6;
  then
A62: k`2 < l`2 by A59,XXREAL_0:1;
  k in Vertical_Line w by A12,A13,A44,JORDAN6:31;
  then k in C /\ Vertical_Line w by A18,A41,XBOOLE_0:def 4;
  then j`2 <= k`2 by JORDAN21:29;
  then d`2 <= k`2 by A1,A7,A18,Lm23,Th84,XXREAL_0:2;
  then
A63: LSeg(l,k) c= LSeg(l,d) by A33,A44,A54,A59,Lm22,GOBOARD7:63;
A64: LSeg(l,k) \ {l,k} c= C`
  proof
    let q be object;
    assume that
A65: q in LSeg(l,k) \ {l,k} and
A66: not q in C`;
A67: q in LSeg(l,k) by A65,XBOOLE_0:def 5;
    reconsider q as Point of T2 by A65;
A68: q in C by A66,SUBSET_1:29;
A69: q`1 = w by A12,A13,A44,A54,A55,A67;
    then
A70: q in Vertical_Line w by JORDAN6:31;
    per cases by A4,A68,XBOOLE_0:def 3;
    suppose q in Jc;
      then q in Jc /\ Vertical_Line w by A70,XBOOLE_0:def 4;
      then
A71:  l`2 <= q`2 by A8,A9,JORDAN21:29;
      q`2 <= l`2 by A22,A44,A56,A57,A59,A67,JGRAPH_6:1;
      then l`2 = q`2 by A71,XXREAL_0:1;
      then l = q by A12,A13,A22,A69,TOPREAL3:6;
      then q in {l,k} by TARSKI:def 2;
      hence contradiction by A65,XBOOLE_0:def 5;
    end;
    suppose q in Jd;
      then
A72:  q in LJ by A63,A67,XBOOLE_0:def 4;
A73:  q`1 = d`1 by A33,A43,A63,A67;
A74:  W-bound LSeg(l,d) <= W-bound LJ by A72,PSCOMP_1:69,XBOOLE_1:17;
A75:  E-bound LJ <= E-bound LSeg(l,d) by A72,PSCOMP_1:67,XBOOLE_1:17;
A76:  W-bound LJ = E-bound LJ by A37,A38,SPRECT_1:15;
A77:  W-bound LSeg(l,d) = d`1 by A22,Lm22,SPRECT_1:54;
      then W-bound LSeg(l,d) = W-bound LJ by A22,A74,A75,A76,Lm22,SPRECT_1:57;
      then q in Vertical_Line ((W-bound LJ + E-bound LJ) / 2)
      by A73,A76,A77,JORDAN6:31;
      then q in LJ /\ Vertical_Line ((W-bound LJ + E-bound LJ) / 2)
      by A72,XBOOLE_0:def 4;
      then
A78:  q`2 <= k`2 by JORDAN21:28;
      k`2 <= q`2 by A22,A44,A56,A57,A59,A67,JGRAPH_6:1;
      then k`2 = q`2 by A78,XXREAL_0:1;
      then k = q by A12,A13,A44,A69,TOPREAL3:6;
      then q in {l,k} by TARSKI:def 2;
      hence contradiction by A65,XBOOLE_0:def 5;
    end;
  end;
  then reconsider X = LSeg(l,k) \ {l,k} as Subset of T2|C` by PRE_TOPC:8;
  now
    assume x in {l,k};
    then x = l or x = k by TARSKI:def 2;
    hence contradiction by A61,Th1;
  end;
  then
A79: x in LSeg(l,k) \ {l,k} by A51,XBOOLE_0:def 5;
  then Component_of(x,C`) = Component_of Down(x,C`) by A64,CONNSP_3:27;
  then
A80: x in Component_of Down(x,C`) by A64,A79,CONNSP_3:26;
  then
A81: X meets Ux by A10,A79,XBOOLE_0:3;
  LSeg(l,k) \ {l,k} is convex by JORDAN1:46;
  then X is connected by CONNSP_1:23;
  then
A82: X c= Component_of Down(x,C`) by A10,A52,A81,CONNSP_1:36;
A83: LSeg(l,k) c= R by A11,A20,A42,JORDAN1:def 1;
A84: the carrier of TR = R by PRE_TOPC:8;
  reconsider AR = a, BR = b, CR = c, DR = d
  as Point of TR by A11,A14,A15,Lm62,Lm63,Lm67,PRE_TOPC:8;
  consider Pcm being Path of c,m, fcm being Function of I[01], T2|LSeg(c,m)
  such that
A85: rng fcm = LSeg(c,m) and
A86: Pcm = fcm by Th43;
A87: LSeg(c,m) c= R by A11,A16,Lm62,Lm67,JORDAN1:def 1;
A88: ml c= R by A11,A31;
  thus Ux is_inside_component_of C
  proof
    thus
A89: Ux is_a_component_of C` by A10,A52;
    assume not Ux is bounded;
    then not Ux c= Ball(x,10) by RLTOPSP1:42;
    then consider u being object such that
A90: u in Ux and
A91: not u in Ball(x,10);
A92: R c= Ball(x,10) by A51,A83,Lm89;
    reconsider u as Point of T2 by A90;
A93: Ux is open by A89,SPRECT_3:8;
    Component_of Down(x,C`) is connected by A52;
    then
A94: Ux is connected by A10,CONNSP_1:23;
    x in Ball(x,10) by Th16;
    then consider P1 being Subset of T2 such that
A95: P1 is_S-P_arc_joining x,u and
