 reserve x,X for set,
         n, m, i for Nat,
         p, q for Point of TOP-REAL n,
         A, B for Subset of TOP-REAL n,
         r, s for Real;
reserve N for non zero Nat,
        u,t for Point of TOP-REAL(N+1);

theorem Th3:
   for TM be metrizable TopSpace st TM is finite-ind second-countable
     for F be closed Subset of TM st ind (F`) <= n
     for f be continuous Function of TM|F,Tunit_circle(n+1)
   ex g being continuous Function of TM,Tunit_circle(n+1) st g|F=f
proof
  defpred P[Nat] means
    for TM be metrizable TopSpace st TM is finite-ind second-countable
      for F be closed Subset of TM st ind F` <= $1
      for f be continuous Function of TM|F,Tunit_circle($1 +1)
    ex g being Function of TM,Tunit_circle($1+1) st g is continuous & g|F=f;
  let TM be metrizable TopSpace such that
A1: TM is finite-ind second-countable;
  let F be closed Subset of TM such that
A2: ind (F`) <= n;
A3:for n st P[n] holds P[n+1]
  proof
    let n;
    set n1=n+1,n2=n1+1;
    assume
A4:   P[n];
    set Tn1=TOP-REAL n1,Tn2=TOP-REAL n2,U=Tunit_circle(n1+1);
    let TM be metrizable TopSpace;
    assume
A5:   TM is finite-ind second-countable;
    let F be closed Subset of TM such that
A6:   ind F` <= n1;
    let f be continuous Function of (TM | F),U;
A7: [#](TM|F) = F by PRE_TOPC:def 5;
A8: dom f = the carrier of (TM|F) by FUNCT_2:def 1;
A9: dom f = the carrier of (TM|F) by FUNCT_2:def 1;
A10: [#](TM|F) = F by PRE_TOPC:def 5;
    per cases;
      suppose
A11:      F is empty;
        take g=TM-->the Point of U;
        g|F={} by A11;
        hence thesis by A11;
      end;
      suppose
A12:      F is non empty;
        set Sn= {p where p is Point of Tn2: p.n2<=0 & |.p.|=1};
        set Sp= {p where p is Point of Tn2: p.n2>=0 & |.p.|=1};
A13:    Sp c= the carrier of Tn2
        proof
          let x be object;
          assume x in Sp;
          then ex p be Point of Tn2 st p=x & p.n2>=0 & |.p.|=1;
          hence thesis;
        end;
        Sn c= the carrier of Tn2
        proof
          let x be object;
          assume x in Sn;
          then ex p be Point of Tn2 st p=x & p.n2<=0 & |.p.|=1;
          hence thesis;
        end;
        then reconsider Sp,Sn as Subset of Tn2 by A13;
A14:    Sn = {t where t is Point of Tn2: t.(n1+1)<=0 & |.t.|=1};
A15:    Sp = {l where l is Point of Tn2: l.(n1+1)>=0 & |.l.|=1};
        then reconsider s1=Sp,s2=Sn as closed Subset of Tn2 by A14,Th2;
A16:    [#] (Tn2|s2) = s2 by PRE_TOPC:def 5;
        U = Tcircle(0.Tn2,1) by TOPREALB:def 7;
        then
A17:    the carrier of U = Sphere(0.Tn2,1) by TOPREALB:9;
A18:    s1 c= the carrier of U
        proof
          let x be object;
          assume x in s1;
          then consider p be Point of Tn2 such that
A19:          x=p
            and
              p.(n1+1)>=0
            and
A20:          |.p.|=1;
          p-0.Tn2 = p by RLVECT_1:13;
          hence thesis by A20,A19,A17;
        end;
A21:    s2 c= the carrier of U
