  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 Th23:
  for T,A st T is normal
  for X be Subset of TOP-REAL n st X is compact non boundary convex
  for f being Function of T|A,(TOP-REAL n) | X st
    f is continuous
  ex g being Function of T,(TOP-REAL n) | X st g is continuous & g|A=f
proof
  set TR=TOP-REAL n;
  let T,A such that
A1: T is normal;
  let S be Subset of TR such that
A2:S is compact non boundary convex;
  let f be Function of T|A,TR | S such that
A3:f is continuous;
A4: [#](T|A) = A by PRE_TOPC:def 5;
A5: [#](TR | S) = S by PRE_TOPC:def 5;
   per cases;
     suppose
A6:      A is empty;
       reconsider TRS=TR|S as non empty TopSpace by A2,A5;
       set g=the continuous Function of T,TRS;
A7:    g|A={} by A6;
       f={} by A6;
       hence thesis by A7;
     end;
     suppose
A8:      A is non empty;
       set H=ClosedHypercube(0.TOP-REAL n,n |-> 1);
       consider h be Function of TR |S,TR |H such that
A9:      h is being_homeomorphism
       and
         h.:Fr S = Fr H by BROUWER2:7,A2;
A10:   rng h = [#](TR|H) by A9,TOPS_2:def 5;
A11:   TR|S is non empty by A5,A2;
       then reconsider hf=h*f as Function of T|A,TR|H;
       consider g be Function of T,TR|H such that
A12:     g is continuous
       and
A13:     g|A=hf by A3, A9, A4,A8,A11,A1,Th22;
       reconsider hg=h"*g as Function of T,TR|S;
       take hg;
       h" is being_homeomorphism by A9,TOPS_2:56;
       hence hg is continuous by A12,TOPS_2:46;
A14:   rng f c= the carrier of (TR|S);
A15:   dom h = [#](TR|S) by A9,TOPS_2:def 5;
       thus hg|A = h"*hf by RELAT_1:83,A13
                .= (h"*h)*f by RELAT_1:36
                .= (id [#](TR|S))*f by A9,A10,A15,TOPS_2:52
                .= f by A14,RELAT_1:53;
     end;
end;
