reserve T for TopStruct;
reserve GX for TopSpace;

theorem
  for A,B,C being TopSpace for f being Function of A,C holds f is
  continuous & C is SubSpace of B implies for h being Function of A,B st h = f
  holds h is continuous
proof
  let A,B,C be TopSpace, f be Function of A,C;
  assume that
A1: f is continuous and
A2: C is SubSpace of B;
  let h be Function of A,B such that
A3: h = f;
  for P being Subset of B holds P is closed implies h"P is closed
  proof
    let P be Subset of B such that
A4: P is closed;
A5: rng h c= the carrier of C by A3,RELAT_1:def 19;
A6: h"P = h"(rng h /\ P) by RELAT_1:133
      .= h"(rng h /\ [#] C /\ P) by A5,XBOOLE_1:28
      .= h"(rng h /\ ([#] C /\ P)) by XBOOLE_1:16
      .= h"(P /\ [#] C) by RELAT_1:133;
    reconsider C as SubSpace of B by A2;
    reconsider Q = P /\ [#] C as Subset of C;
    Q is closed by A4,Th13;
    hence thesis by A1,A3,A6;
  end;
  hence thesis;
end;
