reserve i for Integer,
  a, b, r, s for Real;

theorem
  for f being Function of R^1,R^1, g being Function of REAL,REAL st f =
  g & g is continuous holds f is continuous
proof
  let f be Function of R^1,R^1, g be Function of REAL,REAL such that
A1: f = g and
A2: g is continuous;
  for x being Point of R^1 holds f is_continuous_at x
  proof
    let x be Point of R^1;
    dom f = REAL by A1,FUNCT_2:def 1;
    then x in dom g by A1;
    then g is_continuous_in x by A2;
    hence thesis by A1,Th21;
  end;
  hence thesis by TMAP_1:44;
end;
