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;