        proof
          let x be object;
          assume x in s2;
          then consider p be Point of Tn2 such that
A22:          x=p
            and
              p.(n1+1)<=0
            and
A23:          |.p.|=1;
          p-0.Tn2 = p by RLVECT_1:13;
          hence thesis by A23,A22,A17;
        end;
        then reconsider S1=s1,S2=s2 as Subset of U by A18;
        reconsider A1=f"S1,A2=f"S2 as Subset of TM by A7,XBOOLE_1:1;
A24:    f.:(A1/\A2) c= (f.:A1) /\(f.:A2) by RELAT_1:121;
A25:    f.:A2 c= s2 by FUNCT_1:75;
        S1 is closed by TSEP_1:8;
        then f"S1 is closed by PRE_TOPC:def 6;
        then
A26:      A1 is closed by A7,TSEP_1:8;
        Sphere(0.Tn2,1) c= S1\/S2
        proof
          let x be object;
          assume
A27:        x in Sphere(0.Tn2,1);
          then reconsider p=x as Point of Tn2;
A28:        p.n2 >=0 or p.n2 <=0;
          |.p.| = 1 by A27,TOPREAL9:12;
          then p in Sp or p in Sn by A28;
          hence thesis by XBOOLE_0:def 3;
        end;
        then rng f c= S1\/S2 by A17;
        then
A29:      f"rng f c= f"(S1\/S2) by RELAT_1:143;
        f"rng f = dom f by RELAT_1:134;
        then
A30:      F = f"(S1\/S2) by A29,A7,FUNCT_2:def 1;
        then
A31:      F = A1\/A2 by RELAT_1:140;
        then reconsider TFA12=A1/\A2 as Subset of TM|F by XBOOLE_1:29,A10;
A32:    [#]((TM|F) |TFA12) =TFA12 by PRE_TOPC:def 5;
        reconsider fa=f|TFA12 as Function of (TM|F) |TFA12,U| (f.:TFA12)
          by A10,A12,JORDAN24:12;
A33:    fa is continuous by JORDAN24:13, A10,A12;
        dom f /\ TFA12 = TFA12 by A8,XBOOLE_1:28;
        then
A34:      dom fa=the carrier of ((TM|F) |TFA12) by A32, RELAT_1:61;
A35:    (TM|F) | (f"S1) = TM|A1 by PRE_TOPC:7, A7;
        then reconsider f1=f|A1 as Function of TM|A1,U| (f.:A1)
          by A12,A7,JORDAN24:12;
A36:    f1 is continuous by A35, A12,A10,JORDAN24:13;
A37:    rng f1 c=the carrier of (U| (f.:A1));
A38:    [#](U| (f.:A1))= f.:A1 by PRE_TOPC:def 5;
        then
A39:      rng f1 c= the carrier of U by XBOOLE_1:1;
A40:    (TM|F) | (f"S2) = TM|A2 by PRE_TOPC:7, A7;
        then reconsider f2=f|A2 as Function of TM|A2,U| (f.:A2)
          by A12,A7,JORDAN24:12;
A41:    f2 is continuous by A40, A12,A10,JORDAN24:13;
A42:    [#](U| (f.:A2))= f.:A2 by PRE_TOPC:def 5;
        then
A43:      rng f2 c= the carrier of U by XBOOLE_1:1;
        dom f1 = dom f/\A1 by RELAT_1:61;
        then
A44:      dom f1 = A1 by A9,XBOOLE_1:28;
        [#](TM|A1) = A1 by PRE_TOPC:def 5;
        then reconsider f1 as Function of TM|A1,U by A39,FUNCT_2:2,A44;
A45:    rng f2 c=the carrier of (U| (f.:A2));
        dom f2 = dom f/\A2 by RELAT_1:61;
        then
A46:      dom f2 = A2 by A9,XBOOLE_1:28;
        [#](TM|A2) = A2 by PRE_TOPC:def 5;
        then reconsider f2 as Function of TM|A2,U by A43,A46,FUNCT_2:2;
        f.:A2 c= s2 by FUNCT_1:75;
        then
A47:      rng f2 c= s2 by A42,A45;
        S2 is closed by TSEP_1:8;
