reserve a, r, s for Real;

theorem Th22:
  for Y being non empty TopSpace, F being Function of [:Y,I[01]:],
  Tunit_circle(2), Ft being Function of [:Y,Sspace(0[01]):], R^1 st F is
continuous & Ft is continuous & F | [:the carrier of Y,{0}:] = CircleMap*Ft ex
G being Function of [:Y,I[01]:], R^1 st G is continuous & F = CircleMap*G & G |
  [:the carrier of Y,{0}:] = Ft & for H being Function of [:Y,I[01]:], R^1 st H
is continuous & F = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft holds G = H
proof
  consider UL being Subset-Family of TUC such that
A1: UL is Cover of TUC & UL is open and
A2: for U being Subset of TUC st U in UL ex D being mutually-disjoint
  open Subset-Family of R^1 st union D = (CircleMap)"U & for d being Subset of
R^1 st d in D for f being Function of R^1 | d, TUC | U st f = CircleMap|d holds
  f is being_homeomorphism by Lm13;
  let Y be non empty TopSpace, F be Function of [:Y,I[01]:], TUC, Ft be
  Function of [:Y,Sspace(0[01]):], R^1 such that
A3: F is continuous and
A4: Ft is continuous and
A5: F | [:the carrier of Y,{0}:] = CircleMap*Ft;
  defpred A[set,set] means ex y being Point of Y, t being Point of I[01], N
being non empty open Subset of Y, Fn being Function of [:Y|N,I[01]:], R^1 st $1
= [y,t] & $2 = Fn.$1 & y in N & Fn is continuous & F | [:N,I:] = CircleMap*Fn &
  Fn | [:the carrier of Y,{0}:] = Ft | [:N,I:] & for H being Function of [:Y|N,
I[01]:], R^1 st H is continuous & F | [:N,I:] = CircleMap*H & H | [:the carrier
  of Y,{0}:] = Ft | [:N,I:] holds Fn = H;
A6: dom F = the carrier of [:Y,I[01]:] by FUNCT_2:def 1
    .= [:the carrier of Y,I:] by BORSUK_1:def 2;
A7: the carrier of [:Y,Sspace(0[01]):] = [:the carrier of Y, the carrier of
  Sspace(0[01]):] by BORSUK_1:def 2;
  then
A8: dom Ft = [:the carrier of Y,{0}:] by Lm14,FUNCT_2:def 1;
A9: for x being Point of [:Y,I[01]:] ex z being Point of R^1 st A[x,z]
  proof
    let x be Point of [:Y,I[01]:];
    consider y being Point of Y, t being Point of I[01] such that
A10: x = [y,t] by BORSUK_1:10;
    consider TT being non empty FinSequence of REAL such that
A11: TT.1 = 0 and
A12: TT.len TT = 1 and
A13: TT is increasing and
A14: ex N being open Subset of Y st y in N & for i being Nat
    st i in dom TT & i+1 in dom TT ex Ui being non empty Subset of
Tunit_circle(2) st Ui in UL & F.:[:N,[.TT.i,TT.(i+1).]:] c= Ui by A3,A1,Th21;
    consider N being open Subset of Y such that
A15: y in N and
A16: for i being Nat st i in dom TT & i+1 in dom TT ex Ui
being non empty Subset of Tunit_circle(2) st Ui in UL & F.:[:N,[.TT.i,TT.(i+1)
    .]:] c= Ui by A14;
    reconsider N as non empty open Subset of Y by A15;
    defpred N[Nat] means $1 in dom TT implies ex N2 being non empty
open Subset of Y, S being non empty Subset of I[01], Fn being Function of [:Y|
N2,I[01]|S:], R^1 st S = [.0,TT.$1.] & y in N2 & N2 c= N & Fn is continuous & F
    | [:N2,S:] = CircleMap*Fn & Fn | [:the carrier of Y,{0}:] = Ft | [:N2,I:];
A17: len TT in dom TT by FINSEQ_5:6;
A18: 1 in dom TT by FINSEQ_5:6;
A19: now
      let i be Element of NAT such that
A20:  i in dom TT;
      1 <= i by A20,FINSEQ_3:25;
      then 1 = i or 1 < i by XXREAL_0:1;
      hence
A21:  0 <= TT.i by A11,A13,A18,A20,SEQM_3:def 1;
      assume
A22:  i+1 in dom TT;
A23:  i+0 < i+1 by XREAL_1:8;
      hence
A24:  TT.i < TT.(i+1) by A13,A20,A22,SEQM_3:def 1;
      i+1 <= len TT by A22,FINSEQ_3:25;
      then i+1 = len TT or i+1 < len TT by XXREAL_0:1;
      hence TT.(i+1) <= 1 by A12,A13,A17,A22,SEQM_3:def 1;
      hence TT.i < 1 by A24,XXREAL_0:2;
      thus 0 < TT.(i+1) by A13,A20,A21,A22,A23,SEQM_3:def 1;
    end;
A25: now
      let i be Nat such that
A26:  0 <= TT.i and
A27:  TT.(i+1) <= 1;
      thus [.TT.i,TT.(i+1).] c= I
      proof
        let a be object;
        assume
A28:    a in [.TT.i,TT.(i+1).];
        then reconsider a as Real;
        a <= TT.(i+1) by A28,XXREAL_1:1;
        then
A29:    a <= 1 by A27,XXREAL_0:2;
        0 <= a by A26,A28,XXREAL_1:1;
        hence thesis by A29,BORSUK_1:43;
      end;
    end;
A30: for i being Nat st N[i] holds N[i+1]
    proof
      let i be Nat;
      assume that
A31:  N[i] and
A32:  i+1 in dom TT;
      per cases by A32,TOPREALA:2;
      suppose
A33:    i = 0;
        take N2 = N;
        set Fn = Ft | [:N2,{0}:];
        set S = [.0,TT.(i+1).];
A34:    S = {0} by A11,A33,XXREAL_1:17;
        reconsider S as non empty Subset of I[01] by A11,A33,Lm3,XXREAL_1:17;
A35:    dom Fn = [:N2,S:] by A8,A34,RELAT_1:62,ZFMISC_1:96;
        reconsider K0 = [:N2,S:] as non empty Subset of [:Y,Sspace(0[01]):] by
A7,A34,Lm14,ZFMISC_1:96;
A36:    the carrier of [:Y|N2,I[01]|S:] = [:the carrier of (Y|N2), the
        carrier of ( I[01]|S):] & rng Fn c= R by BORSUK_1:def 2,RELAT_1:def 19;
        the carrier of (Y|N2) = N2 & the carrier of (I[01]|S) = S
        by PRE_TOPC:8;
        then reconsider Fn as Function of [:Y|N2,I[01]|S:], R^1 by A35,A36,
FUNCT_2:2;
A37:    dom (F | [:N2,S:]) = [:N2,S:] by A6,RELAT_1:62,ZFMISC_1:96;
        reconsider S1 = S as non empty Subset of Sspace(0[01]) by A11,A33,Lm14,
XXREAL_1:17;
        take S, Fn;
        thus S = [.0,TT.(i+1).];
        thus y in N2 by A15;
        thus N2 c= N;
        I[01]|S = Sspace(0[01]) by A34,TOPALG_3:5
          .= (Sspace(0[01]))|S1 by A34,Lm14,TSEP_1:3;
        then [:Y|N2,I[01]|S:] = [:Y,Sspace(0[01]):]|K0 by BORSUK_3:22;
        hence Fn is continuous by A4,A34,TOPMETR:7;
        rng Fn c= dom CircleMap by Lm12,TOPMETR:17;
        then
A38:    dom (CircleMap*Fn) = dom Fn by RELAT_1:27;
A39:    [:N2,S:] c= dom Ft by A8,A34,ZFMISC_1:96;
        for x being object st x in dom (F | [:N2,S:]) holds (F | [:N2,S:]).x
        = (CircleMap*Fn).x
        proof
          let x be object such that
A40:      x in dom (F | [:N2,S:]);
          thus (F | [:N2,S:]).x = F.x by A37,A40,FUNCT_1:49
            .= (CircleMap*Ft).x by A5,A7,A35,A37,A40,Lm14,FUNCT_1:49
            .= CircleMap.(Ft.x) by A39,A37,A40,FUNCT_1:13
            .= CircleMap.(Fn.x) by A34,A37,A40,FUNCT_1:49
            .= (CircleMap*Fn).x by A35,A37,A40,FUNCT_1:13;
        end;
        hence F | [:N2,S:] = CircleMap*Fn by A35,A37,A38;
A41:    dom (Fn | [:the carrier of Y,{0}:]) = [:N2,S:] /\ [:the carrier
        of Y,{0}:] by A35,RELAT_1:61;
A42:    for x being object st x in dom (Fn | [:the carrier of Y,{0}:]) holds
        (Fn | [:the carrier of Y,{0}:]).x = (Ft | [:N2,I:]).x
        proof
A43:      [:N2,{0}:] c= [:N2,I:] by Lm3,ZFMISC_1:95;
          let x be object such that
A44:      x in dom (Fn | [:the carrier of Y,{0}:]);
A45:      x in [:N2,{0}:] by A34,A41,A44,XBOOLE_0:def 4;
          x in [:the carrier of Y,{0}:] by A41,A44,XBOOLE_0:def 4;
          hence (Fn | [:the carrier of Y,{0}:]).x = Fn.x by FUNCT_1:49
            .= Ft.x by A45,FUNCT_1:49
            .= (Ft | [:N2,I:]).x by A45,A43,FUNCT_1:49;
        end;
        dom (Ft | [:N2,I:]) = [:the carrier of Y,{0}:] /\ [:N2,I:] by A8,
RELAT_1:61
          .= [:N2,S:] by A34,ZFMISC_1:101;
        hence thesis by A34,A41,A42,ZFMISC_1:101;
      end;
      suppose
A46:    i in dom TT;
        set SS = [.TT.i,TT.(i+1).];
        consider Ui being non empty Subset of TUC such that
A47:    Ui in UL and
A48:    F.:[:N,SS:] c= Ui by A16,A32,A46;
        consider D being mutually-disjoint open Subset-Family of R^1 such that
A49:    union D = (CircleMap)"Ui and
A50:    for d being Subset of R^1 st d in D for f being Function of
R^1 | d, TUC | Ui st f = CircleMap|d holds f is being_homeomorphism by A2,A47;
