
theorem Th7:
  for f being FinSequence of TOP-REAL 2,
  P being non empty Subset of TOP-REAL 2,
  F being Function of I[01], (TOP-REAL 2) | P,
  i being Nat st 1 <= i & i+1 <= len f & f is being_S-Seq &
  P = L~f &
  F is being_homeomorphism & F.0 = f/.1 & F.1 = f/.len f
  ex p1, p2 being Real
   st p1 < p2 & 0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 &
  LSeg (f, i) = F.:[.p1, p2.] & F.p1 = f/.i & F.p2 = f/.(i+1)
proof
  let f be FinSequence of TOP-REAL 2, P be non empty Subset of TOP-REAL 2,
  Ff be Function of I[01], (TOP-REAL 2)|P, i be Nat;
  assume that
A1: 1 <= i and
A2: i+1 <= len f and
A3: f is being_S-Seq and
A4: P = L~f and
A5: Ff is being_homeomorphism and
A6: Ff.0 = f/.1 and
A7: Ff.1 = f/.len f;
A8: f is one-to-one by A3;
A9: the carrier of Closed-Interval-TSpace(1/2,1) = [.1/2,1.] by TOPMETR:18;
A10: [#]Closed-Interval-TSpace(0, 1/2) = [.0,1/2.] by TOPMETR:18;
A11: [#]Closed-Interval-TSpace(1/2,1) = [.1/2,1.] by TOPMETR:18;
A12: len f >= 2 by A3,TOPREAL1:def 8;
  deffunc Q(Nat) = L~(f|($1+2));
  defpred ARC[Nat] means 1 <= $1 + 2 & $1 + 2 <= len f implies
  ex NE being non empty Subset of TOP-REAL 2 st NE = Q($1) &
  for j be Nat for F being Function of I[01], (TOP-REAL 2)|NE st
  1 <= j & j+1 <= $1+2 & F is being_homeomorphism & F.0 = f/.1 &
  F.1 = f/.($1+2)
  ex p1, p2 be Real st p1 < p2 &
  0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 & LSeg (f, j) = F.:[.p1, p2.]
  & F.p1 = f/.j & F.p2 = f/.(j+1);
  reconsider h1 = len f - 2 as Element of NAT by A12,INT_1:5;
A13: f|(len f) = f|(Seg len f) by FINSEQ_1:def 16
    .= f|(dom f) by FINSEQ_1:def 3
    .= f by RELAT_1:68;
A14: ARC[0]
  proof
    assume that
A15: 1 <= 0 + 2 and
A16: 0 + 2 <= len f;
A17: 1 <= len (f|2) by A15,A16,FINSEQ_1:59;
A18: 2 <= len (f|2) by A16,FINSEQ_1:59;
    then reconsider NE = Q(0) as non empty Subset of TOP-REAL 2
    by TOPREAL1:23;
    take NE;
    thus NE = Q(0);
    let j be Nat;
    let F be Function of I[01], (TOP-REAL 2)|NE;
    assume that
A19: 1 <= j and
A20: j+1 <= 0+2 and
A21: F is being_homeomorphism and
A22: F.0 = f/.1 and
A23: F.1 = f/.(0+2);
    j <= 1+1-1 by A20,XREAL_1:19;
    then
A24: j = 1 by A19,XXREAL_0:1;
A25: len (f|2) = 2 by A16,FINSEQ_1:59;
A26: 1 in dom (f|2) by A17,FINSEQ_3:25;
A27: 2 in dom (f|2) by A18,FINSEQ_3:25;
A28: (f|2)/.1 = (f|2).1 by A26,PARTFUN1:def 6;
A29: (f|2)/.2 = (f|2).2 by A27,PARTFUN1:def 6;
A30: (f|2)/.1 = f/.1 by A26,FINSEQ_4:70;
A31: 1 + 1 <= len f by A16;
A32: rng F = [#]((TOP-REAL 2)|NE) by A21,TOPS_2:def 5
      .= L~(f|2) by PRE_TOPC:def 5
      .= L~<* (f|2)/.1, (f|2)/.2 *> by A25,A28,A29,FINSEQ_1:44
      .= LSeg ((f|2)/.1, (f|2)/.2) by SPPOL_2:21
      .= LSeg (f/.1, f/.2) by A27,A30,FINSEQ_4:70
      .= LSeg (f, 1) by A31,TOPREAL1:def 3;
    take 0, jj;
    thus thesis by A22,A23,A24,A32,BORSUK_1:40,RELSET_1:22;
  end;
A33: for n being Nat st ARC[n] holds ARC[n+1]
  proof
    let n be Nat;
    assume
A34: ARC[n];
    assume that
A35: 1 <= n + 1 + 2 and
A36: n + 1 + 2 <= len f;
A37: 2 <= n + 2 by NAT_1:11;
    n+2 <= n+2+1 by NAT_1:11;
    then consider NE being non empty Subset of TOP-REAL 2 such that
A38: NE = Q(n) and
A39: for j be Nat for F being Function of I[01], (TOP-REAL 2)
|NE st 1 <= j & j+1 <= n+2 & F is being_homeomorphism & F.0 = f/.1 & F.1 = f/.(
    n+2)
   ex p1, p2 be Real
     st p1 < p2 & 0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 &
    LSeg (f, j) = F.:[.p1, p2.] & F.p1 = f/.j & F.p2 = f/.(j+1)