        then f"S2 is closed by PRE_TOPC:def 6;
        then
A48:      A2 is closed by A7,TSEP_1:8;
        TM|F` is second-countable by A5;
        then consider X1,X2 be closed Subset of TM such that
A49:        [#]TM = X1\/X2
          and
A50:        A1 c=X1
          and
A51:        A2 c=X2
          and
A52:        A1/\X2=A1/\A2
          and
A53:        A1/\A2=X1/\A2
          and
A54:    ind((X1/\X2)\(A1/\A2))<=n1 qua ExtReal -1
        by A31,TOPDIM_2:24,A5,A6,A26,A48;
        set TX=TM| (X1/\X2);
A55:    [#]TX = X1/\X2 by PRE_TOPC:def 5;
        then reconsider TXA12=A1/\A2 as Subset of TX by A50,A51,XBOOLE_1:27;
        (X1/\X2)\(A1/\A2)=TXA12` by A55,SUBSET_1:def 4;
        then
A56:      ind TXA12` <= n by A5,TOPDIM_1:21,A54;
        set Un=Tunit_circle(n+1);
        set TD=Tdisk(0.Tn1,1);
        deffunc ff(Nat)=PROJ(n2,$1);
        consider FF be FinSequence such that
A57:      len FF = n1 & for k be Nat st k in dom FF holds FF.k=ff(k)
          from FINSEQ_1:sch 2;
A58:    rng FF c= Funcs(the carrier of Tn2,the carrier of R^1)
        proof
          let x be object;
          assume x in rng FF;
          then consider i be object such that
A59:          i in dom FF
            and
A60:          FF.i = x by FUNCT_1:def 3;
          reconsider i as Nat by A59;
          ff(i) in Funcs(the carrier of Tn2,the carrier of R^1)
            by FUNCT_2:8;
          hence thesis by A57,A59,A60;
        end;
A61:    Ball(0.Tn1,1) c= Int cl_Ball(0.Tn1,1) by TOPREAL9:16,TOPS_1:24;
        then
A62:      cl_Ball(0.Tn1,1) is compact non boundary convex;
        reconsider FF as FinSequence of
          Funcs(the carrier of Tn2,the carrier of R^1) by A58,FINSEQ_1:def 4;
        reconsider FF as Element of n1-tuples_on
          Funcs(the carrier of Tn2,the carrier of R^1) by A57,FINSEQ_2:92;
        set FFF=<:FF:>;
A63:    s1/\s2 c= s2 by XBOOLE_1:17;
A64:    FFF.:s2= cl_Ball(0.Tn1,1) by A57,Th1,A15;
        then
A65:      s2 is non empty;
A66:    dom FFF=the carrier of Tn2 by FUNCT_2:def 1;
        then s1/\dom FFF=s1 by XBOOLE_1:28;
        then
A67:      dom (FFF|s1)= s1 by RELAT_1:61;
        s2/\dom FFF=s2 by XBOOLE_1:28,A66;
        then
A68:      dom (FFF|s2)= s2 by RELAT_1:61;
A69:    the carrier of TD = cl_Ball(0.Tn1,1) by BROUWER:3;
        then rng (FFF|s2) c= the carrier of TD by RELAT_1:115,A64;
        then reconsider Fs2=FFF|s2 as Function of Tn2|s2,TD
          by FUNCT_2:2,A16,A68;
A70:    [#](Tn2|s1) = s1 by PRE_TOPC:def 5;
        Fs2 is being_homeomorphism by A57,Th1,A15;
        then
A71:      s2,cl_Ball(0.Tn1,1) are_homeomorphic
          by T_0TOPSP:def 1,METRIZTS:def 1;
A72:    FFF.:s1= cl_Ball(0.Tn1,1) by A57,Th1,A14;
        then
A73:      s1 is non empty;
        rng (FFF|s1) c= the carrier of TD by RELAT_1:115, A72,A69;
        then reconsider Fs1=FFF|s1 as Function of Tn2|s1,TD
          by A67,FUNCT_2:2,A70;
A74:    Fs1.:(s1/\s2) c= FFF.:(s1/\s2) by RELAT_1:128;