A51:    the carrier of (TUC | Ui) = Ui by PRE_TOPC:8;
A52:    TT.i < TT.(i+1) by A19,A32,A46;
        then TT.i in SS by XXREAL_1:1;
        then
A53:    [y,TT.i] in [:N,SS:] by A15,ZFMISC_1:87;
        consider N2 being open Subset of Y, S being non empty Subset of I[01],
        Fn being Function of [:Y|N2,I[01]|S:], R^1 such that
A54:    S = [.0,TT.i.] and
A55:    y in N2 and
A56:    N2 c= N and
A57:    Fn is continuous and
A58:    F | [:N2,S:] = CircleMap*Fn and
A59:    Fn | [:the carrier of Y,{0}:] = Ft | [:N2,I:] by A31,A46;
        reconsider N2 as non empty open Subset of Y by A55;
A60:    the carrier of [:Y|N2,I[01]|S:] = [:the carrier of (Y|N2), the
        carrier of (I[01]|S):] by BORSUK_1:def 2;
        N2 c= N2;
        then reconsider N7 = N2 as non empty Subset of Y|N2 by PRE_TOPC:8;
A61:    dom Fn = the carrier of [:Y|N2,I[01]|S:] by FUNCT_2:def 1;
A62:    0 <= TT.i by A19,A46;
        then
A63:    TT.i in S by A54,XXREAL_1:1;
        then reconsider Ti = {TT.i} as non empty Subset of I[01] by ZFMISC_1:31
;
A64:    the carrier of I[01]|S = S by PRE_TOPC:8;
        then reconsider Ti2 = Ti as non empty Subset of I[01]|S by A63,
ZFMISC_1:31;
        set FnT = Fn | [:N2,Ti:];
A65:    the carrier of [:Y|N2,I[01]|Ti:] = [:the carrier of Y|N2, the
        carrier of I[01]|Ti:] & rng FnT c= REAL by BORSUK_1:def 2;
A66:    [:N2,SS:] c= [:N,SS:] by A56,ZFMISC_1:96;
A67:    the carrier of (Y|N2) = N2 by PRE_TOPC:8;
        {TT.i} c= S by A63,ZFMISC_1:31;
        then
A68:    dom FnT = [:N2,{TT.i}:] by A64,A60,A67,A61,RELAT_1:62,ZFMISC_1:96;
A69:    [:Y|N2|N7,I[01]|S|Ti2:] = [:Y|N2,I[01]|S:] | [:N7,Ti2:] by BORSUK_3:22;
A70:    the carrier of I[01]|Ti = Ti by PRE_TOPC:8;
        rng FnT c= the carrier of R^1 by RELAT_1:def 19;
        then reconsider FnT as Function of [:Y|N2,I[01]|Ti:],R^1
        by A67,A68,A65,A70,FUNCT_2:2;
        Y|N2|N7 = Y|N2 & I[01]|S|Ti2 = I[01]|Ti by GOBOARD9:2;
        then
A71:    FnT is continuous by A57,A69,TOPMETR:7;
A72:     Fn. [y,TT.i] in REAL by XREAL_0:def 1;
        [y,TT.i] in dom F by A6,A63,ZFMISC_1:87;
        then
A73:    F. [y,TT.i] in F.:[:N,SS:] by A53,FUNCT_2:35;
A74:    [y,TT.i] in [:N2,S:] by A55,A63,ZFMISC_1:87;
        then F. [y,TT.i] = (CircleMap*Fn). [y,TT.i] by A58,FUNCT_1:49
          .= CircleMap.(Fn. [y,TT.i]) by A64,A60,A67,A74,FUNCT_2:15;
        then Fn. [y,TT.i] in (CircleMap)"Ui
               by A48,A73,FUNCT_2:38,TOPMETR:17,A72;
        then consider Uit being set such that
A75:    Fn. [y,TT.i] in Uit and
A76:    Uit in D by A49,TARSKI:def 4;
        reconsider Uit as non empty Subset of R^1 by A75,A76;
        [#]R^1 <> {} & Uit is open by A76,TOPS_2:def 1;
        then FnT"Uit is open by A71,TOPS_2:43;
        then consider SF being Subset-Family of [:Y|N2,I[01]|Ti:] such that
A77:    FnT"Uit = union SF and
A78:    for e being set st e in SF ex X1 being Subset of Y|N2, Y1
        being Subset of I[01]|Ti st e = [:X1,Y1:] & X1 is open & Y1 is open by
BORSUK_1:5;
A79:    TT.i in {TT.i} by TARSKI:def 1;
        then
A80:    [y,TT.i] in [:N2,{TT.i}:] by A55,ZFMISC_1:def 2;
        then FnT. [y,TT.i] in Uit by A75,FUNCT_1:49;
        then [y,TT.i] in FnT"Uit by A80,A68,FUNCT_1:def 7;
        then consider N5 being set such that
A81:    [y,TT.i] in N5 and
A82:    N5 in SF by A77,TARSKI:def 4;
        set f = CircleMap|Uit;
A83:    dom f = Uit by Lm12,RELAT_1:62,TOPMETR:17;
A84:    rng f c= Ui
        proof
          let b be object;
          assume b in rng f;
          then consider a being object such that
A85:      a in dom f and
A86:      f.a = b by FUNCT_1:def 3;
          a in union D by A76,A83,A85,TARSKI:def 4;
          then CircleMap.a in Ui by A49,FUNCT_2:38;
          hence thesis by A83,A85,A86,FUNCT_1:49;
        end;
        consider X1 being Subset of Y|N2, Y1 being Subset of I[01]|Ti such
        that
A87:    N5 = [:X1,Y1:] and
A88:    X1 is open and
        Y1 is open by A78,A82;
        the carrier of (R^1 | Uit) = Uit by PRE_TOPC:8;
        then reconsider f as Function of R^1 | Uit, TUC | Ui
        by A51,A83,A84,FUNCT_2:2;
        consider NY being Subset of Y such that
A89:    NY is open and
A90:    NY /\ [#](Y|N2) = X1 by A88,TOPS_2:24;
        consider y1, y2 being object such that
A91:    y1 in X1 and
A92:    y2 in Y1 and
A93:    [y,TT.i] = [y1,y2] by A81,A87,ZFMISC_1:def 2;
        set N1 = NY /\ N2;
        y = y1 by A93,XTUPLE_0:1;
        then
A94:    y in NY by A90,A91,XBOOLE_0:def 4;
        then reconsider N1 as non empty open Subset of Y by A55,A89,
XBOOLE_0:def 4;
A95:    N1 c= N2 by XBOOLE_1:17;
        then [:N1,SS:] c= [:N2,SS:] by ZFMISC_1:96;
        then [:N1,SS:] c= [:N,SS:] by A66;
        then
A96:    F.:[:N1,SS:] c= F.:[:N,SS:] by RELAT_1:123;
        TT.(i+1) <= 1 by A19,A32,A46;
        then reconsider SS as non empty Subset of I[01] by A25,A62,A52,
XXREAL_1:1;
A97:    dom (F| [:N1,SS:]) = [:N1,SS:] by A6,RELAT_1:62,ZFMISC_1:96;
        set Fni1 = f"*(F| [:N1,SS:]);
        f" is being_homeomorphism by A50,A76,TOPS_2:56;
        then
A98:    dom (f") = [#](TUC | Ui);
A99:    rng (F| [:N1,SS:]) c= dom (f")
        proof
          let b be object;
          assume b in rng (F| [:N1,SS:]);
          then consider a being object such that
A100:      a in dom (F| [:N1,SS:]) and
A101:     (F| [:N1,SS:]).a = b by FUNCT_1:def 3;
          b = F.a by A97,A100,A101,FUNCT_1:49;
          then b in F.:[:N1,SS:] by A97,A100,FUNCT_2:35;
          then b in F.:[:N,SS:] by A96;
          then b in Ui by A48;
          hence thesis by A98,PRE_TOPC:8;
        end;
        then
A102:   dom Fni1 = dom (F| [:N1,SS:]) by RELAT_1:27;
        set Fn2 = Fn | [:N1,S:];
A103:   the carrier of (Y|N1) = N1 by PRE_TOPC:8;
        then
A104:   [:N1,S:] = the carrier of [:Y|N1,I[01]|S:] by A64,BORSUK_1:def 2;
        then
A105:   dom Fn2 = the carrier of [:Y|N1,I[01]|S:] by A64,A60,A67,A61,A95,
RELAT_1:62,ZFMISC_1:96;
        reconsider ff = f as Function;
A106:   f is being_homeomorphism by A50,A76;
        then
A107:   f is one-to-one;
A108:   rng Fn2 c= R by RELAT_1:def 19;
AAA:    rng Fni1 c= the carrier of (R^1 | Uit) by RELAT_1:def 19;
        the carrier of (R^1 | Uit) is Subset of R^1 by TSEP_1:1;
        then
A109:   rng Fni1 c= R by AAA,XBOOLE_1:1;
A110:   the carrier of (I[01]|SS) = SS by PRE_TOPC:8;
        then
A111:   [:N1,SS:] = the carrier of [:Y|N1,I[01]|SS:] by A103,BORSUK_1:def 2;
        then reconsider
        Fni1 as Function of [:Y|N1,I[01]|SS:], R^1 by A97,A102,A109,FUNCT_2:2;
        reconsider Fn2 as Function of [:Y|N1,I[01]|S:],R^1
        by A105,A108,FUNCT_2:2;
        set Fn1 = Fn2 +* Fni1;
A112:   rng Fn1 c= rng Fn2 \/ rng Fni1 by FUNCT_4:17;
        dom (Fn | [:N1,S:]) = [:N1,S:]
        by A64,A60,A67,A61,A95,RELAT_1:62,ZFMISC_1:96;
        then
A113:   dom Fn1 = [:N1,S:] \/ [:N1,SS:] by A97,A102,FUNCT_4:def 1;
A114:   rng f = [#](TUC | Ui) by A106;
        then f is onto;
        then
A115:   f" = ff" by A107,TOPS_2:def 4;
A116:   Y1 = Ti
        proof
          thus Y1 c= Ti by A70;
          let a be object;
          assume a in Ti;
          then a = TT.i by TARSKI:def 1;
          hence thesis by A92,A93,XTUPLE_0:1;
        end;
A117:   Fn.:[:N1,{TT.i}:] c= Uit
        proof
          let b be object;
          assume b in Fn.:[:N1,{TT.i}:];
          then consider a being Point of [:Y|N2,I[01]|S:] such that
A118:     a in [:N1,{TT.i}:] and
A119:     Fn.a = b by FUNCT_2:65;
          a in N5 by A87,A90,A116,A118,PRE_TOPC:def 5;
