reserve T for TopStruct;
reserve GX for TopSpace;

theorem
  for A being TopSpace, B being non empty TopSpace for f being Function
  of A,B for C being SubSpace of B holds f is continuous implies for h being
  Function of A,C st h = f holds h is continuous
proof
  let A be TopSpace, B be non empty TopSpace, f be Function of A,B, C be
  SubSpace of B;
  assume
A1: f is continuous;
  let h be Function of A,C such that
A2: h = f;
A3: rng f c= the carrier of C by A2,RELAT_1:def 19;
  for P being Subset of C holds P is closed implies h"P is closed
  proof
    let P be Subset of C;
    assume P is closed;
    then consider Q being Subset of B such that
A4: Q is closed and
A5: Q /\ ([#] C) = P by Th13;
    h"P = f"Q /\ f"([#] C) by A2,A5,FUNCT_1:68
      .= f"Q /\ f"(rng f /\ [#] C) by RELAT_1:133
      .= f"Q /\ f"(rng f) by A3,XBOOLE_1:28
      .= f"Q /\ dom f by RELAT_1:134
      .= f"Q by RELAT_1:132,XBOOLE_1:28;
    hence thesis by A1,A4;
  end;
  hence thesis;
end;