A96: P1 c= Ux by A10,A80,A90,A91,A93,A94,TOPREAL4:29;
A97: P1 is_an_arc_of x,u by A95,TOPREAL4:2;
    reconsider P2 = P1 as Subset of T2|C` by A10,A96,XBOOLE_1:1;
A98: P2 c= Component_of Down(x,C`) by A10,A96;
A99: P2 misses C by A53,SUBSET_1:23;
    then
A100: P2 misses Jc by A4,XBOOLE_1:7,63;
A101: P2 misses Jd by A4,A99,XBOOLE_1:7,63;
A102: x`1 = 1/2*((k+l)`1) by TOPREAL3:4
      .= 1/2*(k`1+l`1) by TOPREAL3:2
      .= 0 by A22,A44;
    then
A103: LSeg(d,x) is vertical by Lm22,SPPOL_1:16;
A104: x = |[x`1,x`2]| by EUCLID:53;
A105: x`2 < l`2 by A62,Th3;
A106: k`2 < x`2 by A62,Th2;
    then
A107: d`2 <= x`2 by A1,A18,A41,Lm23,Th84,XXREAL_0:2;
    d`1 = d`1;
    then
A108: LSeg(d,x) c= LSeg(d,l) by A33,A103,A105,A107,GOBOARD7:63;
A109: LSeg(d,x) misses Jc
    proof
      assume not thesis;
      then consider q being object such that
A110: q in LSeg(d,x) and
A111: q in Jc by XBOOLE_0:3;
      reconsider q as Point of T2 by A110;
      q`2 <= x`2 by A58,A102,A104,A107,A110,Lm22,JGRAPH_6:1;
      then
A112: q`2 < l`2 by A105,XXREAL_0:2;
      q`1 = 0 by A33,A43,A108,A110,Lm22;
      then q in Vertical_Line w by A12,A13,JORDAN6:31;
      then q in Jc /\ Vertical_Line w by A111,XBOOLE_0:def 4;
      hence contradiction by A8,A9,A112,JORDAN21:29;
    end;
    set n = First_Point(P1,x,u,dR);
A113: not u in R by A91,A92;
A114: Fr R = dR by Th52;
    u in P1 by A97,TOPREAL1:1;
    then
A115: P1 \ R <> {}T2 by A113,XBOOLE_0:def 5;
    x in P1 by A97,TOPREAL1:1;
    then P1 meets R by A51,A83,XBOOLE_0:3;
    then
A116: P1 meets dR by A97,A114,A115,CONNSP_1:22,JORDAN6:10;
    P1 is closed by A95,JORDAN6:11,TOPREAL4:2;
    then
A117: n in P1 /\ dR by A97,A116,JORDAN5C:def 1;
    then
A118: n in dR by XBOOLE_0:def 4;
A119: n in P1 by A117,XBOOLE_0:def 4;
    set alpha = Segment(P1,x,u,x,n);
A120: rd < k`2 by A1,A18,A41,Th84;
    l`2 <= m`2 by A36,JORDAN21:28;
    then x`2 < m`2 by A105,XXREAL_0:2;
    then not x in dR by A21,A102,A104,A106,A120,Lm86;
    then
A121: alpha is_an_arc_of x,n by A95,A118,A119,JORDAN16:24,TOPREAL4:2;
A122: alpha misses Jc by A100,JORDAN16:2,XBOOLE_1:63;
A123: alpha misses Jd by A101,JORDAN16:2,XBOOLE_1:63;
    consider Pdx being Path of d,x,
    fdx being Function of I[01], T2|LSeg(d,x) such that
A124: rng fdx = LSeg(d,x) and
A125: Pdx = fdx by Th43;
    consider PJc being Path of a,b, fJc being Function
    of I[01], T2|Jc such that
A126: rng fJc = Jc and
A127: PJc = fJc by A2,Th42;
    consider PJd being Path of a,b, fJd being Function
    of I[01], T2|Jd such that
A128: rng fJd = Jd and
A129: PJd = fJd by A3,Th42;
    consider Palpha being Path of x,n,
    falpha being Function of I[01], T2|alpha such that
A130: rng falpha = alpha and
A131: Palpha = falpha by A121,Th42;
    n in R by A118,Lm67;
    then
A132: ex p st p = n & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
    rng PJc c= the carrier of TR by A11,A19,A84,A126,A127;
    then reconsider h = PJc as Path of AR,BR by Th30;
    rng PJd c= the carrier of TR by A11,A18,A84,A128,A129;
    then reconsider H = PJd as Path of AR,BR by Th30;
A133: LSeg(d,x) c= R by A51,A83,Lm63,Lm67,JORDAN1:def 1;
A134: alpha c= R by A51,A83,A95,A113,Th57,TOPREAL4:2;
A135: ld in LSeg(ld,lg) by RLTOPSP1:68;
A136: pd in LSeg(pd,pg) by RLTOPSP1:68;
    LSeg(lg,c) misses C by A1,Lm78;
    then
A137: LSeg(lg,c) misses Jc by A4,XBOOLE_1:7,63;
A138: LSeg(lg,c) c= R by Lm67,Lm70;
A139: LSeg(pg,c) c= R by Lm67,Lm71;
    LSeg(pg,c) misses C by A1,Lm79;
    then
A140: LSeg(pg,c) misses Jc by A4,XBOOLE_1:7,63;
    consider Plx being Path of l,x, flx being Function of I[01], T2|LSeg(l,x)
