reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem
  for S1, T1, S2, T2 being TopSpace, f being Function of S1, S2, g being
Function of T1, T2 st the TopStruct of S1 = the TopStruct of T1 & the TopStruct
  of S2 = the TopStruct of T2 & f = g & f is continuous holds g is continuous
proof
  let S1, T1, S2, T2 be TopSpace, f be Function of S1, S2, g be Function of T1
  , T2 such that
A1: the TopStruct of S1 = the TopStruct of T1 and
A2: the TopStruct of S2 = the TopStruct of T2 and
A3: f = g and
A4: f is continuous;
  now
    let P2 be Subset of T2 such that
A5: P2 is closed;
    reconsider P1 = P2 as Subset of S2 by A2;
    P1 is closed by A2,A5,TOPS_3:79;
    then f"P1 is closed by A4;
    hence g"P2 is closed by A1,A3,TOPS_3:79;
  end;
  hence thesis;
end;
