reserve r, s, t, g for Real,

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

theorem Th1:
  for T being non empty TopSpace, f being bounded_below RealMap of
  T for p being Point of T holds f.p >= lower_bound f
proof
  let T be non empty TopSpace, f be bounded_below RealMap of T;
  set fc = (f.:the carrier of T);
  let p be Point of T;
  fc is bounded_below & f.p in fc by FUNCT_2:35,MEASURE6:def 10;
  hence thesis by SEQ_4:def 2;
end;