    such that
A141: rng flx = LSeg(l,x) and
A142: Plx = flx by Th43;
    set PCX = Pcm + Pml + Plx;
A143: rng PCX = rng Pcm \/ rng Pml \/ rng Plx by Th40;
A144: ml misses Jd
    proof
      assume ml meets Jd;
      then consider q being object such that
A145: q in ml and
A146: q in Jd by XBOOLE_0:3;
      q in {a,b} by A5,A29,A145,A146,XBOOLE_0:def 4;
      hence contradiction by A30,A145,XBOOLE_0:3;
    end;
A147: LSeg(c,m) /\ C = {m} by A1,Th91;
A148: LSeg(c,m) misses Jd
    proof
      assume LSeg(c,m) meets Jd;
      then consider q being object such that
A149: q in LSeg(c,m) and
A150: q in Jd by XBOOLE_0:3;
      q in {m} by A18,A147,A149,A150,XBOOLE_0:def 4;
      then q = m by TARSKI:def 1;
      then m in {a,b} by A5,A6,A150,XBOOLE_0:def 4;
      hence contradiction by A25,A26,TARSKI:def 2;
    end;
    LSeg(l,x) is vertical by A22,A102,SPPOL_1:16;
    then
A151: LSeg(l,x) c= LSeg(l,k) by A44,A54,A102,A105,A106,GOBOARD7:63;
    l in LSeg(l,x) by RLTOPSP1:68;
    then {l} c= LSeg(l,x) by ZFMISC_1:31;
    then
A152: LSeg(l,x) = LSeg(l,x) \ {l} \/ {l} by XBOOLE_1:45;
    LSeg(l,x) \ {l} c= LSeg(l,k) \ {l,k}
    proof
      let q be object;
      assume
A153: q in LSeg(l,x) \ {l};
      then
A154: q in LSeg(l,x) by ZFMISC_1:56;
A155: q <> l by A153,ZFMISC_1:56;
      q <> k by A22,A56,A102,A104,A105,A106,A154,JGRAPH_6:1;
      then not q in {l,k} by A155,TARSKI:def 2;
      hence thesis by A151,A154,XBOOLE_0:def 5;
    end;
    then LSeg(l,x) \ {l} c= C` by A64;
    then LSeg(l,x) \ {l} misses C by SUBSET_1:23;
    then
A156: LSeg(l,x) \ {l} misses Jd by A4,XBOOLE_1:7,63;
    {l} misses Jd
    proof
      assume {l} meets Jd;
      then l in Jd by ZFMISC_1:50;
      then l in {a,b} by A5,A17,XBOOLE_0:def 4;
      hence thesis by A27,A28,TARSKI:def 2;
    end;
    then LSeg(l,x) misses Jd by A152,A156,XBOOLE_1:70;
    then
A157: rng PCX misses Jd by A85,A86,A141,A142,A143,A144,A148,XBOOLE_1:114;
    LSeg(l,x) c= R by A83,A151;
    then
A158: rng PCX c= R by A85,A86,A87,A88,A141,A142,A143,Lm1;
    LSeg(ld,d) misses C by A1,Lm80;
    then
A159: LSeg(ld,d) misses Jd by A4,XBOOLE_1:7,63;
    LSeg(pd,d) misses C by A1,Lm81;
    then
A160: LSeg(pd,d) misses Jd by A4,XBOOLE_1:7,63;
    per cases;
    suppose
A161: n`2 < 0;
      per cases by A118,A161,Lm77;
      suppose
A162:   n in LSeg(a,ld);
        consider Pnld being Path of n,ld,
        fnld being Function of I[01], T2|LSeg(n,ld) such that
A163:   rng fnld = LSeg(n,ld) and
A164:   Pnld = fnld by Th43;
        consider Pldd being Path of ld,d,
        fldd being Function of I[01], T2|LSeg(ld,d) such that
A165:   rng fldd = LSeg(ld,d) and
A166:   Pldd = fldd by Th43;
A167:   ld`1 = n`1 by A135,A162,Lm45,Lm58;
        then LSeg(n,ld) is vertical by SPPOL_1:16;
        then LSeg(n,ld) c= LSeg(ld,lg) by A132,A167,Lm25,Lm27,Lm45,GOBOARD7:63;
        then
A168:   LSeg(n,ld) c= dR by Lm38;
        set K1 = PCX + Palpha + Pnld + Pldd;
        LSeg(n,ld) misses C by A1,A53,A98,A119,A162,Lm84;
        then
A169:   LSeg(n,ld) misses Jd by A4,XBOOLE_1:7,63;
A170:   rng K1 = rng PCX \/ rng Palpha \/ rng Pnld \/ rng Pldd by Lm9;
        then
A171:   rng PJd misses rng K1 by A123,A128,A129,A130,A131,A157,A159,A163,A164
,A165,A166,A169,Lm3;
A172:   LSeg(ld,d) c= R by Lm67,Lm74;
        LSeg(n,ld) c= R by A168,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A130,A131,A134,A158,A163,A164,A165,A166,A170,A172,Lm2;
        then reconsider v = K1 as Path of CR,DR by Th30;
        consider s, t being Point of I[01] such that