          then
A120:     a in union SF by A82,TARSKI:def 4;
          then a in dom FnT by A77,FUNCT_1:def 7;
          then Fn.a = FnT.a by FUNCT_1:47;
          hence thesis by A77,A119,A120,FUNCT_1:def 7;
        end;
A121:   for p being set st p in ([#][:Y|N1,I[01]|S:]) /\ [#][:Y|N1,I[01]
        |SS:] holds Fn2.p = Fni1.p
        proof
A122:     the carrier of (Y|N2) = N2 by PRE_TOPC:8;
          let p be set such that
A123:     p in ([#][:Y|N1,I[01]|S:]) /\ [#][:Y|N1,I[01]|SS:];
A124:     p in ([#][:Y|N1,I[01]|SS:]) /\ [#][:Y|N1,I[01]|S:] by A123;
          then
A125:     Fn.p = Fn2.p by A104,FUNCT_1:49;
          [:N1,S:] /\ [:N1,SS:] = [:N1,S /\ SS:] by ZFMISC_1:99;
          then
A126:     p in [:N1,{TT.i}:] by A54,A62,A52,A111,A104,A123,XXREAL_1:418;
          then consider
          p1 being Element of N1, p2 being Element of {TT.i} such
          that
A127:     p = [p1,p2] by DOMAIN_1:1;
A128:     p1 in N1;
          S /\ SS = {TT.i} by A54,A62,A52,XXREAL_1:418;
          then p2 in S by XBOOLE_0:def 4;
          then
A129:     p in [:N2,S:] by A95,A127,A128,ZFMISC_1:def 2;
          then
A130:     Fn.p in Fn.:[:N1,{TT.i}:] by A64,A60,A67,A126,FUNCT_2:35;
          (F| [:N1,SS:]).p = F.p by A111,A123,FUNCT_1:49
            .= (F | [:N2,S:]).p by A129,FUNCT_1:49
            .= CircleMap.(Fn.p) by A58,A64,A60,A61,A122,A129,FUNCT_1:13
            .= (CircleMap|Uit).(Fn.p) by A117,A130,FUNCT_1:49
            .= ff.(Fn2.p) by A104,A124,FUNCT_1:49;
          hence Fn2.p = ff".((F| [:N1,SS:]).p) by A117,A83,A107,A125,A130,
FUNCT_1:32
            .= Fni1.p by A115,A97,A111,A123,FUNCT_1:13;
        end;
A131:   [:N1,S:] c= [:N2,S:] by A95,ZFMISC_1:96;
        then reconsider
        K0 = [:N1,S:] as Subset of [:Y|N2,I[01]|S:] by A64,A60,PRE_TOPC:8;
A132:   [:N1,SS:] c= dom F by A6,ZFMISC_1:96;
        reconsider gF = F| [:N1,SS:] as Function of [:Y|N1,I[01]|SS:], TUC by
A97,A99,A111,FUNCT_2:2;
        reconsider fF = F| [:N1,SS:] as Function of [:Y|N1,I[01]|SS:], TUC |
        Ui by A98,A97,A99,A111,FUNCT_2:2;
        [:Y|N1,I[01]|SS:] = [:Y,I[01]:] | [:N1,SS:] by BORSUK_3:22;
        then gF is continuous by A3,TOPMETR:7;
        then
A133:   fF is continuous by TOPMETR:6;
        f" is continuous by A106;
        then f"*fF is continuous by A133;
        then
A134:   Fni1 is continuous by PRE_TOPC:26;
        reconsider aN1 = N1 as non empty Subset of Y|N2 by A95,PRE_TOPC:8;
        S c= S;
        then reconsider aS = S as non empty Subset of I[01]|S by PRE_TOPC:8;
        [:Y|N2,I[01]|S:] | K0 = [:Y|N2|aN1,I[01]|S|aS:] by BORSUK_3:22
          .= [:Y|N1,I[01]|S|aS:] by GOBOARD9:2
          .= [:Y|N1,I[01]|S:] by GOBOARD9:2;
        then
A135:   Fn2 is continuous by A57,TOPMETR:7;
        take N1;
        take S1 = S \/ SS;
A136:   [:N1,S:] \/ [:N1,SS:] = [:N1,S1:] by ZFMISC_1:97;
A137:   the carrier of (I[01]|S1) = S1 by PRE_TOPC:8;
        then [:N1,S1:] = the carrier of [:Y|N1,I[01]|S1:]
        by A103,BORSUK_1:def 2;
        then reconsider Fn1 as Function of [:Y|N1,I[01]|S1:], R^1
        by A136,A113,A112,FUNCT_2:2,XBOOLE_1:1;
        take Fn1;
        thus
A138:   S1 = [.0,TT.(i+1).] by A54,A62,A52,XXREAL_1:165;
        0 <= TT.(i+1) by A19,A32;
        then 0 in S1 by A138,XXREAL_1:1;
        then
A139:   {0} c= S1 by ZFMISC_1:31;
A140:   dom (Fn1 | [:the carrier of Y,{0}:]) = dom Fn1 /\ [:the carrier
        of Y,{0}:] by RELAT_1:61;
        then
A141:   dom (Fn1 | [:the carrier of Y,{0}:]) = [:N1 /\ the carrier of Y,
        S1 /\ {0}:] by A136,A113,ZFMISC_1:100
          .= [:N1,S1 /\ {0}:] by XBOOLE_1:28
          .= [:N1,{0}:] by A139,XBOOLE_1:28;
A142:   for a being object st a in dom (Fn1 | [:the carrier of Y,{0}:])
        holds (Fn1 | [:the carrier of Y,{0}:]).a = (Ft | [:N1,I:]).a
        proof
          let a be object;
A143:     [:N1,I:] c= [:N2,I:] by A95,ZFMISC_1:96;
          assume
A144:     a in dom (Fn1 | [:the carrier of Y,{0}:]);
          then
A145:     a in [:the carrier of Y,{0}:] by A140,XBOOLE_0:def 4;
          then consider a1, a2 being object such that
          a1 in the carrier of Y and
A146:     a2 in {0} and
A147:     a = [a1,a2] by ZFMISC_1:def 2;
A148:     a2 = 0 by A146,TARSKI:def 1;
          0 in S by A54,A62,XXREAL_1:1;
          then {0} c= S by ZFMISC_1:31;
          then
A149:     [:N1,{0}:] c= [:N1,S:] by ZFMISC_1:96;
          then
A150:     a in [:N1,S:] by A141,A144;
A151:     [:N1,S:] c= [:N1,I:] by ZFMISC_1:96;
          then
A152:     a in [:N1,I:] by A150;
          per cases;
          suppose
A153:       not a in dom Fni1;
            thus (Fn1 | [:the carrier of Y,{0}:]).a = Fn1.a by A145,FUNCT_1:49
              .= (Fn | [:N1,S:]).a by A153,FUNCT_4:11
              .= Fn.a by A141,A144,A149,FUNCT_1:49
              .= (Ft | [:N2,I:]).a by A59,A145,FUNCT_1:49
              .= Ft.a by A152,A143,FUNCT_1:49
              .= (Ft | [:N1,I:]).a by A150,A151,FUNCT_1:49;
          end;
          suppose
A154:       a in dom Fni1;
            set e = (Ft | [:N1,I:]).a;
            a in [:N1,SS:] by A6,A102,A154,RELAT_1:62,ZFMISC_1:96;
            then consider b1, b2 being object such that
A155:       b1 in N1 and
A156:       b2 in SS and
A157:       a = [b1,b2] by ZFMISC_1:def 2;
            a2 = b2 by A147,A157,XTUPLE_0:1;
            then
A158:       a2 = TT.i by A62,A148,A156,XXREAL_1:1;
            a1 = b1 by A147,A157,XTUPLE_0:1;
            then
A159:       [a1,TT.i] in [:N1,S:] & [a1,TT.i] in [:N1,{TT.i}:] by A63,A79,A155,
ZFMISC_1:87;
            e = Ft.a by A150,A151,FUNCT_1:49
              .= (Ft | [:N2,I:]).a by A152,A143,FUNCT_1:49
              .= Fn.a by A59,A145,FUNCT_1:49;
            then
A160:       e in Fn.:[:N1,{TT.i}:] by A64,A60,A67,A61,A131,A147,A158,A159,
FUNCT_1:def 6;
            then
A161:       ff.e = CircleMap.e by A117,FUNCT_1:49
              .= CircleMap.(Ft.a) by A150,A151,FUNCT_1:49
              .= (CircleMap*Ft).a by A8,A145,FUNCT_1:13
              .= F.a by A5,A145,FUNCT_1:49;
            thus (Fn1 | [:the carrier of Y,{0}:]).a = Fn1.a by A145,FUNCT_1:49
              .= Fni1.a by A154,FUNCT_4:13
              .= ff".((F| [:N1,SS:]).a) by A115,A102,A154,FUNCT_1:13
              .= ff".(F.a) by A97,A102,A154,FUNCT_1:49
              .= (Ft | [:N1,I:]).a by A117,A83,A107,A160,A161,FUNCT_1:32;
          end;
        end;
A162:   rng Fn1 c= dom CircleMap by Lm12,TOPMETR:17;
        then
A163:   dom (CircleMap*Fn1) = dom Fn1 by RELAT_1:27;
A164:   for a being object st a in dom (CircleMap*Fn1) holds (CircleMap*Fn1
        ).a = F . a
        proof
          let a be object such that
A165:     a in dom (CircleMap*Fn1);
          per cases;
          suppose
A166:       a in dom Fni1;
A167:       [:N1,SS:] c= [:the carrier of Y,I:] by ZFMISC_1:96;
A168:       a in [:N1,SS:] by A6,A102,A166,RELAT_1:62,ZFMISC_1:96;
            then F.a in F.:[:N1,SS:] by A6,A167,FUNCT_1:def 6;
            then
A169:       F.a in F.:[:N,SS:] by A96;
            then a in F"(dom (ff")) by A6,A48,A51,A98,A115,A168,A167,
FUNCT_1:def 7;
            then
A170:       a in dom (ff"*F) by RELAT_1:147;
            thus (CircleMap*Fn1).a = CircleMap.(Fn1.a) by A165,FUNCT_2:15
              .= CircleMap.(Fni1.a) by A166,FUNCT_4:13
              .= CircleMap.(f".((F| [:N1,SS:]).a)) by A102,A166,FUNCT_1:13
              .= CircleMap.(f".(F.a)) by A97,A102,A166,FUNCT_1:49
              .= CircleMap.((ff"*F).a) by A132,A115,A97,A102,A166,FUNCT_1:13
              .= (CircleMap*(ff"*F)).a by A170,FUNCT_1:13