    by A34,A36,A37,XXREAL_0:2;
A40: len (f|(n+1+2)) = n+1+2 by A36,FINSEQ_1:59;
A41: n + 1 + 2 = n + 2 + 1;
A42: 1 <= n + 1 + 1 by NAT_1:11;
A43: n + 1 + 1 <= n + 2 + 1 by NAT_1:11;
    then
A44: n + 1 + 1 <= len f by A36,XXREAL_0:2;
A45: n+2 <= len f by A36,A43,XXREAL_0:2;
A46: n+2 in dom (f|(n+3)) by A40,A42,A43,FINSEQ_3:25;
    then
A47: f/.(n+2) = (f|(n+3))/.(n+2) by FINSEQ_4:70;
    reconsider NE1 = Q(n+1) as non empty Subset of TOP-REAL 2
    by A40,NAT_1:11,TOPREAL1:23;
    take NE1;
    thus NE1 = Q(n+1);
    let j be Nat, G be Function of I[01], (TOP-REAL 2)|NE1;
    assume that
A48: 1 <= j and
A49: j+1 <= n+1+2 and
A50: G is being_homeomorphism and
A51: G.0 = f/.1 and
A52: G.1 = f/.(n+1+2);
A53: G is one-to-one by A50,TOPS_2:def 5;
A54: rng G = [#]((TOP-REAL 2) | Q(n+1)) by A50,TOPS_2:def 5;
A55: dom G = [#]I[01] by A50,TOPS_2:def 5;
A56: rng G = L~(f|(n+3)) by A54,PRE_TOPC:def 5;
    set pp = G".(f/.(n+2));
    G is onto by A54,FUNCT_2:def 3;
    then
A57: pp = (G qua Function)".(f/.(n+2)) by A53,TOPS_2:def 4;
A58: n+2 <= len (f|(n+2)) by A36,A43,FINSEQ_1:59,XXREAL_0:2;
A59: 1 <= len (f|(n+2)) by A36,A42,A43,FINSEQ_1:59,XXREAL_0:2;
A60: n+2 in dom (f|(n+2)) by A42,A58,FINSEQ_3:25;
A61: 1 in dom (f|(n+2)) by A59,FINSEQ_3:25;
A62: f/.(n+2) in rng G by A40,A46,A47,A56,GOBOARD1:1,NAT_1:11;
    then
A63: pp in dom G by A53,A57,FUNCT_1:32;
A64: f/.(n+2) = G.pp by A53,A57,A62,FUNCT_1:32;
    reconsider pp as Real;
A65: n + 1 + 1 <> n + 2 + 1;
A66: n+2 <> n+3;
A67: n+2 in dom f by A42,A45,FINSEQ_3:25;
A68: n+3 in dom f by A35,A36,FINSEQ_3:25;
A69: 1 <> pp
    proof
      assume pp = 1;
      then f/.(n+2) = f/.(n+1+2) by A52,A53,A57,A62,FUNCT_1:32;
      hence thesis by A8,A66,A67,A68,PARTFUN2:10;
    end;
A70: 0 <= pp by A63,BORSUK_1:43;
    pp <= 1 by A63,BORSUK_1:43;
    then
A71: pp < 1 by A69,XXREAL_0:1;
    set pn = f/.(n+2), pn1 = f/.(n+2+1);
A72: pn = (f|(n+2))/.(n+2) by A60,FINSEQ_4:70;
A73: (f|(n+2))/.1 = f/.1 by A61,FINSEQ_4:70;
A74: len (f|(n+2)) = n+2 by A36,A43,FINSEQ_1:59,XXREAL_0:2;
    f|(n+2) is being_S-Seq by A3,JORDAN3:4,NAT_1:11;
    then NE is_an_arc_of (f|(n+2))/.1, pn by A38,A72,A74,TOPREAL1:25;
    then consider F be Function of I[01], (TOP-REAL 2)|NE such that
A75: F is being_homeomorphism and
A76: F.0 = f/.1 and
A77: F.1 = pn by A73,TOPREAL1:def 1;
A78: n + 1 + 1 in dom f by A42,A44,FINSEQ_3:25;
    n + 2 + 1 in dom f by A35,A36,FINSEQ_3:25;
    then LSeg(pn, pn1) is_an_arc_of pn, pn1 by A8,A65,A78,PARTFUN2:10
,TOPREAL1:9;
    then consider F9 be Function of I[01], (TOP-REAL 2)|LSeg(pn, pn1) such that
A79: F9 is being_homeomorphism and
A80: F9.0 = pn and
A81: F9.1 = pn1 by TOPREAL1:def 1;
    set Ex1 = P[01]( 0, 1/2, (#)(0,1),(0,1)(#) ),
    Ex2 = P[01]( 1/2, 1, (#)(0,1),(0,1)(#) );
    set F1 = F * Ex1, F19 = F9 * Ex2;
A82: Ex1 is being_homeomorphism by TREAL_1:17;
A83: Ex2 is being_homeomorphism by TREAL_1:17;
A84: dom Ex1 = [#]Closed-Interval-TSpace(0,1/2) by A82,TOPS_2:def 5;
    then
A85: dom Ex1 = [.0,1/2.] by TOPMETR:18;
    dom F = [#]I[01] by A75,TOPS_2:def 5;
    then
A86: rng Ex1 = dom F by A82,TOPMETR:20,TOPS_2:def 5;
    then rng F1 = rng F by RELAT_1:28;
    then rng F1 = [#] ((TOP-REAL 2) | NE) by A75,TOPS_2:def 5;
    then
A87: rng F1 = Q(n) by A38,PRE_TOPC:def 5;
    dom F1 = the carrier of Closed-Interval-TSpace(0, 1/2) by A84,A86,
RELAT_1:27;
    then reconsider F1 as Function
    of Closed-Interval-TSpace (0,1/2), (TOP-REAL 2)|NE by FUNCT_2:def 1;
