reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem
  for T being set, F being Subset-Family of T holds F <> {} iff
  COMPLEMENT(F) <> {}
proof
  let T be set, F be Subset-Family of T;
  thus F <> {} implies COMPLEMENT(F) <> {} by SETFAM_1:32;
  assume COMPLEMENT(F) <> {};
  then COMPLEMENT(COMPLEMENT(F)) <> {} by SETFAM_1:32;
  hence thesis;
end;