A173:   H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A174:   dom H = the carrier of I[01] by FUNCT_2:def 1;
A175:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A176:   H.s in rng PJd by A174,FUNCT_1:def 3;
        v.t in rng K1 by A175,FUNCT_1:def 3;
        hence contradiction by A171,A173,A176,XBOOLE_0:3;
      end;
      suppose
A177:   n in LSeg(ld,d);
        consider Pnd being Path of n,d,
        fnd being Function of I[01], T2|LSeg(n,d) such that
A178:   rng fnd = LSeg(n,d) and
A179:   Pnd = fnd by Th43;
        set K1 = PCX + Palpha + Pnd;
        ld in LSeg(ld,d) by RLTOPSP1:68;
        then
A180:   ld`2 = n`2 by A177,Lm51;
        then
A181:   LSeg(n,d) is horizontal by Lm23,Lm27,SPPOL_1:15;
A182:   ld`1 <= n`1 by A177,Lm26,JGRAPH_6:3;
        n`1 <= d`1 by A177,Lm22,JGRAPH_6:3;
        then
A183:   LSeg(n,d) c= LSeg(ld,d) by A180,A181,A182,Lm51,GOBOARD7:64;
        then
A184:   LSeg(n,d) c= dR by Lm74;
        LSeg(n,d) misses C by A1,A183,Lm80,XBOOLE_1:63;
        then
A185:   LSeg(n,d) misses Jd by A4,XBOOLE_1:7,63;
A186:   rng K1 = rng PCX \/ rng Palpha \/ rng Pnd by Th40;
        then
A187:   rng K1 misses Jd by A123,A130,A131,A157,A178,A179,A185,XBOOLE_1:114;
        LSeg(n,d) c= R by A184,Lm67;
        then rng K1 c= the carrier of TR by A84,A130,A131,A134,A158,A178,A179
,A186,Lm1;
        then reconsider v = K1 as Path of CR,DR by Th30;
        consider s, t being Point of I[01] such that
A188:   H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A189:   dom H = the carrier of I[01] by FUNCT_2:def 1;
A190:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A191:   H.s in rng PJd by A189,FUNCT_1:def 3;
        v.t in rng K1 by A190,FUNCT_1:def 3;
        hence contradiction by A128,A129,A187,A188,A191,XBOOLE_0:3;
      end;
      suppose
A192:   n in LSeg(d,pd);
        consider Pnd being Path of n,d,
        fnd being Function of I[01], T2|LSeg(n,d) such that
A193:   rng fnd = LSeg(n,d) and
A194:   Pnd = fnd by Th43;
        set K1 = PCX + Palpha + Pnd;
        pd in LSeg(pd,d) by RLTOPSP1:68;
        then pd`2 = n`2 by A192,Lm52;
        then
A195:   LSeg(n,d) is horizontal by Lm23,Lm31,SPPOL_1:15;
A196:   d`2 = d`2;
A197:   d`1 <= n`1 by A192,Lm22,JGRAPH_6:3;
        n`1 <= pd`1 by A192,Lm30,JGRAPH_6:3;
        then
A198:   LSeg(n,d) c= LSeg(pd,d) by A195,A196,A197,Lm52,GOBOARD7:64;
        then
A199:   LSeg(n,d) c= dR by Lm75;
        LSeg(n,d) misses C by A1,A198,Lm81,XBOOLE_1:63;
        then
A200:   LSeg(n,d) misses Jd by A4,XBOOLE_1:7,63;
A201:   rng K1 = rng PCX \/ rng Palpha \/ rng Pnd by Th40;
        then
A202:   rng K1 misses Jd by A123,A130,A131,A157,A193,A194,A200,XBOOLE_1:114;
        LSeg(n,d) c= R by A199,Lm67;
        then rng K1 c= the carrier of TR by A84,A130,A131,A134,A158,A193,A194
,A201,Lm1;
        then reconsider v = K1 as Path of CR,DR by Th30;
        consider s, t being Point of I[01] such that
A203:   H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A204:   dom H = the carrier of I[01] by FUNCT_2:def 1;
A205:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A206:   H.s in rng PJd by A204,FUNCT_1:def 3;
        v.t in rng K1 by A205,FUNCT_1:def 3;
        hence contradiction by A128,A129,A202,A203,A206,XBOOLE_0:3;
      end;
      suppose
A207:   n in LSeg(pd,b);
        consider Pnpd being Path of n,pd,
        fnpd being Function of I[01], T2|LSeg(n,pd) such that
A208:   rng fnpd = LSeg(n,pd) and
A209:   Pnpd = fnpd by Th43;
        consider Ppdd being Path of pd,d,
        fpdd being Function of I[01], T2|LSeg(pd,d) such that
A210:   rng fpdd = LSeg(pd,d) and