A88: F1 is being_homeomorphism by A75,A82,TOPMETR:20,TOPS_2:57;
    then
A89: rng F1 = [#]((TOP-REAL 2)|NE) by TOPS_2:def 5
      .= Q(n) by A38,PRE_TOPC:def 5;
A90: dom Ex2 = [#]Closed-Interval-TSpace(1/2, 1) by A83,TOPS_2:def 5;
    dom F9 = [#]I[01] by A79,TOPS_2:def 5;
    then
A91: rng Ex2 = dom F9 by A83,TOPMETR:20,TOPS_2:def 5;
    then dom F19 = the carrier of Closed-Interval-TSpace(1/2, 1) by A90,
RELAT_1:27;
    then reconsider F19 as Function of Closed-Interval-TSpace (1/2,1),
    (TOP-REAL 2)|LSeg(pn, pn1) by FUNCT_2:def 1;
A92: F19 is being_homeomorphism by A79,A83,TOPMETR:20,TOPS_2:57;
    then
A93: rng F19 = [#]((TOP-REAL 2)|LSeg(pn, pn1)) by TOPS_2:def 5
      .= LSeg(pn, pn1) by PRE_TOPC:def 5;
    set FF = F1 +* F19;
    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;
A94: Q(n+1) = Q(n) \/ LSeg(pn, pn1) by A67,A68,TOPREAL3:38;
A95: the carrier of ((TOP-REAL 2)|Q(n+1)) = [#]((TOP-REAL 2)|Q(n+1))
      .= Q(n+1) by PRE_TOPC:def 5;
    dom F1 = the carrier of T1 by A84,A86,RELAT_1:27;
    then reconsider g11 = F1 as Function
    of T1, ((TOP-REAL 2)|NE1) by A87,A94,A95,RELSET_1:4,XBOOLE_1:7;
    dom F19 = the carrier of T2 by A90,A91,RELAT_1:27;
    then reconsider g22 = F19 as Function
    of T2, ((TOP-REAL 2)|NE1) by A93,A94,A95,RELSET_1:4,XBOOLE_1:7;
A96: [.0,1/2.] = dom F1 by A10,A88,TOPS_2:def 5;
A97: [.1/2, 1.] = dom F19 by A11,A92,TOPS_2:def 5;
A98: 1/2 in { l where l is Real: 0 <= l & l <= 1/2 };
A99: 1/2 in { l where l is Real: 1/2 <= l & l <= 1 };
A100: 1/2 in dom F1 by A96,A98,RCOMP_1:def 1;
A101: 1/2 in dom F19 by A97,A99,RCOMP_1:def 1;
A102: dom FF = [.0,1/2.] \/ [.1/2,1.] by A96,A97,FUNCT_4:def 1
      .= [.0,1.] by XXREAL_1:174;
A103: I[01] is compact by HEINE:4,TOPMETR:20;
A104: (TOP-REAL 2)|NE1 is T_2 by TOPMETR:2;
A105: Ex1.(1/2) = Ex1.(0,1/2)(#) by TREAL_1:def 2
      .= (0,1)(#) by TREAL_1:13
      .= 1 by TREAL_1:def 2;
A106: Ex2.(1/2) = Ex2.(#)(1/2,1) by TREAL_1:def 1
      .= (#)(0,1) by TREAL_1:13
      .= 0 by TREAL_1:def 1;
A107: F1.(1/2) = f/.(n+2) by A77,A100,A105,FUNCT_1:12;
A108: F19.(1/2) = f/.(n+2) by A80,A101,A106,FUNCT_1:12;
A109: [.0,1/2.] /\ [.1/2,1.] = [.1/2,1/2.] by XXREAL_1:143;
    then
A110: dom F1 /\ dom F19 = {1/2} by A96,A97,XXREAL_1:17;
A111: for x being set holds x in Q(n) /\ LSeg(pn,pn1) iff x = pn
    proof
      let x be set;
      thus x in Q(n) /\ LSeg(pn,pn1) implies x = pn
      proof
        assume
A112:   x in Q(n) /\ LSeg(pn,pn1);
        then
A113:   x in LSeg(pn,pn1) by XBOOLE_0:def 4;
        x in Q(n) by A112,XBOOLE_0:def 4;
        then x in union { LSeg(f|(n+2),k) where k is Nat :
        1 <= k & k+1 <= len(f|(n+2)) } by TOPREAL1:def 4;
        then consider X being set such that
A114:   x in X and
A115:   X in { LSeg(f|(n+2),k) where k is Nat :
        1 <= k & k+1 <= len(f|(n+2)) } by TARSKI:def 4;
        consider k be Nat such that
A116:   X = LSeg(f|(n+2),k) and
A117:   1 <= k and
A118:   k+1 <= len(f|(n+2)) by A115;
A119:   len(f|(n+2)) = n+(1+1) by A36,A43,FINSEQ_1:59,XXREAL_0:2;
A120:   len(f|(n+2)) = n+1+1 by A36,A43,FINSEQ_1:59,XXREAL_0:2;
        then
A121:   k <= n+1 by A118,XREAL_1:6;
A122:   f is s.n.c. by A3;
A123:   f is unfolded by A3;
        now
          assume
A124:     k < n+1;
A125:     1 <= 1+k by NAT_1:11;
A126:     k+1 <= len f by A44,A118,A120,XXREAL_0:2;
A127:     k+1 < n+1+1 by A124,XREAL_1:6;
          set p19 = f/.k, p29 = f/.(k+1);