A75:    Fs1 is being_homeomorphism by A57,Th1,A14;
        then
A76:      rng Fs1 = [#]TD by TOPS_2:def 5;
        f.:A1 c= s1 by FUNCT_1:75;
        then WE: (f.:A1)/\(f.:A2) c= s1/\s2 by A25,XBOOLE_1:27;
        then f.:(A1/\A2) c= s1/\s2 by A24;
        then
A78:      Fs1.:(f.:(A1/\A2)) c= Fs1.:(s1/\s2) by RELAT_1:123;
        s1/\s2 c= s1 by XBOOLE_1:17;
        then
A79:      f.:(A1/\A2) c= s1 by WE,A24;
        [#](U| (f.:TFA12))=f.:TFA12 by PRE_TOPC:def 5;
        then
A80:      rng fa c= the carrier of Tn2|s1 by A79,A70;
        then reconsider fa as Function of (TM|F) |TFA12,U
          by XBOOLE_1:1,A70,A34,FUNCT_2:2;
A81:    fa is continuous by A33,PRE_TOPC:26;
        rng fa c= the carrier of Tn2 by A17,XBOOLE_1:1;
        then reconsider fa as Function of (TM|F) |TFA12,Tn2 by FUNCT_2:2,A34;
A82:    fa is continuous by A81,PRE_TOPC:26;
A83:    Fs1" is continuous by A75,TOPS_2:def 5;
A84:    FFF.:(s1/\s2) = Sphere(0.Tn1,1) by A57,Th1;
        then Fs1.:(s1/\s2) = Sphere(0.Tn1,1) by XBOOLE_1:17,RELAT_1:129;
        then
A85:      (Fs1").:Sphere(0.Tn1,1) = Fs1"(Fs1.:(s1/\s2)) by TOPS_2:55,A75,A76
             .= s1/\s2 by FUNCT_1:94,A67,A75,XBOOLE_1:17;
        set A2X=A2 \/ (X1/\X2);
        set A1X=A1 \/ (X1/\X2);
A86:    X2=[#](TM|X2) by PRE_TOPC:def 5;
        (TM|F) |TFA12 = TM| (A1/\A2) by PRE_TOPC:7, A31,XBOOLE_1:29;
        then
A87:    (TM|F) |TFA12 = TX |TXA12 by PRE_TOPC:7,A50,A51,XBOOLE_1:27;
        then reconsider fa as Function of TX |TXA12,Tn2|s1
          by A34,FUNCT_2:2,A80;
        reconsider Ffa=Fs1*fa as Function of TX|TXA12,TD by A73;
A88:    [#](TX |TXA12) = TXA12 by PRE_TOPC:def 5;
        then
A89:      dom Ffa = A1/\A2 by FUNCT_2:def 1;
        rng Ffa = Ffa.:dom Ffa by RELAT_1:113
               .= (Fs1*fa).:(A1/\A2) by A88,FUNCT_2:def 1
               .= Fs1.:(fa.:(A1/\A2)) by RELAT_1:126;
        then E: rng Ffa c= Fs1.:(f.:(A1/\A2)) by RELAT_1:123, RELAT_1:128;
        then
A90:      rng Ffa c= Fs1.:(s1/\s2) by A78;
A91:      rng Ffa c= Sphere(0.Tn1,1) by E,A78,A74,A84;
        fa is continuous by A82,PRE_TOPC:27,A87;
        then
A92:      Ffa is continuous by TOPS_2:46, A75, A73;
        reconsider Ffa as Function of TX |TXA12,Tn1
          by A91,XBOOLE_1:1,FUNCT_2:2,A89,A88;
        Un = Tcircle(0.Tn1,1) by TOPREALB:def 7;
        then
A93:      the carrier of Un = Sphere(0.Tn1,1) by TOPREALB:9;
        then reconsider H=Ffa as Function of TX |TXA12,Un
          by FUNCT_2:2,A89,A88,A90,A74,XBOOLE_1:1,A84;
        Ffa is continuous by A92,PRE_TOPC:26;
        then reconsider H as continuous Function of TX |TXA12,Un
          by PRE_TOPC:27;
        TXA12 is closed by A26,A48,TSEP_1:8;
        then consider g be Function of TX, Un such that
A94:        g is continuous
          and
A95:        g|TXA12 = H by A5,A56, A4;
A96:    rng g c= the carrier of Tn1 by A93,XBOOLE_1:1;
A97:    dom g=the carrier of TX by FUNCT_2:def 1;
A98:    rng g c= the carrier of Un;
