 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
  A is closed & B is closed implies
    for h be Function of (TOP-REAL n) |A,(TOP-REAL n) |B st
      h is being_homeomorphism
    holds h.:Int A = Int B & h.:Fr A = Fr B
proof
  assume that
A1:   A is closed
    and
A2:   B is closed;
  set TR=TOP-REAL n;
A3: [#](TR|A) = A by PRE_TOPC:def 5;
A4: Int B c= B by TOPS_1:16;
A5: [#](TR|B)=B by PRE_TOPC:def 5;
  let h be Function of TR |A,TR |B such that
A6:  h is being_homeomorphism;
A7: dom h = [#](TR|A) by A6,TOPS_2:def 5;
A8: rng h = [#](TR|B) by A6,TOPS_2:def 5;
A9: Fr A \/Int A = (A \Int A) \/Int A by A1,TOPS_1:43
                .= A\/Int A by XBOOLE_1:39
                .= A by TOPS_1:16,XBOOLE_1:12;
  thus
A10:  h.:Int A = Int B
  proof
    thus h.:Int A c= Int B
    proof
      let y be object;
      assume y in h.:Int A;
      then ex x be object st x in dom h & x in Int A & h.x=y
        by FUNCT_1:def 6;
      hence thesis by A2,A6,Th11;
    end;
    let y be object;
    assume
A11:  y in Int B;
    then consider x be object such that
A12:    x in dom h
      and
A13:    h.x=y by A4,A8,A5,FUNCT_1:def 3;
    reconsider x as Point of TR by A7,A3,A12;
    assume
A14:  not y in h.:Int A;
    not x in Int A by A12,FUNCT_1:def 6,A14,A13;
    then x in Fr A by A12,A9,A3,XBOOLE_0:def 3;
    then h.x in Fr B by Th10,A1,A6;
    hence thesis by A11,A13, TOPS_1:39,XBOOLE_0:3;
  end;
  Fr A = A \Int A by A1,TOPS_1:43;
  then h.:Fr A = (h.:A) \ (h.:Int A) by A6,FUNCT_1:64
               .= B\ (h.:Int A) by RELAT_1:113,A7,A8,A3,A5
               .= Fr B by A10,A2,TOPS_1:43;
  hence thesis;
end;
