reserve r, s, t, g for Real,

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

theorem
  for T being non empty TopSpace, f being RealMap of T st (for p being
  Point of T holds f.p >= r) & for t st for p being Point of T holds f.p >= t
  holds r >= t holds r = lower_bound f
proof
  let T be non empty TopSpace, f be RealMap of T;
  set c = the carrier of T;
  set fc = (f.:the carrier of T);
  assume that
A1: for p being Point of T holds f.p >= r and
A2: for t st for p being Point of T holds f.p >= t holds r >= t;
A3: now
    let t;
    assume for s st s in fc holds s >= t;
    then for s being Point of T holds f.s >= t by FUNCT_2:35;
    hence r >= t by A2;
  end;
  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 >= r by A1;
  end;
  hence thesis by A3,SEQ_4:44;
end;
