reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th16:
  for T being set, F, G being Subset-Family of T st F c= G & G is
  compl-closed holds COMPLEMENT F c= G
proof
  let T be set, F, G be Subset-Family of T;
  assume
A1: F c= G & G is compl-closed;
  let x be object;
  assume
A2: x in COMPLEMENT F;
  then reconsider x9 = x as Subset of T;
  x9` in F by A2,SETFAM_1:def 7;
  then x9`` in G by A1;
  hence thesis;
end;