          n+1 <= n+1+1 by NAT_1:11;
          then k <= n+2 by A124,XXREAL_0:2;
          then
A128:     k in Seg len(f|(n+2)) by A117,A119,FINSEQ_1:1;
A129:     k+1 in Seg len(f|(n+2)) by A118,A125,FINSEQ_1:1;
A130:     k in dom(f|(n+2)) by A128,FINSEQ_1:def 3;
A131:     k+1 in dom(f|(n+2)) by A129,FINSEQ_1:def 3;
A132:     (f|(n+2))/.k = p19 by A130,FINSEQ_4:70;
A133:     (f|(n+2))/.(k+1) = p29 by A131,FINSEQ_4:70;
A134:     LSeg(f,k) = LSeg(p19,p29) by A117,A126,TOPREAL1:def 3
            .= LSeg(f|(n+2),k) by A117,A118,A132,A133,TOPREAL1:def 3;
          LSeg(f,n+2) = LSeg(pn,pn1) by A36,A42,TOPREAL1:def 3;
          then LSeg(f|(n+2),k) misses LSeg(pn,pn1)
          by A122,A127,A134,TOPREAL1:def 7;
          then LSeg(f|(n+2),k) /\ LSeg(pn,pn1) = {};
          hence contradiction by A113,A114,A116,XBOOLE_0:def 4;
        end;
        then
A135:   k = n + 1 by A121,XXREAL_0:1;
A136:   1 <= n+1 by A117,A121,XXREAL_0:2;
A137:   n+1+1 <= len f by A36,A43,XXREAL_0:2;
        set p19 = f/.(n+1), p29 = f/.(n+1+1);
A138:   n+1 <= n+1+1 by NAT_1:11;
A139:   1 <= 1+n by NAT_1:11;
A140:   1 <= 1+(n+1) by NAT_1:11;
A141:   n+1 in Seg len(f|(n+2)) by A119,A138,A139,FINSEQ_1:1;
A142:   n+1+1 in Seg len(f|(n+2)) by A119,A140,FINSEQ_1:1;
A143:   n+1 in dom(f|(n+2)) by A141,FINSEQ_1:def 3;
A144:   n+1+1 in dom(f|(n+2)) by A142,FINSEQ_1:def 3;
A145:   (f|(n+2))/.(n+1) = p19 by A143,FINSEQ_4:70;
A146:   (f|(n+2))/.(n+1+1) = p29 by A144,FINSEQ_4:70;
A147:   LSeg(f,n+1) = LSeg(p19,p29) by A136,A137,TOPREAL1:def 3
          .= LSeg(f|(n+2),n+1) by A119,A139,A145,A146,TOPREAL1:def 3;
        LSeg(pn,pn1) = LSeg(f,n+1+1) by A36,A42,TOPREAL1:def 3;
        then
A148:   x in LSeg(f,n+1) /\ LSeg(f,n+1+1)
        by A113,A114,A116,A135,A147,XBOOLE_0:def 4;
        1 <= n+1 by NAT_1:11;
        then LSeg(f,n+1) /\ LSeg(f,n+1+1) = {f/.(n+1+1)}
        by A36,A123,TOPREAL1:def 6;
        hence thesis by A148,TARSKI:def 1;
      end;
      assume
A149: x = pn;
      then
A150: x in LSeg(pn,pn1) by RLTOPSP1:68;
A151: len(f|(n+2)) = n+2 by A36,A43,FINSEQ_1:59,XXREAL_0:2;
      then
A152: dom(f|(n+2)) = Seg(n+2) by FINSEQ_1:def 3;
      n + 2 in Seg(n+2) by A42,FINSEQ_1:1;
      then
A153: x = (f|(n+2))/.(n+1+1) by A149,A152,FINSEQ_4:70;
      1 <= n+1 by NAT_1:11;
      then
A154: x in LSeg(f|(n+2),n+1) by A151,A153,TOPREAL1:21;
A155: 1 <= n+1 by NAT_1:11;
      n+1+1 <= len(f|(n+2)) by A36,A43,FINSEQ_1:59,XXREAL_0:2;
      then LSeg(f|(n+2),n+1) in {LSeg(f|(n+2),k) where k is Nat :
      1 <= k & k+1 <= len(f|(n+2))} by A155;
      then x in union {LSeg(f|(n+2),k) where k is Nat :
      1 <= k & k+1 <= len(f|(n+2))} by A154,TARSKI:def 4;
      then x in Q(n) by TOPREAL1:def 4;
      hence thesis by A150,XBOOLE_0:def 4;
    end;
    f/.(n+2) in rng F19 by A101,A108,FUNCT_1:def 3;
    then
A156: {f/.(n+2)} c= rng F19 by ZFMISC_1:31;
A157: F1.:(dom F1 /\ dom F19) = Im(F1,1/2) by A96,A97,A109,XXREAL_1:17
      .= {f/.(n+2)} by A100,A107,FUNCT_1:59;
    then
A158: rng FF = Q(n) \/ LSeg(pn, pn1) by A89,A93,A156,TOPMETR2:2;
    then
A159: rng FF = [#]((TOP-REAL 2)|Q(n+1)) by A67,A68,A95,TOPREAL3:38;
    rng FF c= the carrier of ((TOP-REAL 2)|Q(n+1)) by A89,A93,A94,A95,A156,A157
,TOPMETR2:2;
    then reconsider FF as Function of I[01], (TOP-REAL 2)|NE1 by A102,
BORSUK_1:40,FUNCT_2:2;
A160: rng Ex1 = [#] Closed-Interval-TSpace(0,1) by A82,TOPS_2:def 5;
A161: 0 in { l where l is Real: 0 <= l & l <= 1/2 };
