End filter_from_subbasis.
Arguments Build_Filter_from_subbasis {X}.
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.