              .= (CircleMap*ff"*F).a by RELAT_1:36
              .= (CircleMap*ff").(F.a) by A132,A97,A102,A166,FUNCT_1:13
              .= F.a by A48,A51,A114,A107,A169,TOPALG_3:2;
          end;
          suppose
A171:       not a in dom Fni1;
            then
A172:       a in [:N1,S:] by A97,A102,A113,A163,A165,XBOOLE_0:def 3;
            thus (CircleMap*Fn1).a = CircleMap.(Fn1.a) by A165,FUNCT_2:15
              .= CircleMap.(Fn | [:N1,S:].a) by A171,FUNCT_4:11
              .= CircleMap.(Fn.a) by A172,FUNCT_1:49
              .= (CircleMap*Fn).a by A64,A60,A67,A131,A172,FUNCT_2:15
              .= F.a by A58,A131,A172,FUNCT_1:49;
          end;
        end;
A173:   S c= S1 by XBOOLE_1:7;
        then
A174:   [#](I[01]|S1) = the carrier of (I[01]|S1) & I[01]|S is SubSpace
        of I[01]|S1 by A64,A137,TSEP_1:4;
A175:   SS c= S1 by XBOOLE_1:7;
        then reconsider
        F1 = [#](I[01]|S), F2 = [#](I[01]|SS) as Subset of I[01]|S1
        by A137,A173,PRE_TOPC:8;
        reconsider hS = F1, hSS = F2 as Subset of I[01] by PRE_TOPC:8;
        hS is closed by A54,BORSUK_4:23,PRE_TOPC:8;
        then
A176:   F1 is closed by TSEP_1:8;
        thus y in N1 by A55,A94,XBOOLE_0:def 4;
        thus N1 c= N by A56,A95;
        hSS is closed by BORSUK_4:23,PRE_TOPC:8;
        then
A177:   F2 is closed by TSEP_1:8;
        I[01]|SS is SubSpace of I[01]|S1 by A110,A137,A175,TSEP_1:4;
        then ex h being Function of [:Y|N1,I[01]|S1:],R^1 st h = Fn2+*Fni1 &
h is continuous by A64,A110,A137,A174,A176,A177,A135,A134,A121,TOPALG_3:19;
        hence Fn1 is continuous;
        dom Fn1 = dom F /\ [:N1,S1:] by A6,A136,A113,XBOOLE_1:28,ZFMISC_1:96;
        hence F | [:N1,S1:] = CircleMap*Fn1 by A162,A164,FUNCT_1:46,RELAT_1:27;
        dom (Ft | [:N1,I:]) = dom Ft /\ [:N1,I:] by RELAT_1:61
          .= [:(the carrier of Y) /\ N1,{0} /\ I:] by A8,ZFMISC_1:100
          .= [:N1,{0} /\ I:] by XBOOLE_1:28
          .= [:N1,{0}:] by Lm3,XBOOLE_1:28;
        hence thesis by A141,A142;
      end;
    end;
A178: N[ 0 ] by FINSEQ_3:24;
    for i being Nat holds N[i] from NAT_1:sch 2(A178,A30);
    then consider
    N2 being non empty open Subset of Y, S being non empty Subset of
    I[01], Fn1 being Function of [:Y|N2,I[01]|S:], R^1 such that
A179: S = [.0,TT.len TT.] and
A180: y in N2 and
A181: N2 c= N and
A182: Fn1 is continuous and
A183: F | [:N2,S:] = CircleMap*Fn1 and
A184: Fn1 | [:the carrier of Y,{0}:] = Ft | [:N2,I:] by A17;
    Fn1.x in REAL by XREAL_0:def 1;
    then reconsider z = Fn1.x as Point of R^1 by TOPMETR:17;
A185: I[01]|S = I[01] by A12,A179,Lm6,BORSUK_1:40,TSEP_1:3;
    then reconsider Fn1 as Function of [:Y|N2,I[01]:], R^1;
    take z, y, t, N2;
    take Fn1;
    thus x = [y,t] & z = Fn1.x & y in N2 & Fn1 is continuous & F | [:N2,I:] =
CircleMap*Fn1 & Fn1 | [:the carrier of Y,{0}:] = Ft | [:N2,I:] by A10,A12,A179
,A180,A182,A183,A184,A185,BORSUK_1:40;
    let H be Function of [:Y|N2,I[01]:], R^1 such that
A186: H is continuous and
A187: F | [:N2,I:] = CircleMap*H and
A188: H | [:the carrier of Y,{0}:] = Ft | [:N2,I:];
    defpred M[Nat] means $1 in dom TT implies ex Z being non empty
    Subset of I[01] st Z = [.0,TT.$1.] & Fn1 | [:N2,Z:] = H | [:N2,Z:];
A189: dom Fn1 = the carrier of [:Y|N2,I[01]:] by FUNCT_2:def 1;
A190: the carrier of [:Y|N2,I[01]:] = [:the carrier of Y|N2,I:] & the
    carrier of Y |N2 = N2 by BORSUK_1:def 2,PRE_TOPC:8;
A191: dom H = the carrier of [:Y|N2,I[01]:] by FUNCT_2:def 1;
A192: for i being Nat st M[i] holds M[i+1]
    proof
      let i be Nat such that
A193: M[i] and
A194: i+1 in dom TT;
      per cases by A194,TOPREALA:2;
      suppose
A195:   i = 0;
        set Z = [.0,TT.(i+1).];
A196:   Z = {0} by A11,A195,XXREAL_1:17;
        reconsider Z as non empty Subset of I[01] by A11,A195,Lm3,XXREAL_1:17;
A197:   [:N2,Z:] c= [:N2,I:] by ZFMISC_1:95;
        then
A198:   dom (Fn1 | [:N2,Z:]) = [:N2,Z:] by A190,A189,RELAT_1:62;
A199:   for x being object st x in dom (Fn1 | [:N2,Z:]) holds (Fn1 | [:N2,Z
        :]).x = (H | [:N2,Z:]).x
        proof
          let x be object;
A200:     [:N2,Z:] c= [:the carrier of Y,Z:] by ZFMISC_1:95;
          assume
A201:     x in dom (Fn1 | [:N2,Z:]);
          hence (Fn1 | [:N2,Z:]).x = Fn1.x by A198,FUNCT_1:49
            .= (Fn1 | [:the carrier of Y,{0}:]).x by A196,A198,A201,A200,
FUNCT_1:49
            .= H.x by A184,A188,A196,A198,A201,A200,FUNCT_1:49
            .= (H | [:N2,Z:]).x by A198,A201,FUNCT_1:49;
        end;
        take Z;
        thus Z = [.0,TT.(i+1).];
        dom (H | [:N2,Z:]) = [:N2,Z:] by A191,A190,A197,RELAT_1:62;
        hence thesis by A189,A199,RELAT_1:62;
      end;
      suppose
A202:   i in dom TT;
        set ZZ = [.TT.i,TT.(i+1).];
A203:   0 <= TT.i by A19,A202;
A204:   TT.(i+1) <= 1 by A19,A194,A202;
        then reconsider ZZ as Subset of I[01] by A25,A203;
        consider Z being non empty Subset of I[01] such that
A205:   Z = [.0,TT.i.] and
A206:   Fn1 | [:N2,Z:] = H | [:N2,Z:] by A193,A202;
        take Z1 = Z \/ ZZ;
A207:   TT.i < TT.(i+1) by A19,A194,A202;
        hence Z1 = [.0,TT.(i+1).] by A205,A203,XXREAL_1:165;
A208:   [:N2,Z1:] c= [:N2,I:] by ZFMISC_1:95;
        then
A209:   dom (Fn1 | [:N2,Z1:]) = [:N2,Z1:] by A190,A189,RELAT_1:62;
A210:   for x being object st x in dom (Fn1 | [:N2,Z1:]) holds (Fn1 | [:N2,
        Z1:]).x = (H | [:N2,Z1:]).x
        proof
          0 <= TT.(i+1) by A19,A194;
          then
A211:     TT.(i+1) is Point of I[01] by A204,BORSUK_1:43;
          0 <= TT.i & TT.i <= 1 by A19,A194,A202;
          then TT.i is Point of I[01] by BORSUK_1:43;
          then
A212:     ZZ is connected by A207,A211,BORSUK_4:24;
          consider Ui being non empty Subset of TUC such that
A213:     Ui in UL and
A214:     F.:[:N,ZZ:] c= Ui by A16,A194,A202;
          consider D being mutually-disjoint open Subset-Family of R^1 such
          that
A215:     union D = (CircleMap)"Ui and
A216:     for d being Subset of R^1 st d in D for f being Function
of R^1 | d, TUC | Ui st f = CircleMap|d holds f is being_homeomorphism by A2
,A213;
          let x be object such that
A217:     x in dom (Fn1 | [:N2,Z1:]);
          consider x1, x2 being object such that
A218:     x1 in N2 and
A219:     x2 in Z1 and
A220:     x = [x1,x2] by A209,A217,ZFMISC_1:def 2;
A221:     TT.i in ZZ by A207,XXREAL_1:1;
          then [x1,TT.i] in [:N,ZZ:] by A181,A218,ZFMISC_1:87;
          then
A222:     F. [x1,TT.i] in F.:[:N,ZZ:] by FUNCT_2:35;
          reconsider xy = {x1} as non empty Subset of Y by A218,ZFMISC_1:31;
A223:     xy c= N2 by A218,ZFMISC_1:31;
          then reconsider xZZ = [:xy,ZZ:] as Subset of [:Y|N2,I[01]:] by A190,
ZFMISC_1:96;
A224:     dom (H | [:xy,ZZ:]) = [:xy,ZZ:] by A191,A190,A223,RELAT_1:62
,ZFMISC_1:96;
A225:     D is Cover of Fn1.:xZZ
          proof
            let b be object;
A226:       [:N,ZZ:] c= [:the carrier of Y,I:] by ZFMISC_1:96;
            assume b in Fn1.:xZZ;
            then consider a being Point of [:Y|N2,I[01]:] such that
A227:       a in xZZ and
A228:       Fn1.a = b by FUNCT_2:65;
            xy c= N by A181,A218,ZFMISC_1:31;
            then [:xy,ZZ:] c= [:N,ZZ:] by ZFMISC_1:95;
            then a in [:N,ZZ:] by A227;
            then
A229:       F.a in F.:[:N,ZZ:] by A6,A226,FUNCT_1:def 6;
            CircleMap.(Fn1.a) = (CircleMap*Fn1).a by FUNCT_2:15
              .= F.a by A12,A179,A183,A190,BORSUK_1:40,FUNCT_1:49;
            hence b in union D by A214,A215,A228,A229,Lm12,FUNCT_1:def 7