        reconsider g as Function of TX, Tn1 by A97,A96,FUNCT_2:2;
A99:    g is continuous by A94,PRE_TOPC:26;
        the carrier of Un c= the carrier of TD by A93,A69,TOPREAL9:17;
        then reconsider g as Function of TX,TD by A98,XBOOLE_1:1,FUNCT_2:2,A97;
        reconsider G=(Fs1")*g as Function of TX,Tn2|s1;
A100:   dom G = the carrier of TX by FUNCT_2:def 1,A73;
        g is continuous by A99,PRE_TOPC:27;
        then
A101:     G is continuous by A83,TOPS_2:46;
A102:   rng G c= s1 by A70;
        then reconsider G as Function of TX,Tn2 by XBOOLE_1:1,FUNCT_2:2,A100;
A103:   G is continuous by PRE_TOPC:26,A101;
        reconsider G as Function of TX,U by A18,A102,XBOOLE_1:1,FUNCT_2:2,A100;
A104:   G is continuous by PRE_TOPC:27,A103;
A105:   rng fa c= dom Fs1 by A67,A70;
A106:   G|TXA12 = (Fs1")*(g|TXA12) by RELAT_1:83
               .= (Fs1"*Fs1)*fa by RELAT_1:36, A95
               .= (id dom Fs1)*fa by TOPS_2:52, A75,A76
               .= fa by RELAT_1:53,A105;
A107:   now
          let xx be object;
          assume
A108:       xx in dom f1/\dom G;
          then
A109:       xx in A1 by A44,XBOOLE_0:def 4;
          xx in X2 by A108,A55,XBOOLE_0:def 4;
          then
A110:       xx in A1/\A2 by A109,A52,XBOOLE_0:def 4;
          hence G.xx = (G|TXA12).xx by FUNCT_1:49
                    .= f.xx by A110,FUNCT_1:49,A106
                    .= f1.xx by A109,FUNCT_1:49;
        end;
A111:   now
          let xx be object;
          assume
A112:       xx in dom f2/\dom G;
          then
A113:       xx in A2 by A46,XBOOLE_0:def 4;
          xx in X1 by A112,A55,XBOOLE_0:def 4;
          then
A114:        xx in A1/\A2 by A113,A53,XBOOLE_0:def 4;
          hence G.xx = (G|TXA12).xx by FUNCT_1:49
                    .= f.xx by A114,FUNCT_1:49, A106
                    .= f2.xx by A113,FUNCT_1:49;
        end;
        rng G = G.:dom G by RELAT_1:113
             .= (Fs1").:(g.:dom G) by RELAT_1:126
             .= (Fs1").:(rng g) by A97,A100,RELAT_1:113;
        then rng G c= s1/\s2 by A85,A93,A98,RELAT_1:123;
        then rng G c= s2 by A63;
        then
A115:     rng f2 \/rng G c= s2 by XBOOLE_1:8, A47;
        f2 is continuous by A41,PRE_TOPC:26;
        then reconsider f2G=f2+*G as continuous Function of TM|A2X,U
          by A111,PARTFUN1:def 4,A104,A48,TOPGEN_5:10;
A116:   dom f2G = the carrier of TM|A2X by FUNCT_2:def 1;
A117:   rng f2G c= rng f2 \/rng G by FUNCT_4: 17;
        then rng f2G c= s2 by A115;
        then reconsider f2G as Function of TM|A2X,Tn2
          by XBOOLE_1:1,FUNCT_2:2,A116;
A118:   f2G is continuous by PRE_TOPC:26;
        reconsider f2G as Function of TM|A2X,Tn2|s2
          by FUNCT_2:2,A116,A115,A117,XBOOLE_1:1,A16;
        cl_Ball(0.Tn1,1) is compact non boundary convex by A61;
        then consider H2 being Function of TM,Tn2|s2 such that
A119:       H2 is continuous
          and
A120:       H2|A2X=f2G by A118,PRE_TOPC:27,A71, TIETZE_2:24, A48;
A121:   TM is non empty by A12;