A162: 1/2 in { l where l is Real: 0 <= l & l <= 1/2 };
A163: 0 in dom Ex1 by A85,A161,RCOMP_1:def 1;
A164: 1/2 in dom Ex1 by A85,A162,RCOMP_1:def 1;
A165: Ex1 is one-to-one continuous by A82,TOPS_2:def 5;
A166: Ex1.0 = Ex1.(#)(0,1/2) by TREAL_1:def 1
      .= (#)(0,1) by TREAL_1:13
      .= 0 by TREAL_1:def 1;
A167:  Ex1 is onto by A160,FUNCT_2:def 3;
      then
A168: Ex1".0 = (Ex1 qua Function)".0 by A165,TOPS_2:def 4
      .= 0 by A163,A165,A166,FUNCT_1:32;
A169: Ex2.(1/2) = Ex2.(#)(1/2,1) by TREAL_1:def 1
      .= (#)(0,1) by TREAL_1:13
      .= 0 by TREAL_1:def 1;
A170: Ex1".1 = (Ex1 qua Function)".1 by A167,A165,TOPS_2:def 4
      .= 1/2 by A105,A164,A165,FUNCT_1:32;
A171: Ex2.1 = Ex2.(1/2,1)(#) by TREAL_1:def 2
      .= (0,1)(#) by TREAL_1:13
      .= 1 by TREAL_1:def 2;
A172: LSeg(pn, pn1) = F19.:[.1/2,1.] by A9,A93,RELSET_1:22;
A173: FF.(1/2) = f/.(n+2) by A101,A108,FUNCT_4:13;
A174: for x be set st x in [.0,1/2.] & x <> 1/2 holds not x in dom F19
    proof
      let x be set;
      assume that
A175: x in [.0,1/2.] and
A176: x <> 1/2;
      assume x in dom F19;
      then x in dom F1 /\ dom F19 by A96,A175,XBOOLE_0:def 4;
      hence thesis by A110,A176,TARSKI:def 1;
    end;
A177: FF.:[.1/2,1.] c= F19.:[.1/2,1.]
    proof
      let a be object;
      assume a in FF.:[.1/2,1.];
      then consider x be object such that
      x in dom FF and
A178: x in [.1/2,1.] and
A179: a = FF.x by FUNCT_1:def 6;
      FF.x = F19.x by A97,A178,FUNCT_4:13;
      hence thesis by A97,A178,A179,FUNCT_1:def 6;
    end;
    F19.:[.1/2,1.] c= FF.:[.1/2,1.]
    proof
      let a be object;
      assume a in F19.:[.1/2,1.];
      then consider x be object such that
A180: x in dom F19 and
A181: x in [.1/2,1.] and
A182: a = F19.x by FUNCT_1:def 6;
A183: x in dom FF by A180,FUNCT_4:12;
      a = FF.x by A180,A182,FUNCT_4:13;
      hence thesis by A181,A183,FUNCT_1:def 6;
    end;
    then
A184: FF.:[.1/2,1.] = F19.:[.1/2,1.] by A177;
    set GF = G" * FF;
    reconsider GF as Function of I[01],I[01];
A185: 0 in dom FF by A102,BORSUK_1:40,43;
A186: 1 in dom FF by A102,BORSUK_1:40,43;
    0 in { l where l is Real: 0 <= l & l <= 1/2 };
    then 0 in [.0,1/2.] by RCOMP_1:def 1;
    then
A187: FF.0 = F1.0 by A174,FUNCT_4:11
      .= f/.1 by A76,A163,A166,FUNCT_1:13;
    1 in { l where l is Real: 1/2 <= l & l <= 1 };
    then
A188: 1 in dom F19 by A97,RCOMP_1:def 1;
    then
A189: FF.1 = F19.1 by FUNCT_4:13
      .= pn1 by A81,A171,A188,FUNCT_1:12;
A190: 0 in dom G by A55,BORSUK_1:43;
A191:  G is onto by A54,FUNCT_2:def 3;
      then
A192: G".(f/.1) = (G qua Function)".(f/.1) by A53,TOPS_2:def 4
      .= 0 by A51,A53,A190,FUNCT_1:32;
    then
A193: GF.0 = 0 by A185,A187,FUNCT_1:13;
A194: 1 in dom G by A55,BORSUK_1:43;
A195: G".pn1 = (G qua Function)".pn1 by A191,A53,TOPS_2:def 4
      .= 1 by A52,A53,A194,FUNCT_1:32;
    then
A196: GF.1 = 1 by A186,A189,FUNCT_1:13;
    reconsider ppp = 1/2 as Point of I[01] by BORSUK_1:43;
    TopSpaceMetr RealSpace is T_2 by PCOMPS_1:34;
    then
A197: I[01] is T_2 by TOPMETR:2,20,def 6;
A198: T1 is compact by HEINE:4;
A199: T2 is compact by HEINE:4;
A200: F1 is continuous by A88,TOPS_2:def 5;
A201: F19 is continuous by A92,TOPS_2:def 5;
A202: (TOP-REAL 2)|NE is SubSpace of ((TOP-REAL 2)|NE1) by A38,A94,TOPMETR:4;
A203: (TOP-REAL 2)|LSeg(pn, pn1) is SubSpace of (TOP-REAL 2)|NE1 by A94,
TOPMETR:4;
A204: g11 is continuous by A200,A202,PRE_TOPC:26;
A205: g22 is continuous by A201,A203,PRE_TOPC:26;
