
theorem
  for T being TopStruct, f being RealMap of T, g being Function of
  T, R^1 st f = g holds f is continuous iff g is continuous
proof
  let T be TopStruct, f be RealMap of T, g be Function of T, R^1 such that
A1: f = g;
  thus f is continuous implies g is continuous
  proof
    assume
A2: for Y being Subset of REAL st Y is closed holds f"Y is closed;
    let P be Subset of R^1 such that
A3: P is closed;
    reconsider R = P as Subset of REAL by TOPMETR:17;
    R is closed by A3,Th23;
    hence thesis by A1,A2;
  end;
  assume
A4: for Y being Subset of R^1 st Y is closed holds g"Y is closed;
  let P be Subset of REAL such that
A5: P is closed;
  reconsider R = P as Subset of R^1 by TOPMETR:17;
  R is closed by A5,Th23;
  hence thesis by A1,A4;
end;