A211:   Ppdd = fpdd by Th43;
A212:   pd`1 = n`1 by A136,A207,Lm46,Lm60;
        then LSeg(n,pd) is vertical by SPPOL_1:16;
        then LSeg(n,pd) c= LSeg(pd,pg) by A132,A212,Lm29,Lm31,Lm46,GOBOARD7:63;
        then
A213:   LSeg(n,pd) c= dR by Lm42;
        set K1 = PCX + Palpha + Pnpd + Ppdd;
        LSeg(n,pd) misses C by A1,A53,A98,A119,A207,Lm85;
        then
A214:   LSeg(n,pd) misses Jd by A4,XBOOLE_1:7,63;
A215:   rng K1 = rng PCX \/ rng Palpha \/ rng Pnpd \/ rng Ppdd by Lm9;
        then
A216:   rng PJd misses rng K1 by A123,A128,A129,A130,A131,A157,A160,A208,A209
,A210,A211,A214,Lm3;
A217:   LSeg(pd,d) c= R by Lm67,Lm75;
        LSeg(n,pd) c= R by A213,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A130,A131,A134,A158,A208,A209,A210,A211,A215,A217,Lm2;
        then reconsider v = K1 as Path of CR,DR by Th30;
        consider s, t being Point of I[01] such that
A218:   H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A219:   dom H = the carrier of I[01] by FUNCT_2:def 1;
A220:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A221:   H.s in rng PJd by A219,FUNCT_1:def 3;
        v.t in rng K1 by A220,FUNCT_1:def 3;
        hence contradiction by A216,A218,A221,XBOOLE_0:3;
      end;
    end;
    suppose
A222: n`2 >= 0;
      per cases by A118,A222,Lm76;
      suppose
A223:   n in LSeg(a,lg);
        consider Pnlg being Path of n,lg,
        fnlg being Function of I[01], T2|LSeg(n,lg) such that
A224:   rng fnlg = LSeg(n,lg) and
A225:   Pnlg = fnlg by Th43;
        consider Plgc being Path of lg,c,
        flgc being Function of I[01], T2|LSeg(lg,c) such that
A226:   rng flgc = LSeg(lg,c) and
A227:   Plgc = flgc by Th43;
A228:   ld`1 = n`1 by A135,A223,Lm45,Lm57;
        then LSeg(n,lg) is vertical by Lm24,Lm26,SPPOL_1:16;
        then LSeg(n,lg) c= LSeg(ld,lg) by A132,A228,Lm25,Lm27,Lm45,GOBOARD7:63;
        then
A229:   LSeg(n,lg) c= dR by Lm38;
        set K1 = Pdx + Palpha + Pnlg + Plgc;
        LSeg(n,lg) misses C by A1,A53,A98,A119,A223,Lm82;
        then
A230:   LSeg(n,lg) misses Jc by A4,XBOOLE_1:7,63;
A231:   rng K1 = rng Pdx \/ rng Palpha \/ rng Pnlg \/ rng Plgc by Lm9;
        then
A232:   rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A137,A224,A225,A226
,A227,A230,Lm3;
A233:   rng K1 = rng -K1 by Th32;
        LSeg(n,lg) c= R by A229,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A124,A125,A130,A131,A133,A134,A138,A224,A225,A226,A227,A231,Lm2;
        then reconsider v = -K1 as Path of CR,DR by A233,Th30;
        consider s, t being Point of I[01] such that
A234:   h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A235:   dom h = the carrier of I[01] by FUNCT_2:def 1;
A236:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A237:   h.s in rng PJc by A235,FUNCT_1:def 3;
        v.t in rng -K1 by A236,FUNCT_1:def 3;
        hence contradiction by A126,A127,A232,A233,A234,A237,XBOOLE_0:3;
      end;
      suppose
A238:   n in LSeg(lg,c);
        consider Pnc being Path of n,c,
        fnc being Function of I[01], T2|LSeg(n,c) such that
A239:   rng fnc = LSeg(n,c) and
A240:   Pnc = fnc by Th43;
        set K1 = Pdx + Palpha + Pnc;
        lg in LSeg(lg,c) by RLTOPSP1:68;
        then
A241:   lg`2 = n`2 by A238,Lm53;
        then
A242:   LSeg(n,c) is horizontal by Lm21,Lm25,SPPOL_1:15;
A243:   lg`1 <= n`1 by A238,Lm24,JGRAPH_6:3;
        n`1 <= c`1 by A238,Lm20,JGRAPH_6:3;
        then
A244:   LSeg(n,c) c= LSeg(lg,c) by A241,A242,A243,Lm53,GOBOARD7:64;
        then
A245:   LSeg(n,c) c= dR by Lm70;
        LSeg(n,c) misses C by A1,A244,Lm78,XBOOLE_1:63;
        then