        then reconsider H2X=H2|X2 as Function of TM|X2,(Tn2|s2) | (H2.:X2)
          by JORDAN24:12,A65;
A122:   H2X is continuous by JORDAN24:13,A65,A121,A119;
        dom H2 = the carrier of TM by FUNCT_2:def 1,A65;
        then
A123:     dom H2X = X2/\the carrier of TM by RELAT_1:61;
        then
A124:     dom H2X= the carrier of TM|X2 by XBOOLE_1:28,A86;
        f1 is continuous by A36,PRE_TOPC:26;
        then reconsider f1G=f1+*G as continuous Function of TM|A1X,U
          by A107,PARTFUN1:def 4,A104,A26,TOPGEN_5:10;
A125:   dom f1G = the carrier of TM|A1X by FUNCT_2:def 1;
        f.:A1 c= s1 by FUNCT_1:75;
        then rng f1 c= s1 by A38,A37;
        then
A126:     rng f1 \/rng G c= s1 by A102,XBOOLE_1:8;
A127:   rng f1G c= rng f1 \/rng G by FUNCT_4:17;
        then rng f1G c= s1 by A126;
        then reconsider f1G as Function of TM|A1X,Tn2
          by XBOOLE_1:1,FUNCT_2:2,A125;
A128:   f1G is continuous by PRE_TOPC:26;
        reconsider f1G as Function of TM|A1X,Tn2|s1
          by FUNCT_2:2,A125, A126,A127,XBOOLE_1:1,A70;
        s1,cl_Ball(0.Tn1,1) are_homeomorphic
          by T_0TOPSP:def 1,A75,METRIZTS:def 1;
        then consider H1 being Function of TM,Tn2|s1 such that
A129:       H1 is continuous
          and
A130:       H1|A1X=f1G by A62,A26,A128,PRE_TOPC:27,TIETZE_2:24;
        reconsider H1X=H1|X1 as Function of TM|X1,(Tn2|s1) | (H1.:X1)
          by A121,JORDAN24:12,A73;
A131:   H1X is continuous by JORDAN24:13,A73,A121,A129;
        [#]((Tn2|s1) | (H1.:X1)) = (H1.:X1) by PRE_TOPC:def 5;
        then
A132:     rng H1X c= the carrier of Tn2|s1 by XBOOLE_1:1;
        dom H1 = the carrier of TM by FUNCT_2:def 1,A73;
        then
A133:     dom H1X = X1/\the carrier of TM by RELAT_1:61;
        then
A134:     dom H1X= X1 by XBOOLE_1:28;
        X1=[#](TM|X1) by PRE_TOPC:def 5;
        then
A135:     dom H1X= the carrier of TM|X1 by A133,XBOOLE_1:28;
        then reconsider H1X as Function of TM|X1,Tn2|s1 by A132, FUNCT_2:2;
A136:   H1X is continuous by A131,PRE_TOPC:26;
A137:   rng H1X c= s1 by A70;
        [#]((Tn2|s2) | (H2.:X2)) = (H2.:X2) by PRE_TOPC:def 5;
        then rng H2X c= the carrier of Tn2|s2 by XBOOLE_1:1;
        then reconsider H2X as Function of TM|X2,Tn2|s2 by A124, FUNCT_2:2;
A138:   H2X is continuous by A122,PRE_TOPC:26;
        reconsider H1X as Function of TM|X1,Tn2
          by A137,XBOOLE_1:1,A135,FUNCT_2:2;
A139:   rng H2X c= s2 by A16;
        then reconsider H2X as Function of TM|X2,Tn2
          by XBOOLE_1:1,A124,FUNCT_2:2;
A140:   H2X is continuous by A138,PRE_TOPC:26;
        A141: now
          let xx be object;
          assume
A142:       xx in dom H1X/\dom H2X;
          then
A143:       H2X.xx=H2.xx by A86,FUNCT_1:49;
          xx in X1 by A142,A134,XBOOLE_0:def 4;
          then
A144:       H1X.xx=H1.xx by FUNCT_1:49;
A145:     xx in X1/\X2 by A142, A123,XBOOLE_1:28,A134;
          then
A146:       xx in A2X by XBOOLE_0:def 3;