,TOPMETR:17;
          end;
A230:     D is Cover of H.:xZZ
          proof
            let b be object;
A231:       [:N,ZZ:] c= [:the carrier of Y,I:] by ZFMISC_1:96;
            assume b in H.:xZZ;
            then consider a being Point of [:Y|N2,I[01]:] such that
A232:       a in xZZ and
A233:       H.a = b by FUNCT_2:65;
A234:       CircleMap.(H.a) = (CircleMap*H).a by FUNCT_2:15
              .= F.a by A187,A190,FUNCT_1:49;
            xy c= N by A181,A218,ZFMISC_1:31;
            then [:xy,ZZ:] c= [:N,ZZ:] by ZFMISC_1:95;
            then a in [:N,ZZ:] by A232;
            then F.a in F.:[:N,ZZ:] by A6,A231,FUNCT_1:def 6;
            hence b in union D by A214,A215,A233,A234,Lm12,FUNCT_1:def 7
,TOPMETR:17;
          end;
A235:       H. [x1,TT.i] in REAL by XREAL_0:def 1;
          TT.i in Z by A205,A203,XXREAL_1:1;
          then
A236:     [x1,TT.i] in [:N2,Z:] by A218,ZFMISC_1:87;
          then
A237:     Fn1. [x1,TT.i] = (Fn1 | [:N2,Z:]). [x1,TT.i] by FUNCT_1:49
            .= H. [x1,TT.i] by A206,A236,FUNCT_1:49;
A238:     [:N2,Z:] c= [:N2,I:] by ZFMISC_1:95;
          then F. [x1,TT.i] = (CircleMap*H). [x1,TT.i] by A187,A236,FUNCT_1:49
            .= CircleMap.(H. [x1,TT.i]) by A191,A190,A236,A238,FUNCT_1:13;
          then H. [x1,TT.i] in (CircleMap)"Ui by A214,A222,FUNCT_2:38,A235
,TOPMETR:17;
          then consider Uith being set such that
A239:     H. [x1,TT.i] in Uith and
A240:     Uith in D by A215,TARSKI:def 4;
A241:       Fn1. [x1,TT.i] in REAL by XREAL_0:def 1;
          F. [x1,TT.i] = (CircleMap*Fn1). [x1,TT.i] by A12,A179,A183,A236,A238,
BORSUK_1:40,FUNCT_1:49
            .= CircleMap.(Fn1. [x1,TT.i]) by A190,A189,A236,A238,FUNCT_1:13;
          then Fn1. [x1,TT.i] in (CircleMap)"Ui by A214,A222,FUNCT_2:38,A241
,TOPMETR:17;
          then consider Uit being set such that
A242:     Fn1. [x1,TT.i] in Uit and
A243:     Uit in D by A215,TARSKI:def 4;
          I[01] is SubSpace of I[01] by TSEP_1:2;
          then
A244:     [:Y|N2,I[01]:] is SubSpace of [:Y,I[01]:] by BORSUK_3:21;
          xy is connected by A218;
          then [:xy,ZZ:] is connected by A212,TOPALG_3:16;
          then
A245:     xZZ is connected by A244,CONNSP_1:23;
          reconsider Uith as non empty Subset of R^1 by A239,A240;
A246:     x1 in xy by TARSKI:def 1;
          then
A247:     [x1,TT.i] in xZZ by A221,ZFMISC_1:87;
          then H. [x1,TT.i] in H.:xZZ by FUNCT_2:35;
          then Uith meets H.:xZZ by A239,XBOOLE_0:3;
          then
A248:     H.:xZZ c= Uith by A186,A245,A240,A230,TOPALG_3:12,TOPS_2:61;
          reconsider Uit as non empty Subset of R^1 by A242,A243;
          set f = CircleMap|Uit;
A249:     dom f = Uit by Lm12,RELAT_1:62,TOPMETR:17;
A250:     rng f c= Ui
          proof
            let b be object;
            assume b in rng f;
            then consider a being object such that
A251:       a in dom f and
A252:       f.a = b by FUNCT_1:def 3;
            a in union D by A243,A249,A251,TARSKI:def 4;
            then CircleMap.a in Ui by A215,FUNCT_2:38;
            hence thesis by A249,A251,A252,FUNCT_1:49;
          end;
          the carrier of (TUC | Ui) = Ui & the carrier of (R^1 | Uit) =
          Uit by PRE_TOPC:8;
          then reconsider f as Function of R^1 | Uit, TUC | Ui by A249,A250,
FUNCT_2:2;
A253:     dom (Fn1 | [:xy,ZZ:]) = [:xy,ZZ:] by A190,A189,A223,RELAT_1:62
,ZFMISC_1:96;
          H. [x1,TT.i] in H.:xZZ by A191,A247,FUNCT_1:def 6;
          then Uit meets Uith by A242,A248,A237,XBOOLE_0:3;
          then
A254:     Uit = Uith by A243,A240,TAXONOM2:def 5;
A255:     rng (H | [:xy,ZZ:]) c= dom f
          proof
            let b be object;
            assume b in rng (H | [:xy,ZZ:]);
            then consider a being object such that
A256:       a in dom (H | [:xy,ZZ:]) and
A257:       (H | [:xy,ZZ:]).a = b by FUNCT_1:def 3;
            H.a = b by A224,A256,A257,FUNCT_1:49;
            then b in H.:xZZ by A191,A224,A256,FUNCT_1:def 6;
            hence thesis by A248,A254,A249;
          end;
          Fn1. [x1,TT.i] in Fn1.:xZZ by A247,FUNCT_2:35;
          then Uit meets Fn1.:xZZ by A242,XBOOLE_0:3;
          then
A258:     Fn1.:xZZ c= Uit by A182,A185,A243,A245,A225,TOPALG_3:12,TOPS_2:61;
A259:     rng (Fn1 | [:xy,ZZ:]) c= dom f
          proof
            let b be object;
            assume b in rng (Fn1 | [:xy,ZZ:]);
            then consider a being object such that
A260:       a in dom (Fn1 | [:xy,ZZ:]) and
A261:       (Fn1 | [:xy,ZZ:]).a = b by FUNCT_1:def 3;
            Fn1.a = b by A253,A260,A261,FUNCT_1:49;
            then b in Fn1.:xZZ by A189,A253,A260,FUNCT_1:def 6;
            hence thesis by A258,A249;
          end;
          then
A262:     dom (f*(Fn1 | [:xy,ZZ:])) = dom (Fn1 | [:xy,ZZ:]) by RELAT_1:27;
A263:     for x being object st x in dom (f*(Fn1 | [:xy,ZZ:])) holds (f*(
          Fn1 | [:xy,ZZ:])).x = (f*(H | [:xy,ZZ:])).x
          proof
            let x be object such that
A264:       x in dom (f*(Fn1 | [:xy,ZZ:]));
A265:       Fn1.x in Fn1.:[:xy,ZZ:] by A189,A253,A262,A264,FUNCT_1:def 6;
A266:       H.x in H.:[:xy,ZZ:] by A191,A253,A262,A264,FUNCT_1:def 6;
            thus (f*(Fn1 | [:xy,ZZ:])).x = ((f*Fn1) | [:xy,ZZ:]).x by
RELAT_1:83
              .= (f*Fn1).x by A253,A262,A264,FUNCT_1:49
              .= f.(Fn1.x) by A189,A264,FUNCT_1:13
              .= CircleMap.(Fn1.x) by A258,A265,FUNCT_1:49
              .= (CircleMap*Fn1).x by A189,A264,FUNCT_1:13
              .= CircleMap.(H.x) by A12,A179,A183,A187,A191,A264,BORSUK_1:40
,FUNCT_1:13
              .= f.(H.x) by A248,A254,A266,FUNCT_1:49
              .= (f*H).x by A191,A264,FUNCT_1:13
              .= ((f*H) | [:xy,ZZ:]).x by A253,A262,A264,FUNCT_1:49
              .= (f*(H | [:xy,ZZ:])).x by RELAT_1:83;
          end;
          f is being_homeomorphism by A216,A243;
          then
A267:     f is one-to-one;
          dom (f*(H | [:xy,ZZ:])) = dom (H | [:xy,ZZ:]) by A255,RELAT_1:27;
          then
A268:     f*(Fn1 | [:xy,ZZ:]) = f*(H | [:xy,ZZ:]) by A253,A224,A259,A263,
RELAT_1:27;
          per cases;
          suppose
            x2 in ZZ;
            then
A269:       x in [:xy,ZZ:] by A220,A246,ZFMISC_1:87;
            thus (Fn1 | [:N2,Z1:]).x = Fn1.x by A209,A217,FUNCT_1:49
              .= (Fn1 | [:xy,ZZ:]).x by A269,FUNCT_1:49
              .= (H | [:xy,ZZ:]).x by A267,A253,A224,A259,A255,A268,FUNCT_1:27
              .= H.x by A269,FUNCT_1:49
              .= (H | [:N2,Z1:]).x by A209,A217,FUNCT_1:49;
          end;
          suppose
            not x2 in ZZ;
            then x2 in Z by A219,XBOOLE_0:def 3;
            then
A270:       x in [:N2,Z:] by A218,A220,ZFMISC_1:87;
            thus (Fn1 | [:N2,Z1:]).x = Fn1.x by A209,A217,FUNCT_1:49
              .= (Fn1 | [:N2,Z:]).x by A270,FUNCT_1:49
              .= H.x by A206,A270,FUNCT_1:49
              .= (H | [:N2,Z1:]).x by A209,A217,FUNCT_1:49;
          end;
        end;
        dom (H | [:N2,Z1:]) = [:N2,Z1:] by A191,A190,A208,RELAT_1:62;
        hence thesis by A189,A210,RELAT_1:62;
      end;
    end;
A271: M[ 0 ] by FINSEQ_3:24;
    for i being Nat holds M[i] from NAT_1:sch 2(A271,A192);
    then consider Z being non empty Subset of I[01] such that
A272: Z = [.0,TT.len TT.] and
A273: Fn1 | [:N2,Z:] = H | [:N2,Z:] by A17;
    thus Fn1 = Fn1 | [:N2,Z:] by A12,A190,A189,A272,BORSUK_1:40,RELAT_1:69
      .= H by A12,A191,A190,A272,A273,BORSUK_1:40,RELAT_1:69;
  end;
  consider G being Function of [:Y,I[01]:],R^1 such that