A246:   LSeg(n,c) misses Jc by A4,XBOOLE_1:7,63;
A247:   rng K1 = rng Pdx \/ rng Palpha \/ rng Pnc by Th40;
        then
A248:   rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A239,A240,A246,
XBOOLE_1:114;
A249:   rng K1 = rng -K1 by Th32;
        LSeg(n,c) c= R by A245,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A124,A125,A130,A131,A133,A134,A239,A240,A247,Lm1;
        then reconsider v = -K1 as Path of CR,DR by A249,Th30;
        consider s, t being Point of I[01] such that
A250:   h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A251:   dom h = the carrier of I[01] by FUNCT_2:def 1;
A252:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A253:   h.s in rng PJc by A251,FUNCT_1:def 3;
        v.t in rng -K1 by A252,FUNCT_1:def 3;
        hence contradiction by A126,A127,A248,A249,A250,A253,XBOOLE_0:3;
      end;
      suppose
A254:   n in LSeg(c,pg);
        consider Pnc being Path of n,c,
        fnc being Function of I[01], T2|LSeg(n,c) such that
A255:   rng fnc = LSeg(n,c) and
A256:   Pnc = fnc by Th43;
        set K1 = Pdx + Palpha + Pnc;
        pg in LSeg(pg,c) by RLTOPSP1:68;
        then pg`2 = n`2 by A254,Lm54;
        then
A257:   LSeg(n,c) is horizontal by Lm21,Lm29,SPPOL_1:15;
A258:   c`2 = c`2;
A259:   c`1 <= n`1 by A254,Lm20,JGRAPH_6:3;
        n`1 <= pg`1 by A254,Lm28,JGRAPH_6:3;
        then
A260:   LSeg(c,n) c= LSeg(c,pg) by A257,A258,A259,Lm54,GOBOARD7:64;
        then
A261:   LSeg(n,c) c= dR by Lm71;
        LSeg(n,c) misses C by A1,A260,Lm79,XBOOLE_1:63;
        then
A262:   LSeg(n,c) misses Jc by A4,XBOOLE_1:7,63;
A263:   rng K1 = rng Pdx \/ rng Palpha \/ rng Pnc by Th40;
        then
A264:   rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A255,A256,A262,
XBOOLE_1:114;
A265:   rng K1 = rng -K1 by Th32;
        LSeg(n,c) c= R by A261,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A124,A125,A130,A131,A133,A134,A255,A256,A263,Lm1;
        then reconsider v = -K1 as Path of CR,DR by A265,Th30;
        consider s, t being Point of I[01] such that
A266:   h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A267:   dom h = the carrier of I[01] by FUNCT_2:def 1;
A268:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A269:   h.s in rng PJc by A267,FUNCT_1:def 3;
        v.t in rng -K1 by A268,FUNCT_1:def 3;
        hence contradiction by A126,A127,A264,A265,A266,A269,XBOOLE_0:3;
      end;
      suppose
A270:   n in LSeg(pg,b);
        consider Pnpg being Path of n,pg,
        fnpg being Function of I[01], T2|LSeg(n,pg) such that
A271:   rng fnpg = LSeg(n,pg) and
A272:   Pnpg = fnpg by Th43;
        consider Ppgc being Path of pg,c,
        fpgc being Function of I[01], T2|LSeg(pg,c) such that
A273:   rng fpgc = LSeg(pg,c) and
A274:   Ppgc = fpgc by Th43;
A275:   pd`1 = n`1 by A136,A270,Lm46,Lm59;
        then LSeg(n,pg) is vertical by Lm28,Lm30,SPPOL_1:16;
        then LSeg(n,pg) c= LSeg(pd,pg) by A132,A275,Lm29,Lm31,Lm46,GOBOARD7:63;
        then
A276:   LSeg(n,pg) c= dR by Lm42;
        set K1 = Pdx + Palpha + Pnpg + Ppgc;
        LSeg(n,pg) misses C by A1,A53,A98,A119,A270,Lm83;
        then
A277:   LSeg(n,pg) misses Jc by A4,XBOOLE_1:7,63;
A278:   rng K1 = rng Pdx \/ rng Palpha \/ rng Pnpg \/ rng Ppgc by Lm9;
        then
A279:   rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A140,A271,A272,A273
,A274,A277,Lm3;
A280:   rng K1 = rng -K1 by Th32;
        LSeg(n,pg) c= R by A276,Lm67;
        then rng K1 c= the carrier of TR
        by A84,A124,A125,A130,A131,A133,A134,A139,A271,A272,A273,A274,A278,Lm2;
        then reconsider v = -K1 as Path of CR,DR by A280,Th30;
