reserve T for TopStruct;
reserve GX for TopSpace;
reserve T for TopStruct,
  x,y for Point of T;

theorem Th32:
  for S,T being TopSpace, f being Function of S,T, g being
  Function of the TopStruct of S, T st f = g holds f is continuous iff g is
  continuous
proof
  let S,T be TopSpace, f be Function of S,T, g be Function of the TopStruct of
  S, T such that
A1: f = g;
  thus f is continuous implies g is continuous
  by A1,Th31;
  assume
A2: g is continuous;
  let P1 being Subset of T;
  assume P1 is closed;
  then g"P1 is closed by A2;
  hence f" P1 is closed by A1,Th31;
end;
