theorem
  for f be real-valued Function st rng f is bounded_above holds
    f is bounded_above
proof
  let f be real-valued Function;
  set X = dom f;
  assume rng f is bounded_above;
  then consider a be Real such that
A1: a is UpperBound of rng f;
AA: f|X = f;
  for x1 being object st x1 in X /\ dom f holds f.x1 <= a
  proof
    let x1 be object;
    assume x1 in X /\ dom f;
    then f.x1 in rng f by FUNCT_1:def 3;
    hence thesis by A1,XXREAL_2:def 1;
  end;
  hence thesis by AA,RFUNCT_1:70;
end;
