reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;

theorem Th16:
  for T being non empty TopSpace, X being non empty Subset of T, Y
being compact Subset of T, f being continuous RealMap of T st X c= Y
holds lower_bound
  (f|Y) <= lower_bound (f|X)
proof
  let T be non empty TopSpace, X be non empty Subset of T, Y be compact Subset
  of T, f be continuous RealMap of T;
A1: (f|Y).:the carrier of (T|Y) = (f|Y).:Y by PRE_TOPC:8
    .= f.:Y by RELAT_1:129;
  assume
A2: X c= Y;
  then reconsider Y1 = Y as non empty compact Subset of T;
A3: (f|X).:the carrier of (T|X) = (f|X).:X by PRE_TOPC:8
    .= f.:X by RELAT_1:129;
  (f|Y1).:the carrier of (T|Y1) is bounded_below by MEASURE6:def 10;
  hence thesis by A2,A1,A3,RELAT_1:123,SEQ_4:47;
end;
