Definition ultrafilter {X:Type} (F:Filter X) : Prop :=
  forall S:Ensemble X, In (filter_family F) S \/
                       In (filter_family F) (Ensembles.Complement S).

Lemma ultrafilter_extension: forall {X:Type} (F:Filter X), exists U:Filter X, Included (filter_family F) (filter_family U) /\ ultrafilter U.