          xx in A1X by A145, XBOOLE_0:def 3;
          then
A147:       H1.xx = (H1|A1X).xx by FUNCT_1:49;
A148:     f2G.xx=G.xx by A145,A100,A55,FUNCT_4:13;
          f1G.xx = G.xx by A145,A100,A55,FUNCT_4:13;
          hence H1X.xx = H2X.xx
            by A148,A147,A146,FUNCT_1:49,A144,A143,A120,A130;
        end;
        H1X is continuous by A136,PRE_TOPC:26;
        then reconsider H12=H1X+*H2X as continuous Function of
          TM| (X1\/X2),Tn2 by A140,A141,PARTFUN1:def 4,TOPGEN_5:10;
A149:   TM| (X1\/X2) = the TopStruct of TM by A49,TSEP_1:93;
        then reconsider H12 as Function of TM,Tn2;
A150:   rng H12 c= rng H1X \/ rng H2X by FUNCT_4:17;
        F/\X1 = (A1\/A2)/\X1 by A30,RELAT_1:140
             .= (A1/\X1)\/(A2/\X1) by XBOOLE_1:23
             .= A1 \/(A2/\X1) by A50,XBOOLE_1:28
             .= A1 by A53,XBOOLE_1:17,XBOOLE_1:12;
        then
A151:     H1X|F= H1|A1 by RELAT_1:71;
        f1G=G+*f1 by A107,PARTFUN1:def 4,FUNCT_4:34;
        then
A152:     f1G|A1 =f1 by FUNCT_4:23, A44;

A153:   F/\X2 = (A1\/A2)/\X2 by A30,RELAT_1:140
             .= (A1/\X2)\/(A2/\X2) by XBOOLE_1:23
             .= (A1/\X2)\/A2 by A51,XBOOLE_1:28
             .= A2 by A52,XBOOLE_1:17,XBOOLE_1:12;
A154:   H2|A2 = f2G|A2 by A120,RELAT_1:74, XBOOLE_1:7;
        rng H1X \/ rng H2X c= s1\/s2 by A139,A137,XBOOLE_1:13;
        then
A155:     rng H12 c= s1\/s2 by A150;
A156:   dom H12 = the carrier of TM by FUNCT_2:def 1;
A157:   H12 is continuous by PRE_TOPC:32,A149;
        s1\/s2 c= Sphere(0.Tn2,1) by A18, A21,A17,XBOOLE_1:8;
        then reconsider H12 as Function of TM,U
          by A155,XBOOLE_1:1,A17,A156,FUNCT_2:2;
        take H12;
        thus H12 is continuous by PRE_TOPC:27,A157;
A158:   H1|A1 = f1G|A1 by A130,XBOOLE_1:7,RELAT_1:74;
        f2G=G+*f2 by A111,PARTFUN1:def 4,FUNCT_4:34;
        then
A159:     f2G|A2 =f2 by FUNCT_4:23,A46;
        thus H12|F = H1X|F +*H2X|F by FUNCT_4:71
                  .= f1+*f2 by A152,A159,A158,A154,A151, RELAT_1:71,A153
                  .= f| (A1\/A2) by FUNCT_4:78
                  .= f by A31,A10;
      end;
    end;
    let f be continuous Function of TM|F,Tunit_circle(n+1);
A160:P[0 qua Nat]
    proof
      reconsider Z=0 as Real;
      set TR=TOP-REAL 1,U=Tunit_circle(0+1);
      let TM be metrizable TopSpace;
      assume
A161:   TM is finite-ind second-countable;
      let F be closed Subset of TM;
      assume
A162:   ind F` <= 0;
      let f be continuous Function of TM|F,U;
A164:   f"rng f = f"(the carrier of U) by RELAT_1:143,135;
      U = Tcircle(0.TR,1) by TOPREALB:def 7;
      then
A165:   the carrier of U = Sphere(0.TR,1) by TOPREALB:9;
  0.TR = 0*(0+1) by EUCLID:70
          .= <*0 *> by FINSEQ_2:59;
      then
A166:   {<* Z qua ExtReal-1 *>,<* Z qua ExtReal+1 *>} =Fr Ball(0.TR,1)
        by TOPDIM_2:18;
A167: Fr Ball(0.TR,1)= Sphere(0.TR,1) by JORDAN:24;
      then reconsider mONE = <*-1*>,ONE=<*1*> as Point of U
        by A165,TARSKI:def 2,A166;
