reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;

theorem
  for T being non empty TopSpace, f being bounded_above RealMap of T for
  t st for p being Point of T holds f.p <= t holds upper_bound f <= t
proof
  let T be non empty TopSpace, f be bounded_above RealMap of T;
  set c = the carrier of T;
  set fc = (f.:the carrier of T);
  let t;
  assume
A1: for p being Point of T holds f.p <= t;
  now
    let s;
    assume s in fc;
    then ex x being object st x in c & x in c & s = f.x by FUNCT_2:64;
    hence s <= t by A1;
  end;
  hence thesis by SEQ_4:45;
end;
