reserve r, s, t, g for Real,

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

theorem
  for T being non empty TopSpace, f being bounded_below RealMap of T for
  s being Real st for t being Point of T holds f.t >= s
  holds lower_bound f >= s
proof
  let T be non empty TopSpace, f be bounded_below RealMap of T;
  set c = the carrier of T;
  set fc = (f.:the carrier of T);
  let s be Real;
  assume
A1: for t being Point of T holds f.t >= s;
  now
    let p1 be Real;
    assume p1 in fc;
    then ex x being object st x in c & x in c & p1 = f.x by FUNCT_2:64;
    hence p1 >= s by A1;
  end;
  hence thesis by SEQ_4:43;
end;