      reconsider Q={ONE},Q1={mONE} as closed Subset of U;
      set F1=f"Q,F2=f"Q1;
A168: [#](TM|F)=F by PRE_TOPC:def 5;
      then reconsider A1=F1,A2=F2 as Subset of TM by XBOOLE_1:1;
A169: dom f = F by A168,FUNCT_2:def 1;
      Q\/Q1 = the carrier of U by A166,A167,A165,ENUMSET1:1;
      then
A170:   F1\/F2 = f"(the carrier of U) by RELAT_1:140
              .= F by A164,RELAT_1:134,A169;
      F2 is closed by PRE_TOPC:def 6;
      then
A171:   A2 is closed by TSEP_1:8,A168;
      F1 is closed by PRE_TOPC:def 6;
      then
A172:   A1 is closed by TSEP_1:8,A168;
      ONE<>mONE
      proof
        assume ONE=mONE;
        then <*1*>.1 = -1 by FINSEQ_1:40;
        hence thesis;
      end;
      then W: F1 misses F2 by ZFMISC_1:11,FUNCT_1:71;
      TM|F` is second-countable by A161;
      then consider X1,X2 be closed Subset of TM such that
A174:     [#]TM = X1\/X2
        and
A175:     A1 c=X1
        and
A176:     A2 c=X2
        and A1/\X2=A1/\A2 & A1/\A2=X1/\A2
        and
A177:     ind((X1/\X2)\(A1/\A2))<=0 qua ExtReal-1
          by A161,A162,A170,TOPDIM_2:24,A172,A171;
      ind (X1/\X2) >= -1 by TOPDIM_1:5,A161;
      then
A178:   X1/\X2 is empty by A177,W,XXREAL_0:1,TOPDIM_1:6,A161;
      set h1=(TM|X1)-->ONE,h2=(TM|X2)-->mONE;
A179: [#](TM|X1)=X1 by PRE_TOPC:def 5;
A180: dom h2 = the carrier of (TM|X2);
A181: [#](TM|X2)=X2 by PRE_TOPC:def 5;
      dom h1 = the carrier of (TM|X1);
      then h1 tolerates h2
        by A178,A179,A180,A181,XBOOLE_0:def 7,PARTFUN1:56;
      then reconsider h12=h1+*h2 as continuous Function of TM| ([#]TM),U
        by A174, TOPGEN_5:10;
      [#] (TM| ([#]TM)) = [#]TM by PRE_TOPC:def 5;
      then reconsider H12=h12 as Function of TM,U;
A182: for x be object st x in F holds (H12|F).x = f.x
      proof
        let x be object;
        assume
A183:     x in F;
        then
A184:     (H12|F).x=h12.x by FUNCT_1:49;
        per cases by A183,A170,XBOOLE_0:def 3;
        suppose
A185:       x in F1;
          then not x in dom h2 by A175,A178,XBOOLE_0:def 4,A181;
          then
A186:       H12.x=h1.x by FUNCT_4:11;
A187:     f.x in {ONE} by A185,FUNCT_1:def 7;
          h1.x = ONE by A185,A175,A179,FUNCOP_1:7;
          hence thesis by A187,TARSKI:def 1,A186,A184;
        end;
        suppose
A188:       x in F2;
          then
A189:       f.x in {mONE} by FUNCT_1:def 7;
A190:     h2.x = mONE by A188,A176,A181,FUNCOP_1:7;
          H12.x=h2.x by A188,A176,A180,A181,FUNCT_4:13;
          hence thesis by A189,TARSKI:def 1,A190,A184;
        end;
      end;
      take H12;
      TM| ([#]TM) = the TopStruct of TM by TSEP_1:93;
      hence H12 is continuous by PRE_TOPC:32;
      dom H12 = the carrier of TM by FUNCT_2:def 1;
      then dom (H12|F)=F by RELAT_1:62;
      hence thesis by A182,A168,FUNCT_2:def 1;
    end;
    for n holds P[n] from NAT_1:sch 2(A160,A3);
    then ex g be Function of TM, Tunit_circle(n+1) st
      g is continuous & g|F=f by A1,A2;
    hence thesis;
  end;