A206: [#] T1 = [.0,1/2.] by TOPMETR:18;
A207: [#] T2 = [.1/2,1.] by TOPMETR:18;
    then
A208: ([#] T1) \/ ([#]T2) = [#] I[01] by A206,BORSUK_1:40,XXREAL_1:174;
    ([#] T1) /\ ([#]T2) = {ppp} by A206,A207,XXREAL_1:418;
    then reconsider FF as continuous Function of I[01],(TOP-REAL 2)|NE1
    by A107,A108,A197,A198,A199,A204,A205,A208,COMPTS_1:20;
A209: F1 is one-to-one by A88,TOPS_2:def 5;
A210: F19 is one-to-one by A92,TOPS_2:def 5;
    for x1,x2 be set st x1 in dom F19 & x2 in dom F1 \ dom F19 holds
    F19.x1 <> F1.x2
    proof
      let x1,x2 be set;
      assume that
A211: x1 in dom F19 and
A212: x2 in dom F1 \ dom F19;
      assume
A213: F19.x1 = F1.x2;
A214: F19.x1 in LSeg(pn, pn1) by A93,A211,FUNCT_2:4;
A215: x2 in dom F1 by A212,XBOOLE_0:def 5;
A216: not x2 in dom F19 by A212,XBOOLE_0:def 5;
      F1.x2 in NE by A38,A89,A212,FUNCT_2:4;
      then F1.x2 in NE /\ LSeg(pn, pn1) by A213,A214,XBOOLE_0:def 4;
      then F1.x2 = pn by A38,A111;
      hence thesis by A100,A101,A107,A209,A215,A216,FUNCT_1:def 4;
    end;
    then
A217: FF is one-to-one by A209,A210,TOPMETR2:1;
A218: G" is being_homeomorphism by A50,TOPS_2:56;
    FF is being_homeomorphism by A103,A104,A159,A217,COMPTS_1:17;
    then
A219: GF is being_homeomorphism by A218,TOPS_2:57;
    then
A220: GF is continuous by TOPS_2:def 5;
A221: dom GF = [#]I[01] by A219,TOPS_2:def 5;
A222: rng GF = [#]I[01] by A219,TOPS_2:def 5;
A223: GF is one-to-one by A219,TOPS_2:def 5;
    then
A224: dom (GF") = [#]I[01] by A222,TOPS_2:49;
A225: rng G = [#]((TOP-REAL 2)|Q(n+1)) by A50,TOPS_2:def 5
      .= rng FF by A67,A68,A95,A158,TOPREAL3:38;
A226: G * (G" * FF) = (FF qua Relation) * ( G * (G" qua Function) )
      by RELAT_1:36
      .= id rng G * FF by A53,A54,TOPS_2:52
      .= FF by A225,RELAT_1:54;
A227: 1/2 in dom GF by A221,BORSUK_1:43;
    then
A228: GF.(1/2) in rng GF by FUNCT_1:def 3;
A229: GF.(1/2) = G".(FF.(1/2)) by A227,FUNCT_1:12
      .= pp by A101,A108,FUNCT_4:13;
A230: [.pp,1.] c= GF.:[.1/2,1.]
    proof
      let a be object;
      assume a in [.pp,1.];
      then a in { l where l is Real: pp <= l & l <= 1 }
          by RCOMP_1:def 1;
      then consider l1 be Real such that
A231: l1 = a and
A232: pp <= l1 and
A233: l1 <= 1;
A234: 0 <= pp by A228,A229,BORSUK_1:43;
      set cos = GF".l1;
      l1 in dom (GF") by A224,A232,A233,A234,BORSUK_1:43;
      then
A235: cos in rng (GF") by FUNCT_1:def 3;
A236: l1 in rng GF by A222,A232,A233,A234,BORSUK_1:43;
      GF is onto by A222,FUNCT_2:def 3;
      then
A237: GF.cos = GF.((GF qua Function)".l1) by A223,TOPS_2:def 4
        .= l1 by A223,A236,FUNCT_1:35;
      reconsider cos as Real;
      reconsider A3 = cos, A4 = 1, A5 = 1/2 as Point of I[01]
      by A235,BORSUK_1:43;
      reconsider A1 = GF.A3, A2 = GF.A4 as Point of I[01];
      reconsider Fhc = A1, Fh0 = A2, Fh12 = GF.A5 as Real;
A238: cos <= 1
      proof
        assume cos > 1;
        then Fhc > Fh0 by A193,A196,A220,A223,JORDAN5A:16;
        hence thesis by A102,A189,A195,A233,A237,BORSUK_1:40,FUNCT_1:13;
      end;
      cos >= 1/2
      proof
        assume cos < 1/2;
        then Fhc < Fh12 by A193,A196,A220,A223,JORDAN5A:16;
        hence thesis by A102,A173,A232,A237,BORSUK_1:40,FUNCT_1:13;
      end;
      then cos in { l where l is Real: 1/2 <= l & l <= 1 } by A238;
      then cos in [.1/2,1.] by RCOMP_1:def 1;
      hence thesis by A221,A231,A235,A237,FUNCT_1:def 6;
    end;
    GF.:[.1/2,1.] c= [.pp,1.]