A274: for x being Point of [:Y,I[01]:] holds A[x,G.x] from FUNCT_2:sch 3
  (A9 );
  take G;
A275: now
    let N be Subset of Y, F be Function of [:Y|N,I[01]:],R^1;
    thus dom F = the carrier of [:Y|N,I[01]:] by FUNCT_2:def 1
      .= [:the carrier of Y|N,I:] by BORSUK_1:def 2
      .= [:N,I:] by PRE_TOPC:8;
  end;
A276: for p being Point of [:Y,I[01]:], y being Point of Y, t being Point of
  I[01], N1, N2 being non empty open Subset of Y, Fn1 being Function of [:Y|N1,
I[01]:],R^1, Fn2 being Function of [:Y|N2,I[01]:],R^1 st p = [y,t] & y in N1 &
y in N2 & Fn2 is continuous & Fn1 is continuous & F | [:N2,I:] = CircleMap*Fn2
& Fn2 | [:the carrier of Y,{0}:] = Ft | [:N2,I:] & F | [:N1,I:] = CircleMap*Fn1
& Fn1 | [:the carrier of Y,{0}:] = Ft | [:N1,I:] holds Fn1 | [:{y},I:] = Fn2 |
  [:{y},I:]
  proof
    let p be Point of [:Y,I[01]:], y be Point of Y, t be Point of I[01], N1,
N2 be non empty open Subset of Y, Fn1 be Function of [:Y|N1,I[01]:],R^1, Fn2 be
    Function of [:Y|N2,I[01]:],R^1 such that
    p = [y,t] and
A277: y in N1 and
A278: y in N2 and
A279: Fn2 is continuous and
A280: Fn1 is continuous and
A281: F | [:N2,I:] = CircleMap*Fn2 and
A282: Fn2 | [:the carrier of Y,{0}:] = Ft | [:N2,I:] and
A283: F | [:N1,I:] = CircleMap*Fn1 and
A284: Fn1 | [:the carrier of Y,{0}:] = Ft | [:N1,I:];
A285: {y} c= N1 by A277,ZFMISC_1:31;
    consider TT being non empty FinSequence of REAL such that
A286: TT.1 = 0 and
A287: TT.len TT = 1 and
A288: TT is increasing and
A289: ex N being open Subset of Y st y in N & for i being Nat
    st i in dom TT & i+1 in dom TT ex Ui being non empty Subset of
Tunit_circle(2) st Ui in UL & F.:[:N,[.TT.i,TT.(i+1).]:] c= Ui by A3,A1,Th21;
    consider N being open Subset of Y such that
A290: y in N and
A291: for i being Nat st i in dom TT & i+1 in dom TT ex Ui
being non empty Subset of Tunit_circle(2) st Ui in UL & F.:[:N,[.TT.i,TT.(i+1)
    .]:] c= Ui by A289;
    reconsider N as non empty open Subset of Y by A290;
    defpred M[Nat] means $1 in dom TT implies ex Z being non empty
    Subset of I[01] st Z = [.0,TT.$1.] & Fn1 | [:{y},Z:] = Fn2 | [:{y},Z:];
A292: len TT in dom TT by FINSEQ_5:6;
A293: dom Fn2 = the carrier of [:Y|N2,I[01]:] by FUNCT_2:def 1;
A294: dom Fn2 = [:N2,I:] by A275;
A295: {y} c= N2 by A278,ZFMISC_1:31;
A296: the carrier of [:Y|N1,I[01]:] = [:the carrier of Y|N1,I:] & the
    carrier of Y |N1 = N1 by BORSUK_1:def 2,PRE_TOPC:8;
A297: the carrier of [:Y|N2,I[01]:] = [:the carrier of Y|N2,I:] & the
    carrier of Y |N2 = N2 by BORSUK_1:def 2,PRE_TOPC:8;
A298: dom Fn1 = [:N1,I:] by A275;
A299: dom Fn1 = the carrier of [:Y|N1,I[01]:] by FUNCT_2:def 1;
A300: 1 in dom TT by FINSEQ_5:6;
A301: for i being Nat st M[i] holds M[i+1]
    proof
      let i be Nat such that
A302: M[i] and
A303: i+1 in dom TT;
      per cases by A303,TOPREALA:2;
      suppose
A304:   i = 0;
        set Z = [.0,TT.(i+1).];
A305:   Z = {0} by A286,A304,XXREAL_1:17;
        reconsider Z as non empty Subset of I[01] by A286,A304,Lm3,XXREAL_1:17;
A306:   [:{y},Z:] c= [:N2,I:] by A295,ZFMISC_1:96;
A307:   dom (Fn1 | [:{y},Z:]) = [:{y},Z:] by A285,A298,RELAT_1:62,ZFMISC_1:96;
A308:   [:{y},Z:] c= [:N1,I:] by A285,ZFMISC_1:96;
A309:   for x being object st x in dom (Fn1 | [:{y},Z:]) holds (Fn1 | [:{y}
        ,Z:]).x = (Fn2 | [:{y},Z:]).x
        proof
          let x be object;
A310:     [:{y},Z:] c= [:the carrier of Y,Z:] by ZFMISC_1:95;
          assume
A311:     x in dom (Fn1 | [:{y},Z:]);
          hence (Fn1 | [:{y},Z:]).x = Fn1.x by A307,FUNCT_1:49
            .= (Fn1 | [:the carrier of Y,{0}:]).x by A305,A307,A311,A310,
FUNCT_1:49
            .= Ft.x by A284,A308,A307,A311,FUNCT_1:49
            .= (Ft | [:N2,I:]).x by A307,A306,A311,FUNCT_1:49
            .= Fn2.x by A282,A305,A307,A311,A310,FUNCT_1:49
            .= (Fn2 | [:{y},Z:]).x by A307,A311,FUNCT_1:49;
        end;
        take Z;
        thus Z = [.0,TT.(i+1).];
        dom (Fn2 | [:{y},Z:]) = [:{y},Z:] by A295,A294,RELAT_1:62,ZFMISC_1:96;
        hence thesis by A307,A309;
      end;
      suppose
A312:   i in dom TT;
A313:   now
          let i be Element of NAT such that
A314:     i in dom TT;
          1 <= i by A314,FINSEQ_3:25;
          then 1 = i or 1 < i by XXREAL_0:1;
          hence
A315:     0 <= TT.i by A286,A288,A300,A314,SEQM_3:def 1;
          assume
A316:     i+1 in dom TT;
A317:     i+0 < i+1 by XREAL_1:8;
          hence
A318:     TT.i < TT.(i+1) by A288,A314,A316,SEQM_3:def 1;
          i+1 <= len TT by A316,FINSEQ_3:25;
          then i+1 = len TT or i+1 < len TT by XXREAL_0:1;
          hence TT.(i+1) <= 1 by A287,A288,A292,A316,SEQM_3:def 1;
          hence TT.i < 1 by A318,XXREAL_0:2;
          thus 0 < TT.(i+1) by A288,A314,A315,A316,A317,SEQM_3:def 1;
        end;
        then
A319:   0 <= TT.i by A312;
A320:   TT.(i+1) <= 1 by A303,A312,A313;
        set ZZ = [.TT.i,TT.(i+1).];
        consider Z being non empty Subset of I[01] such that
A321:   Z = [.0,TT.i.] and
A322:   Fn1 | [:{y},Z:] = Fn2 | [:{y},Z:] by A302,A312;
        now
          let i be Nat such that
A323:     0 <= TT.i and
A324:     TT.(i+1) <= 1;
          thus [.TT.i,TT.(i+1).] c= I
          proof
            let a be object;
            assume
A325:       a in [.TT.i,TT.(i+1).];
            then reconsider a as Real;
            a <= TT.(i+1) by A325,XXREAL_1:1;
            then
A326:       a <= 1 by A324,XXREAL_0:2;
            0 <= a by A323,A325,XXREAL_1:1;
            hence thesis by A326,BORSUK_1:43;
          end;
        end;
        then reconsider ZZ as Subset of I[01] by A319,A320;
        take Z1 = Z \/ ZZ;
A327:   TT.i < TT.(i+1) by A303,A312,A313;
        hence Z1 = [.0,TT.(i+1).] by A321,A319,XXREAL_1:165;
A328:   dom (Fn1 | [:{y},Z1:]) = [:{y},Z1:] by A285,A298,RELAT_1:62,ZFMISC_1:96
;
A329:   for x being object st x in dom (Fn1 | [:{y},Z1:]) holds (Fn1 | [:{y
        },Z1:]).x = (Fn2 | [:{y},Z1:]).x
        proof
          0 <= TT.(i+1) by A303,A313;
          then
A330:     TT.(i+1) is Point of I[01] by A320,BORSUK_1:43;
          0 <= TT.i & TT.i <= 1 by A303,A312,A313;
          then TT.i is Point of I[01] by BORSUK_1:43;
          then
A331:     ZZ is connected by A327,A330,BORSUK_4:24;
A332:     TT.i in ZZ by A327,XXREAL_1:1;
          consider Ui being non empty Subset of TUC such that
A333:     Ui in UL and
A334:     F.:[:N,ZZ:] c= Ui by A291,A303,A312;
          consider D being mutually-disjoint open Subset-Family of R^1 such
          that
A335:     union D = (CircleMap)"Ui and
A336:     for d being Subset of R^1 st d in D for f being Function
of R^1 | d, TUC | Ui st f = CircleMap|d holds f is being_homeomorphism by A2
,A333;
          let x be object such that
A337:     x in dom (Fn1 | [:{y},Z1:]);
          consider x1, x2 being object such that
A338:     x1 in {y} and
A339:     x2 in Z1 and
A340:     x = [x1,x2] by A328,A337,ZFMISC_1:def 2;
          reconsider xy = {x1} as non empty Subset of Y by A338,ZFMISC_1:31;
A341:     xy c= N2 by A295,A338,ZFMISC_1:31;
          then
A342:     [:xy,ZZ:] c= [:N2,I:] by ZFMISC_1:96;
A343:     x1 = y by A338,TARSKI:def 1;
          then [x1,TT.i] in [:N,ZZ:] by A290,A332,ZFMISC_1:87;
          then
A344:     F. [x1,TT.i] in F.:[:N,ZZ:] by FUNCT_2:35;
A345:     xy c= N1 by A285,A338,ZFMISC_1:31;
          then reconsider xZZ = [:xy,ZZ:] as Subset of [:Y|N1,I[01]:] by A296,
ZFMISC_1:96;
          xy is connected by A338;
          then
A346:     [:xy,ZZ:] is connected by A331,TOPALG_3:16;
A347:     xy c= N by A290,A343,ZFMISC_1:31;
