  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,S,A,B st T is normal
  for X be Subset of TOP-REAL n st
    X is compact non boundary convex & B,X are_homeomorphic
  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
proof
  let T,S,A,B such that
A1: T is normal;
A2: [#](T|A)=A by PRE_TOPC:def 5;
A3: [#](S|B)=B by PRE_TOPC:def 5;
  set TR=TOP-REAL n;
  let X be Subset of TR such that
A4: X is compact non boundary convex
  and
A5: B,X are_homeomorphic;
  consider h be Function of S|B,TR|X such that
A6:h is being_homeomorphism by METRIZTS:def 1,A5,T_0TOPSP:def 1;
A7: h" is continuous by TOPS_2:def 5,A6;
  let f being Function of T|A,S|B such that
A8: f is continuous;
A9:rng f c= the carrier of (S|B);
A10: dom h = [#](S|B) by TOPS_2:def 5,A6;
A11:X is non empty by A4;
A12: rng h=[#](TR|X) by TOPS_2:def 5,A6;
  then
A13: B is non empty by A11;
  per cases;
  suppose
A14:  A is empty;
    reconsider SB=S|B as non empty TopSpace by A11,A12;
    set h=the continuous Function of T,SB;
    reconsider h as Function of T,S|B;
    take h;
    f={} by A14;
    hence thesis by A14;
  end;
  suppose
A15:  A is non empty;
    reconsider hf=h*f as Function of T|A,TR|X by A3,A13;
    consider g be Function of T,TR | X such that
A16:  g is continuous
    and
A17:  g|A=hf by A3,A13,A2,A11,A15,A8,A6,A1,A4,Th23;
    reconsider hg= h"*g as Function of T,S| B by A11;
    take hg;
    hg |A = h"*(g|A) by RELAT_1:83
         .= (h"*h)*f by A17,RELAT_1:36
         .= (id dom h)*f by TOPS_2:52,A12, A6
         .= f by A10,A9,RELAT_1:53;
    hence thesis by A7,A11, A16,TOPS_2:46;
  end;
end;