    proof
      let a be object;
      assume a in GF.:[.1/2,1.];
      then consider x be object such that
      x in dom GF and
A239: x in [.1/2,1.] and
A240: a = GF.x by FUNCT_1:def 6;
      x in { l where l is Real: 1/2 <= l & l <= 1 }
               by A239,RCOMP_1:def 1;
      then consider l1 be Real such that
A241: l1 = x and
A242: 1/2 <= l1 and
A243: l1 <= 1;
      reconsider ll = l1, pol = 1/2 as Point of I[01] by A242,A243,BORSUK_1:43;
      reconsider A1 = GF.1[01], A2 = GF.ll, A3 = GF.pol as Point of I[01];
      reconsider A4 = A1, A5 = A2, A6 = A3 as Real;
A244: A4 >= A5
      proof
        per cases;
        suppose 1 <> l1;
          then 1 > l1 by A243,XXREAL_0:1;
          hence thesis by A193,A196,A220,A223,BORSUK_1:def 15,JORDAN5A:16;
        end;
        suppose 1 = l1;
          hence thesis by BORSUK_1:def 15;
        end;
      end;
      A5 >= A6
      proof
        per cases;
        suppose l1 <> 1/2;
          then l1 > 1/2 by A242,XXREAL_0:1;
          hence thesis by A193,A196,A220,A223,JORDAN5A:16;
        end;
        suppose l1 = 1/2;
          hence thesis;
        end;
      end;
      then A5 in { l where l is Real: pp <= l & l <= 1 }
         by A196,A229,A244,BORSUK_1:def 15;
      hence thesis by A240,A241,RCOMP_1:def 1;
    end;
    then [.pp,1.] = GF.:[.1/2,1.] by A230;
    then
A245: G.:[.pp,1.] = LSeg (pn, pn1) by A172,A184,A226,RELAT_1:126;
    ex p1, p2 be Real st p1 < p2 &
    0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 & LSeg (f, j) = G.:[.p1, p2.]
    & G.p1 = f/.j & G.p2 = f/.(j+1)
    proof
      per cases;
      suppose j+1 <= n+2;
        then consider r1, r2 be Real such that
A246:   r1 < r2 and
A247:   0 <= r1 and
A248:   r1 <= 1 and 0 <= r2 and
A249:   r2 <= 1 and
A250:   LSeg (f, j) = F.:[.r1, r2.] and
A251:   F.r1 = f/.j and
A252:   F.r2 = f/.(j+1) by A39,A48,A75,A76,A77;
        set f1 = Ex1".r1, f2 = Ex1".r2;
A253:   Ex1 is continuous one-to-one by A82,TOPS_2:def 5;
A254:   dom Ex1 = [#]Closed-Interval-TSpace(0,1/2) by A82,TOPS_2:def 5;
A255:   rng Ex1 = [#]Closed-Interval-TSpace(0,1) by A82,TOPS_2:def 5;
        then
A256:    Ex1 is onto by FUNCT_2:def 3;
        then
A257:   f1 = (Ex1 qua Function)".r1 by A253,TOPS_2:def 4;
A258:   f2 = (Ex1 qua Function)".r2 by A253,A256,TOPS_2:def 4;
A259:   rng Ex1 = [#]I[01] by A82,TOPMETR:20,TOPS_2:def 5
          .= the carrier of I[01];
        then
A260:   r1 in rng Ex1 by A247,A248,BORSUK_1:43;
A261:   r2 in rng Ex1 by A246,A247,A249,A259,BORSUK_1:43;
A262:   f1 in the carrier of Closed-Interval-TSpace(0,1/2) by A253,A254,A257
,A260,FUNCT_1:32;
A263:   f2 in the carrier of Closed-Interval-TSpace(0,1/2) by A253,A254,A258
,A261,FUNCT_1:32;
A264:   f1 in [.0,1/2.] by A262,TOPMETR:18;
A265:   f2 in [.0,1/2.] by A263,TOPMETR:18;
        reconsider f1, f2 as Real;
        reconsider r19 = r1, r29 = r2 as Point of Closed-Interval-TSpace(0,1)
        by A246,A247,A248,A249,BORSUK_1:43,TOPMETR:20;
A266:   Ex1" is being_homeomorphism by A82,TOPS_2:56;
A267:   f1 = Ex1".r19;
A268:   f2 = Ex1".r29;
A269:   Ex1" is continuous one-to-one by A266,TOPS_2:def 5;
        then
A270:   f1 < f2 by A168,A170,A246,A267,A268,JORDAN5A:15;
A271:   [.0,1/2.] c= [.0,1.] by XXREAL_1:34;
A272:   r1 = Ex1.f1 by A253,A257,A260,FUNCT_1:32;
A273:   r2 = Ex1.f2 by A253,A258,A261,FUNCT_1:32;
A274:   f1 in { l where l is Real: 0 <= l & l <= 1/2 }
             by A264,RCOMP_1:def 1;
A275:   f2 in { l where l is Real: 0 <= l & l <= 1/2 }
               by A265,RCOMP_1:def 1;
A276:   ex ff1 be Real st ( ff1 = f1)&( 0 <= ff1)&( ff1 <= 1/2) by A274;
        ex ff2 be Real st ( ff2 = f2)&( 0 <= ff2)&( ff2 <= 1/2) by A275;
        then
A277:   Ex1.:
        [.f1,f2.] = [.r1,r2.] by A82,A105,A166,A270,A272,A273,A276,JORDAN5A:20;
A278:   F1.:[.f1,f2.] c= FF.:[.f1, f2.]
        proof
          let a be object;
          assume a in F1.:[.f1, f2.];
          then consider x be object such that
A279:     x in dom F1 and
A280:     x in [.f1, f2.] and
A281:     a = F1.x by FUNCT_1:def 6;
A282:     now per cases;
            suppose x <> 1/2;
              hence FF.x = F1.x by A10,A174,A279,FUNCT_4:11;
            end;
            suppose x = 1/2;
              hence FF.x = F1.x by A101,A107,A108,FUNCT_4:13;
            end;
          end;
          x in dom FF by A279,FUNCT_4:12;
          hence thesis by A280,A281,A282,FUNCT_1:def 6;
        end;
        FF.:[.f1,f2.] c= F1.:[.f1, f2.]