        consider s, t being Point of I[01] such that
A281:   h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A282:   dom h = the carrier of I[01] by FUNCT_2:def 1;
A283:   dom v = the carrier of I[01] by FUNCT_2:def 1;
A284:   h.s in rng PJc by A282,FUNCT_1:def 3;
        v.t in rng -K1 by A283,FUNCT_1:def 3;
        hence contradiction by A126,A127,A279,A280,A281,A284,XBOOLE_0:3;
      end;
    end;
  end;

:: uniqueness
  let V be Subset of T2;
  assume
A285: V is_inside_component_of C;
  assume
A286: V <> Ux;
  consider VP being Subset of T2|C` such that
A287: VP = V and
A288: VP is a_component and
  VP is bounded Subset of Euclid 2 by A285,JORDAN2C:13;
  reconsider T2C = T2|C` as non empty SubSpace of T2;
  VP <> {}(T2|C`) by A288,CONNSP_1:32;
  then reconsider VP as non empty Subset of T2C;
A289: V misses C by A53,A287,SUBSET_1:23;
  consider Pjd being Path of j,d,
  fjd being Function of I[01], T2|LSeg(j,d) such that
A290: rng fjd = LSeg(j,d) and
A291: Pjd = fjd by Th43;
  consider Plk being Path of l,k,
  flk being Function of I[01], T2|LSeg(l,k) such that
A292: rng flk = LSeg(l,k) and
A293: Plk = flk by Th43;
  set beta = Pcm + Pml + Plk + Pkj + Pjd;
A294: rng beta = rng Pcm \/ rng Pml \/ rng Plk \/ rng Pkj \/ rng Pjd by Lm11;
  dom beta = [#]I[01] by FUNCT_2:def 1;
  then beta.:dom beta is compact by WEIERSTR:8;
  then
A295: rng beta is closed by RELAT_1:113;
A296: ml misses V by A19,A29,A289,XBOOLE_1:1,63;
  {l,k} c= LSeg(l,k)
  proof
    let x be object;
    assume x in {l,k};
    then x = l or x = k by TARSKI:def 2;
    hence thesis by RLTOPSP1:68;
  end;
  then
A297: LSeg(l,k) = LSeg(l,k) \ {l,k} \/ {l,k} by XBOOLE_1:45;
A298: LSeg(l,k) \ {l,k} misses V
  proof
    assume not thesis;
    then ex q being object st ( q in LSeg(l,k) \ {l,k})&( q in V)
by XBOOLE_0:3;
    then V meets Ux by A10,A82,XBOOLE_0:3;
    hence contradiction by A10,A52,A286,A287,A288,CONNSP_1:35;
  end;
A299: not l in V by A17,A19,A289,XBOOLE_0:3;
  not k in V by A18,A41,A289,XBOOLE_0:3;
  then {l,k} misses V by A299,ZFMISC_1:51;
  then
A300: LSeg(l,k) misses V by A297,A298,XBOOLE_1:70;
A301: kj misses V by A50,A289,XBOOLE_1:63;
A302: LSeg(j,d) misses V by A1,A285,Th90;
  LSeg(c,m) misses V by A1,A285,Th89;
  then LSeg(c,m) \/ ml \/ LSeg(l,k) misses V by A296,A300,XBOOLE_1:114;
  then
A303: rng beta misses V by A85,A86,A290,A291,A292,A293,A294,A301,A302,
XBOOLE_1:114;
A304: m = |[m`1,m`2]| by EUCLID:53;
A305: c = |[c`1,c`2]| by EUCLID:53;
A306: j = |[j`1,j`2]| by EUCLID:53;
A307: not a in LSeg(c,m) by A12,A13,A21,A23,A24,A304,A305,Lm16,JGRAPH_6:1;
  not a in ml by A30,ZFMISC_1:49;
  then
A308: not a in LSeg(c,m) \/ ml by A307,XBOOLE_0:def 3;
  not a in LSeg(l,k) by A22,A44,A56,A57,A59,Lm16,JGRAPH_6:1;
  then
A309: not a in LSeg(c,m) \/ ml \/ LSeg(l,k) by A308,XBOOLE_0:def 3;
  not a in kj by A49,ZFMISC_1:49;
  then
A310: not a in LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj by A309,XBOOLE_0:def 3;
  not a in LSeg(j,d) by A34,A35,A58,A306,Lm16,Lm22,JGRAPH_6:1;
  then not a in rng beta by A85,A86,A290,A291,A292,A293,A294,A310,
XBOOLE_0:def 3;
  then consider ra being positive Real such that
A311: Ball(a,ra) misses rng beta by A295,Th25;
A312: not b in LSeg(c,m) by A12,A13,A21,A23,A24,A304,A305,Lm17,JGRAPH_6:1;
  not b in ml by A30,ZFMISC_1:49;
  then
A313: not b in LSeg(c,m) \/ ml by A312,XBOOLE_0:def 3;
  not b in LSeg(l,k) by A22,A44,A56,A57,A59,Lm17,JGRAPH_6:1;
  then
A314: not b in LSeg(c,m) \/ ml \/ LSeg(l,k) by A313,XBOOLE_0:def 3;
  not b in kj by A49,ZFMISC_1:49;
  then
