  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;
reserve T1,T2,S1,S2 for non empty TopSpace,
        t1 for Point of T1, t2 for Point of T2,
        pn,qn for Point of TOP-REAL n,
        pm,qm for Point of TOP-REAL m;
reserve T,S for TopSpace,
        A for closed Subset of T,
        B for Subset of S;

theorem
  for T be non empty TopSpace, n st n>=1 &
     for S be TopSpace,A be non empty closed Subset of T,B being Subset of S
       st ex X be Subset of TOP-REAL n st
          X is compact non boundary convex & B,X are_homeomorphic
     holds
       for f being Function of T|A,S|B st f is continuous
         ex g being Function of T,S|B st g is continuous & g|A = f
  holds T is normal
proof
  let T be non empty TopSpace,n;
  set TR= TOP-REAL n;
  assume that
A1:  n>=1 and
A2:  for S be TopSpace,A be non empty closed Subset of T,B be Subset of S st
           ex X be Subset of TR st
             X is compact non boundary convex & B,X are_homeomorphic
     holds
       for f being Function of T|A,S|B st f is continuous
         ex g being Function of T,S|B st g is continuous & g|A = f;
  set CC=Closed-Interval-TSpace(-1,1);
  for A be non empty closed Subset of T
    for f be continuous Function of T|A, CC
      ex g being continuous Function of T, Closed-Interval-TSpace(-1,1)
  st g|A = f
    proof
      let A be non empty closed Subset of T;
      let f be continuous Function of T|A, CC;
A3:     the carrier of CC = [.-1,1.] by TOPMETR:18;
A4:     rng f c= REAL;
      dom f =the carrier of (T|A) by FUNCT_2:def 1;
      then reconsider F=f as Function of T|A, R^1 by A4,TOPMETR:17,FUNCT_2:2;
      reconsider F as continuous Function of T|A, R^1 by PRE_TOPC: 26;
      set IF1=incl(F,n);
      set n1=n|->1;
      set CH=ClosedHypercube(0.TR,n1);
A5:   dom IF1 = the carrier of (T|A) by FUNCT_2:def 1;
A6:   [#](TR|CH)=CH by PRE_TOPC:def 5;
      0.TR = 0*n by EUCLID:70;
      then
A7:     0.TR = n |-> 0 by EUCLID:def 4;
A8:   rng IF1 c= CH
        proof
          let y be object;
          assume
A9:         y in rng IF1;
          then reconsider y as Point of TR;
          consider x be object such that
A10:        x in dom IF1 & IF1.x = y by A9,FUNCT_1:def 3;
          reconsider x as Point of T|A by A10;
          now let i;
            assume
A11:          i in Seg n;
            then
A12:          y.i = f.x by A10,TOPREALC:47;
A13:        (0.TR) .i = 0 by A7;
            n1.i = 1 by A11,FINSEQ_2:57;
            hence y.i in [. 0.TR.i - n1.i,0.TR.i+n1.i .] by A3,A13,A12;
          end;
          hence thesis by Def2;
        end;
      then reconsider IF=IF1 as Function of T|A, TR|CH by A6,A5,FUNCT_2:2;
A14:  IF is continuous by PRE_TOPC:27;
A15:  n in Seg n by A1,FINSEQ_1:1;
      consider g be Function of T,TR|CH such that
A16:    g is continuous & g|A = IF by A2,A14,METRIZTS:def 1;
      set P=PROJ(n,n);
A17:  P|CH = P|(TR|CH) by A6, TMAP_1:def 4;
      reconsider Pch=P|CH as Function of TR|CH,R^1 by PRE_TOPC:9;
      reconsider Pg=Pch*g as Function of T,R^1;
A18:      P is continuous by A15,TOPREALC:57;
A19:  dom Pg = the carrier of T by FUNCT_2:def 1;
A20:  rng Pg c= rng Pch by RELAT_1:26;
A21:  (0.TR) .n = 0 by A7;
A22:  n1.n = 1 by A1,FINSEQ_1:1,FINSEQ_2:57;
      P.:CH = [. 0.TR.n - n1.n, 0.TR.n + n1.n .] by A1,FINSEQ_1:1,Th7;
      then rng Pg c= [. -1,1 .] by RELAT_1:115,A20,A21,A22;
      then reconsider Pg as Function of T,CC by A19,A3,FUNCT_2:2;
A23:  Pg is continuous by A18,A17,A16, PRE_TOPC:27;
A24:  dom Pch = CH by FUNCT_2:def 1;
      (Pch*g) |A = (Pch) * (g|A) by RELAT_1:83
                .= P*IF by A16,A8,A24,RELAT_1:165
                .= F by TOPREALC:56,A1;
      hence thesis by A23;
    end;
  hence thesis by TIETZE:24;
end;