        proof
          let a be object;
          assume a in FF.:[.f1, f2.];
          then consider x be object such that
          x in dom FF and
A283:     x in [.f1, f2.] and
A284:     a = FF.x by FUNCT_1:def 6;
A285:     [.f1, f2.] c= [.0,1/2.] by A264,A265,XXREAL_2:def 12;
          now per cases;
            suppose x <> 1/2;
              hence FF.x = F1.x by A174,A283,A285,FUNCT_4:11;
            end;
            suppose x = 1/2;
              hence FF.x = F1.x by A101,A107,A108,FUNCT_4:13;
            end;
          end;
          hence thesis by A96,A283,A284,A285,FUNCT_1:def 6;
        end;
        then F1.:[.f1,f2.] = FF.:[.f1, f2.] by A278;
        then
A286:   F.:[.r1, r2.] = FF.:[.f1, f2.] by A277,RELAT_1:126;
        set e1 = GF.f1, e2 = GF.f2;
A287:   e1 in the carrier of I[01] by A264,A271,BORSUK_1:40,FUNCT_2:5;
A288:   e2 in the carrier of I[01] by A265,A271,BORSUK_1:40,FUNCT_2:5;
A289:   e1 in { l where l is Real: 0 <= l & l <= 1 }
         by A287,BORSUK_1:40,RCOMP_1:def 1;
A290:   e2 in { l where l is Real : 0 <= l & l <= 1 }
         by A288,BORSUK_1:40,RCOMP_1:def 1;
        consider l1 be Real such that
A291:   l1 = e1 and
A292:   0 <= l1 and
A293:   l1 <= 1 by A289;
        consider l2 be Real such that
A294:   l2 = e2 and 0 <= l2 and
A295:   l2 <= 1 by A290;
reconsider f19 = f1, f29 = f2 as Point of I[01] by A264,A265,A271,BORSUK_1:40;
A296:   GF.0 = 0 by A185,A187,A192,FUNCT_1:13;
A297:   GF.1 = 1 by A186,A189,A195,FUNCT_1:13;
A298:   l1 = GF.f19 by A291;
        l2 = GF.f29 by A294;
        then
A299:   l1 < l2 by A220,A223,A270,A296,A297,A298,JORDAN5A:16;
A300:   f1 < f2 by A168,A170,A246,A267,A268,A269,JORDAN5A:15;
A301:   0 <= f1 by A264,A271,BORSUK_1:40,43;
        f2 <= 1 by A265,A271,BORSUK_1:40,43;
        then
A302:   G.:[.l1, l2.] = G.:(GF.:[.f1, f2.])
        by A193,A196,A219,A291,A294,A300,A301,JORDAN5A:20,TOPMETR:20
          .= FF.:[.f1, f2.] by A226,RELAT_1:126;
A303:   FF.f19 = F.r19
        proof
          per cases;
          suppose
A304:       f19 = 1/2;
            then FF.f19 = F19.(1/2) by A101,FUNCT_4:13
              .= F9.0 by A101,A169,FUNCT_1:12
              .= F.r19 by A77,A80,A105,A253,A255,A257,A304,FUNCT_1:32;
            hence thesis;
          end;
          suppose f19 <> 1/2;
            then FF.f19 = F1.f19 by A174,A264,FUNCT_4:11
              .= F.(Ex1.f19) by A85,A264,FUNCT_1:13
              .= F.r19 by A253,A255,A257,FUNCT_1:32;
            hence thesis;
          end;
        end;
A305:   FF.f29 = F.r29
        proof
          per cases;
          suppose
A306:       f29 = 1/2;
            then FF.f29 = F19.(1/2) by A101,FUNCT_4:13
              .= F9.0 by A101,A169,FUNCT_1:12
              .= F.r29 by A77,A80,A105,A253,A255,A258,A306,FUNCT_1:32;
            hence thesis;
          end;
          suppose f29 <> 1/2;
            then FF.f29 = F1.f29 by A174,A265,FUNCT_4:11
              .= F.(Ex1.f29) by A85,A265,FUNCT_1:13
              .= F.r29 by A253,A255,A258,FUNCT_1:32;
            hence thesis;
          end;
        end;
A307:   G.l1 = f/.j by A102,A226,A251,A291,A303,BORSUK_1:40,FUNCT_1:12;
        G.l2 = f/.(j+1) by A102,A226,A252,A294,A305,BORSUK_1:40,FUNCT_1:12;
        hence thesis by A250,A286,A292,A293,A295,A299,A302,A307;
      end;
      suppose j+1 > n+2;
        then
A308:   j+1 = n+3 by A41,A49,NAT_1:9;
        then LSeg(f,j) = LSeg (pn, pn1) by A36,A42,TOPREAL1:def 3;
        hence thesis by A52,A64,A70,A71,A245,A308;
      end;
    end;
    hence thesis;
  end;
A309: for n be Nat holds ARC[n] from NAT_1:sch 2(A14,A33);
  1 <= h1 + 2 by A12,XXREAL_0:2;
  then ex NE being non empty Subset of TOP-REAL 2 st ( NE = Q(h1))
&( for j be Nat for F being Function of I[01], (TOP-REAL 2 )|NE st 1
  <= j & j+1 <= h1+2 & F is being_homeomorphism & F.0 = f/.1 & F.1 = f /.(h1+2)
ex p1, p2 be Real
     st p1 < p2 & 0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 & LSeg (f,
  j) = F.:[.p1, p2.] & F.p1 = f/.j & F.p2 = f/.(j+1)) by A309;
  hence thesis by A1,A2,A4,A5,A6,A7,A13;
end;