A315: not b in LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj by A314,XBOOLE_0:def 3;
  not b in LSeg(j,d) by A34,A35,A58,A306,Lm17,Lm22,JGRAPH_6:1;
  then not b in rng beta by A85,A86,A290,A291,A292,A293,A294,A315,
XBOOLE_0:def 3;
  then consider rb being positive Real such that
A316: Ball(b,rb) misses rng beta by A295,Th25;
  set A = Ball(a,ra), B = Ball(b,rb);
A317: a in A by Th16;
A318: b in B by Th16;
  VP is non empty;
  then consider t being object such that
A319: t in V by A287;
  V in {W where W is Subset of T2: W is_inside_component_of C} by A285;
  then t in BDD C by A319,TARSKI:def 4;
  then
A320: C = Fr V by A287,A288,Lm15;
  then a in Cl V by A14,XBOOLE_0:def 4;
  then A meets V by A317,PRE_TOPC:def 7;
  then consider u being object such that
A321: u in A and
A322: u in V by XBOOLE_0:3;
  b in Cl V by A15,A320,XBOOLE_0:def 4;
  then B meets V by A318,PRE_TOPC:def 7;
  then consider v being object such that
A323: v in B and
A324: v in V by XBOOLE_0:3;
  reconsider u, v as Point of T2 by A321,A323;
A325: the carrier of T2C|VP = VP by PRE_TOPC:8;
  reconsider u1 = u, v1 = v as Point of T2C|VP by A287,A322,A324,PRE_TOPC:8;
  T2C|VP is pathwise_connected by A288,Th69;
  then
A326: u1,v1 are_connected;
  then consider fuv being Function of I[01], T2C|VP such that
A327: fuv is continuous and
A328: fuv.0 = u1 and
A329: fuv.1 = v1;
A330: T2C|VP = T2|V by A287,GOBOARD9:2;
  fuv is Path of u1,v1 by A326,A327,A328,A329,BORSUK_2:def 2;
  then reconsider uv = fuv as Path of u,v by A326,A330,TOPALG_2:1;
A331: rng fuv c= the carrier of T2C|VP;
  then
A332: rng uv misses rng beta by A287,A303,A325,XBOOLE_1:63;
  consider au being Path of a,u,
  fau being Function of I[01], T2|LSeg(a,u) such that
A333: rng fau = LSeg(a,u) and
A334: au = fau by Th43;
  consider vb being Path of v,b,
  fvb being Function of I[01], T2|LSeg(v,b) such that
A335: rng fvb = LSeg(v,b) and
A336: vb = fvb by Th43;
  set AB = au + uv + vb;
A337: rng AB = rng au \/ rng uv \/ rng vb by Th40;
  a in A by Th16;
  then LSeg(a,u) c= A by A321,JORDAN1:def 1;
  then
A338: LSeg(a,u) misses rng beta by A311,XBOOLE_1:63;
  b in B by Th16;
  then LSeg(v,b) c= B by A323,JORDAN1:def 1;
  then LSeg(v,b) misses rng beta by A316,XBOOLE_1:63;
  then
A339: rng AB misses rng beta by A332,A333,A334,A335,A336,A337,A338,XBOOLE_1:114
;
A340: a,b are_connected by BORSUK_2:def 3;
A341: V c= R by A1,A285,Th93;
  then
A342: LSeg(a,u) c= R by A11,A14,A322,JORDAN1:def 1;
A343: LSeg(v,b) c= R by A11,A15,A324,A341,JORDAN1:def 1;
  rng uv c= R by A287,A325,A331,A341;
  then LSeg(a,u) \/ rng uv c= R by A342,XBOOLE_1:8;
  then rng AB c= the carrier of TR by A84,A333,A334,A335,A336,A337,A343,
XBOOLE_1:8;
  then reconsider h = AB as Path of AR,BR by A340,Th29;
A344: c,d are_connected by BORSUK_2:def 3;
  LSeg(c,m) \/ ml c= R by A87,A88,XBOOLE_1:8;
  then
A345: LSeg(c,m) \/ ml \/ LSeg(l,k) c= R by A83,XBOOLE_1:8;
  kj c= R by A11,A50;
  then
A346: LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj c= R by A345,XBOOLE_1:8;
  LSeg(j,d) c= R by A11,A32,Lm63,Lm67,JORDAN1:def 1;
  then rng beta c= the carrier of TR by A84,A85,A86,A290,A291,A292,A293,A294
,A346,XBOOLE_1:8;
  then reconsider v = beta as Path of CR,DR by A344,Th29;
  consider s, t being Point of I[01] such that
A347: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A348: dom h = the carrier of I[01] by FUNCT_2:def 1;
A349: dom v = the carrier of I[01] by FUNCT_2:def 1;
A350: h.s in rng AB by A348,FUNCT_1:def 3;
  v.t in rng beta by A349,FUNCT_1:def 3;
  hence contradiction by A339,A347,A350,XBOOLE_0:3;
end;