A348:     D is Cover of Fn1.:xZZ
          proof
            let b be object;
A349:       [:N,ZZ:] c= [:the carrier of Y,I:] by ZFMISC_1:96;
            assume b in Fn1.:xZZ;
            then consider a being Point of [:Y|N1,I[01]:] such that
A350:       a in xZZ and
A351:       Fn1.a = b by FUNCT_2:65;
A352:       CircleMap.(Fn1.a) = (CircleMap*Fn1).a by FUNCT_2:15
              .= F.a by A283,A296,FUNCT_1:49;
            [:xy,ZZ:] c= [:N,ZZ:] by A347,ZFMISC_1:95;
            then a in [:N,ZZ:] by A350;
            then F.a in F.:[:N,ZZ:] by A6,A349,FUNCT_1:def 6;
            hence b in union D by A334,A335,A351,A352,Lm12,FUNCT_1:def 7
,TOPMETR:17;
          end;
A353:     I[01] is SubSpace of I[01] by TSEP_1:2;
          then [:Y|N1,I[01]:] is SubSpace of [:Y,I[01]:] by BORSUK_3:21;
          then
A354:     xZZ is connected by A346,CONNSP_1:23;
          reconsider XZZ = [:xy,ZZ:] as Subset of [:Y|N2,I[01]:] by A297,A341,
ZFMISC_1:96;
          [:Y|N2,I[01]:] is SubSpace of [:Y,I[01]:] by A353,BORSUK_3:21;
          then
A355:     XZZ is connected by A346,CONNSP_1:23;
A356:     D is Cover of Fn2.:xZZ
          proof
            let b be object;
A357:       [:N,ZZ:] c= [:the carrier of Y,I:] by ZFMISC_1:96;
            assume b in Fn2.:xZZ;
            then consider a being Point of [:Y|N2,I[01]:] such that
A358:       a in xZZ and
A359:       Fn2.a = b by FUNCT_2:65;
A360:       CircleMap.(Fn2.a) = (CircleMap*Fn2).a by FUNCT_2:15
              .= F.a by A281,A297,FUNCT_1:49;
            [:xy,ZZ:] c= [:N,ZZ:] by A347,ZFMISC_1:95;
            then a in [:N,ZZ:] by A358;
            then F.a in F.:[:N,ZZ:] by A6,A357,FUNCT_1:def 6;
            hence b in union D by A334,A335,A359,A360,Lm12,FUNCT_1:def 7
,TOPMETR:17;
          end;
A361:     TT.i in Z by A321,A319,XXREAL_1:1;
          then
A362:     [x1,TT.i] in [:{y},Z:] by A338,ZFMISC_1:87;
A363:       Fn1. [x1,TT.i] in REAL by XREAL_0:def 1;
A364:     [:{y},Z:] c= [:N1,I:] by A285,ZFMISC_1:96;
          then F. [x1,TT.i] = (CircleMap*Fn1). [x1,TT.i] by A283,A362,
FUNCT_1:49
            .= CircleMap.(Fn1. [x1,TT.i]) by A298,A362,A364,FUNCT_1:13;
          then Fn1. [x1,TT.i] in (CircleMap)"Ui by A334,A344,FUNCT_2:38,A363
,TOPMETR:17;
          then consider Uit being set such that
A365:     Fn1. [x1,TT.i] in Uit and
A366:     Uit in D by A335,TARSKI:def 4;
          reconsider Uit as non empty Subset of R^1 by A365,A366;
          set f = CircleMap|Uit;
A367:     dom f = Uit by Lm12,RELAT_1:62,TOPMETR:17;
A368:     rng f c= Ui
          proof
            let b be object;
            assume b in rng f;
            then consider a being object such that
A369:       a in dom f and
A370:       f.a = b by FUNCT_1:def 3;
            a in union D by A366,A367,A369,TARSKI:def 4;
            then CircleMap.a in Ui by A335,FUNCT_2:38;
            hence thesis by A367,A369,A370,FUNCT_1:49;
          end;
          the carrier of (TUC | Ui) = Ui & the carrier of (R^1 | Uit) =
          Uit by PRE_TOPC:8;
          then reconsider f as Function of R^1 | Uit, TUC | Ui by A367,A368,
FUNCT_2:2;
A371:     [:N2,Z:] c= [:N2,I:] by ZFMISC_1:95;
A372:       Fn2. [x1,TT.i] in REAL by XREAL_0:def 1;
A373:     [x1,TT.i] in [:N2,Z:] by A295,A338,A361,ZFMISC_1:87;
          then F. [x1,TT.i] = (CircleMap*Fn2). [x1,TT.i] by A281,A371,
FUNCT_1:49
            .= CircleMap.(Fn2. [x1,TT.i]) by A293,A297,A373,A371,FUNCT_1:13;
          then Fn2. [x1,TT.i] in (CircleMap)"Ui by A334,A344,FUNCT_2:38,A372
,TOPMETR:17;
          then consider Uith being set such that
A374:     Fn2. [x1,TT.i] in Uith and
A375:     Uith in D by A335,TARSKI:def 4;
          reconsider Uith as non empty Subset of R^1 by A374,A375;
A376:     dom (Fn1 | [:xy,ZZ:]) = [:xy,ZZ:] by A296,A299,A345,RELAT_1:62
,ZFMISC_1:96;
A377:     x1 in xy by TARSKI:def 1;
          then
A378:     [x1,TT.i] in xZZ by A332,ZFMISC_1:87;
          then Fn1. [x1,TT.i] in Fn1.:xZZ by FUNCT_2:35;
          then Uit meets Fn1.:xZZ by A365,XBOOLE_0:3;
          then
A379:     Fn1.:xZZ c= Uit by A280,A366,A354,A348,TOPALG_3:12,TOPS_2:61;
A380:     rng (Fn1 | [:xy,ZZ:]) c= dom f
          proof
            let b be object;
            assume b in rng (Fn1 | [:xy,ZZ:]);
            then consider a being object such that
A381:       a in dom (Fn1 | [:xy,ZZ:]) and
A382:       (Fn1 | [:xy,ZZ:]).a = b by FUNCT_1:def 3;
            Fn1.a = b by A376,A381,A382,FUNCT_1:49;
            then b in Fn1.:xZZ by A299,A376,A381,FUNCT_1:def 6;
            hence thesis by A379,A367;
          end;
          then
A383:     dom (f*(Fn1 | [:xy,ZZ:])) = dom (Fn1 | [:xy,ZZ:]) by RELAT_1:27;
          [x1,TT.i] in [:xy,ZZ:] by A338,A343,A332,ZFMISC_1:87;
          then [x1,TT.i] in dom Fn2 by A294,A342;
          then
A384:     Fn2. [x1,TT.i] in Fn2.:xZZ by A378,FUNCT_2:35;
          then Uith meets Fn2.:xZZ by A374,XBOOLE_0:3;
          then
A385:     Fn2.:xZZ c= Uith by A279,A375,A355,A356,TOPALG_3:12,TOPS_2:61;
          Fn1. [x1,TT.i] = (Fn1 | [:{y},Z:]). [x1,TT.i] by A362,FUNCT_1:49
            .= Fn2. [x1,TT.i] by A322,A362,FUNCT_1:49;
          then Uit meets Uith by A365,A384,A385,XBOOLE_0:3;
          then
A386:     Uit = Uith by A366,A375,TAXONOM2:def 5;
A387:     for x being object st x in dom (f*(Fn1 | [:xy,ZZ:])) holds (f*(
          Fn1 | [:xy,ZZ:])).x = (f*(Fn2 | [:xy,ZZ:])).x
          proof
A388:       dom (Fn1 | [:xy,ZZ:]) c= dom Fn1 by RELAT_1:60;
            let x be object such that
A389:       x in dom (f*(Fn1 | [:xy,ZZ:]));
A390:       Fn1.x in Fn1.:[:xy,ZZ:] by A299,A376,A383,A389,FUNCT_1:def 6;
A391:       Fn2.x in Fn2.:[:xy,ZZ:] by A294,A342,A376,A383,A389,FUNCT_1:def 6;
            dom (Fn1 | [:xy,ZZ:]) = dom Fn1 /\ [:xy,ZZ:] by RELAT_1:61;
            then
A392:       x in [:xy,ZZ:] by A383,A389,XBOOLE_0:def 4;
            thus (f*(Fn1 | [:xy,ZZ:])).x = ((f*Fn1) | [:xy,ZZ:]).x by
RELAT_1:83
              .= (f*Fn1).x by A376,A383,A389,FUNCT_1:49
              .= f.(Fn1.x) by A299,A389,FUNCT_1:13
              .= CircleMap.(Fn1.x) by A379,A390,FUNCT_1:49
              .= (CircleMap*Fn1).x by A299,A389,FUNCT_1:13
              .= F.x by A283,A298,A383,A389,A388,FUNCT_1:49
              .= (CircleMap*Fn2).x by A281,A342,A392,FUNCT_1:49
              .= CircleMap.(Fn2.x) by A294,A342,A392,FUNCT_1:13
              .= f.(Fn2.x) by A385,A386,A391,FUNCT_1:49
              .= (f*Fn2).x by A294,A342,A392,FUNCT_1:13
              .= ((f*Fn2) | [:xy,ZZ:]).x by A376,A383,A389,FUNCT_1:49
              .= (f*(Fn2 | [:xy,ZZ:])).x by RELAT_1:83;
          end;
A393:     dom (Fn2 | [:xy,ZZ:]) = [:xy,ZZ:] by A293,A297,A341,RELAT_1:62
,ZFMISC_1:96;
A394:     rng (Fn2 | [:xy,ZZ:]) c= dom f
          proof
            let b be object;
            assume b in rng (Fn2 | [:xy,ZZ:]);
            then consider a being object such that
A395:       a in dom (Fn2 | [:xy,ZZ:]) and
A396:       (Fn2 | [:xy,ZZ:]).a = b by FUNCT_1:def 3;
            Fn2.a = b by A393,A395,A396,FUNCT_1:49;
            then b in Fn2.:xZZ by A293,A393,A395,FUNCT_1:def 6;
            hence thesis by A385,A386,A367;
          end;
          then dom (f*(Fn2 | [:xy,ZZ:])) = dom (Fn2 | [:xy,ZZ:]) by RELAT_1:27;
          then
A397:     f*(Fn1 | [:xy,ZZ:]) = f*(Fn2 | [:xy,ZZ:]) by A376,A393,A380,A387,
RELAT_1:27;
          f is being_homeomorphism by A336,A366;
          then
A398:     f is one-to-one;
          per cases;
          suppose
            x2 in ZZ;
            then
A399:       x in [:xy,ZZ:] by A340,A377,ZFMISC_1:87;
