theorem Th18:
  for f being PartFunc of A,REAL st f|A is bounded_above holds rng
  (f|X) is bounded_above
proof
  let f be PartFunc of A,REAL;
  assume f|A is bounded_above;
  then rng f is bounded_above by INTEGRA1:13;
  hence thesis by RELAT_1:70,XXREAL_2:43;
end;