            thus (Fn1 | [:{y},Z1:]).x = Fn1.x by A328,A337,FUNCT_1:49
              .= (Fn1 | [:xy,ZZ:]).x by A399,FUNCT_1:49
              .= (Fn2 | [:xy,ZZ:]).x by A398,A376,A393,A380,A394,A397,
FUNCT_1:27
              .= Fn2.x by A399,FUNCT_1:49
              .= (Fn2 | [:{y},Z1:]).x by A328,A337,FUNCT_1:49;
          end;
          suppose
            not x2 in ZZ;
            then x2 in Z by A339,XBOOLE_0:def 3;
            then
A400:       x in [:{y},Z:] by A338,A340,ZFMISC_1:87;
            thus (Fn1 | [:{y},Z1:]).x = Fn1.x by A328,A337,FUNCT_1:49
              .= (Fn1 | [:{y},Z:]).x by A400,FUNCT_1:49
              .= Fn2.x by A322,A400,FUNCT_1:49
              .= (Fn2 | [:{y},Z1:]).x by A328,A337,FUNCT_1:49;
          end;
        end;
        dom (Fn2 | [:{y},Z1:]) = [:{y},Z1:] by A295,A293,A297,RELAT_1:62
,ZFMISC_1:96;
        hence thesis by A328,A329;
      end;
    end;
A401: M[ 0 ] by FINSEQ_3:24;
    for i being Nat holds M[i] from NAT_1:sch 2(A401,A301);
    then ex Z being non empty Subset of I[01] st Z = [.0,TT.len TT.] & Fn1 |
    [:{y},Z:] = Fn2 | [:{y},Z:] by A292;
    hence thesis by A287,BORSUK_1:40;
  end;
  for p being Point of [:Y,I[01]:], V being Subset of R^1 st G.p in V &
  V is open holds ex W being Subset of [:Y,I[01]:] st p in W & W is open & G.:W
  c= V
  proof
    let p be Point of [:Y,I[01]:], V be Subset of R^1 such that
A402: G.p in V & V is open;
    consider y being Point of Y, t being Point of I[01], N being non empty
    open Subset of Y, Fn being Function of [:Y|N,I[01]:], R^1 such that
A403: p = [y,t] and
A404: G.p = Fn.p and
A405: y in N and
A406: Fn is continuous and
A407: F | [:N,I:] = CircleMap*Fn & Fn | [:the carrier of Y,{0}:] = Ft
    | [:N,I:] and
    for H being Function of [:Y|N,I[01]:], R^1 st H is continuous & F | [:
N,I:] = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft | [:N,I:] holds Fn = H
    by A274;
A408: the carrier of [:Y|N,I[01]:] = [:the carrier of (Y|N),I:] by
BORSUK_1:def 2
      .= [:N,I:] by PRE_TOPC:8;
    then p in the carrier of [:Y|N,I[01]:] by A403,A405,ZFMISC_1:87;
    then consider W being Subset of [:Y|N,I[01]:] such that
A409: p in W and
A410: W is open and
A411: Fn.:W c= V by A402,A404,A406,JGRAPH_2:10;
A412: dom Fn = [:N,I:] by A408,FUNCT_2:def 1;
A413: [#](Y|N) = N by PRE_TOPC:def 5;
    then
A414: [#][:Y|N,I[01]:] = [:N,[#]I[01]:] by BORSUK_3:1;
    [:Y|N,I[01]:] = [:Y,I[01]:] | [:N,[#]I[01]:] by Lm7,BORSUK_3:22;
    then consider C being Subset of [:Y,I[01]:] such that
A415: C is open and
A416: C /\ [#][:Y|N,I[01]:] = W by A410,TOPS_2:24;
    take WW = C /\ [:N,[#]I[01]:];
    thus p in WW by A409,A416,A413,BORSUK_3:1;
    [:N,[#]I[01]:] is open by BORSUK_1:6;
    hence WW is open by A415;
    let y be object;
    assume y in G.:WW;
    then consider x being Point of [:Y,I[01]:] such that
A417: x in WW and
A418: y = G.x by FUNCT_2:65;
    consider y0 being Point of Y, t0 being Point of I[01], N0 being non empty
    open Subset of Y, Fn0 being Function of [:Y|N0,I[01]:], R^1 such that
A419: x = [y0,t0] and
A420: G.x = Fn0.x and
A421: y0 in N0 & Fn0 is continuous & F | [:N0,I:] = CircleMap*Fn0 &
    Fn0 | [:the carrier of Y,{0}:] = Ft | [:N0,I :] and
    for H being Function of [:Y|N0,I[01]:], R^1 st H is continuous & F |
[:N0,I:] = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft | [:N0,I:] holds Fn0
    = H by A274;
    x in [:N,[#]I[01]:] by A417,XBOOLE_0:def 4;
    then
A422: y0 in N by A419,ZFMISC_1:87;
A423: x in [:{y0},I:] by A419,ZFMISC_1:105;
    then Fn.x = (Fn | [:{y0},I:]).x by FUNCT_1:49
      .= (Fn0 | [:{y0},I:]).x by A276,A406,A407,A419,A421,A422
      .= Fn0.x by A423,FUNCT_1:49;
    then y in Fn.:W by A412,A416,A414,A417,A418,A420,FUNCT_1:def 6;
    hence thesis by A411;
  end;
  hence G is continuous by JGRAPH_2:10;
  for x being Point of [:Y,I[01]:] holds F.x = (CircleMap*G).x
  proof
    let x be Point of [:Y,I[01]:];
    consider y being Point of Y, t being Point of I[01], N being non empty
    open Subset of Y, Fn being Function of [:Y|N,I[01]:], R^1 such that
A424: x = [y,t] and
A425: G.x = Fn.x and
A426: y in N and
    Fn is continuous and
A427: F | [:N,I:] = CircleMap*Fn and
    Fn | [:the carrier of Y,{0}:] = Ft | [:N,I:] and
    for H being Function of [:Y|N,I[01]:], R^1 st H is continuous & F | [:
N,I:] = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft | [:N,I:] holds Fn = H
    by A274;
A428: the carrier of [:Y|N,I[01]:] = [:the carrier of (Y|N),I:] by
BORSUK_1:def 2
      .= [:N,I:] by PRE_TOPC:8;
    then
A429: x in the carrier of [:Y|N,I[01]:] by A424,A426,ZFMISC_1:87;
    thus (CircleMap*G).x = CircleMap.(G.x) by FUNCT_2:15
      .= (CircleMap*Fn).x by A425,A429,FUNCT_2:15
      .= F.x by A427,A428,A429,FUNCT_1:49;
  end;
  hence F = CircleMap*G;
A430: [:the carrier of Y,{0}:] c= [:the carrier of Y,I:] by Lm3,ZFMISC_1:95;
A431: the carrier of [:Y,I[01]:] = [:the carrier of Y,I:] by BORSUK_1:def 2;
  then
A432: dom G = [:the carrier of Y,I:] by FUNCT_2:def 1;
A433: for x being object st x in dom Ft holds Ft.x = G.x
  proof
    let x be object;
    assume
A434: x in dom Ft;
    then x in dom G by A8,A432,A430;
    then consider
    y being Point of Y, t being Point of I[01], N being non empty
    open Subset of Y, Fn being Function of [:Y|N,I[01]:], R^1 such that
A435: x = [y,t] and
A436: G.x = Fn.x and
A437: y in N and
    Fn is continuous and
    F | [:N,I:] = CircleMap*Fn and
A438: Fn | [:the carrier of Y,{0}:] = Ft | [:N,I:] and
    for H being Function of [:Y|N,I[01]:], R^1 st H is continuous & F | [:
N,I:] = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft | [:N,I:] holds Fn = H
    by A274;
    x in [:N,I:] by A435,A437,ZFMISC_1:87;
    hence Ft.x = (Ft | [:N,I:]).x by FUNCT_1:49
      .= G.x by A7,A434,A436,A438,Lm14,FUNCT_1:49;
  end;
  dom Ft = dom G /\ [:the carrier of Y,{0}:] by A8,A432,A430,XBOOLE_1:28;
  hence G | [:the carrier of Y,{0}:] = Ft by A433,FUNCT_1:46;
  let H be Function of [:Y,I[01]:], R^1 such that
A439: H is continuous & F = CircleMap*H and
A440: H | [:the carrier of Y,{0}:] = Ft;
  for x being Point of [:Y,I[01]:] holds G.x = H.x
  proof
    let x be Point of [:Y,I[01]:];
    consider y being Point of Y, t being Point of I[01], N being non empty
    open Subset of Y, Fn being Function of [:Y|N,I[01]:], R^1 such that
A441: x = [y,t] and
A442: G.x = Fn.x and
A443: y in N and
    Fn is continuous and
    F | [:N,I:] = CircleMap*Fn and
    Fn | [:the carrier of Y,{0}:] = Ft | [:N,I:] and
A444: for H being Function of [:Y|N,I[01]:], R^1 st H is continuous &
F | [:N,I:] = CircleMap*H & H | [:the carrier of Y,{0}:] = Ft | [:N,I:] holds
    Fn = H by A274;
A445: the carrier of [:Y|N,I[01]:] = [:the carrier of (Y|N),I:] by
BORSUK_1:def 2
      .= [:N,I:] by PRE_TOPC:8;
    then
A446: x in the carrier of [:Y|N,I[01]:] by A441,A443,ZFMISC_1:87;
    dom H = the carrier of [:Y,I[01]:] by FUNCT_2:def 1;
    then [:N,I:] c= dom H by A431,ZFMISC_1:95;
    then
A447: dom (H | [:N,I:]) = [:N,I:] by RELAT_1:62;
    rng (H | [:N,I:]) c= R by RELAT_1:def 19;
    then reconsider
    H1 = H | [:N,I:] as Function of [:Y|N,I[01]:], R^1 by A445,A447,FUNCT_2:2;
A448: H | [:N,I:] | [:the carrier of Y,{0}:] = H | ([:the carrier of Y,{0}
    :] /\ [:N,I:]) by RELAT_1:71
      .= Ft | [:N,I:] by A440,RELAT_1:71;
    H1 is continuous & F | [:N,I:] = CircleMap*(H| [:N,I:]) by A439,RELAT_1:83
,TOPALG_3:17;
    hence G.x = (H| [:N,I:]).x by A442,A444,A448
      .= H.x by A445,A446,FUNCT_1:49;
  end;
  hence thesis;
end;
