proposition: Lemma FilterBase_Generated {X : UU} (base : BaseOfFilter X) Hbase : FilterBase base = FilterGenerated base Hbase.
proof: Proof. simple refine (subtypePath_prop (B := λ _, make_hProp _ _) _). - apply isapropdirprod. + apply isapropdirprod. * apply isaprop_isfilter_imply. * apply isaprop_isfilter_finite_intersection. + apply isaprop_isfilter_notempty. - simpl. apply filterbase_genetated. + intros A B. apply (BaseOfFilter_and base). + apply (BaseOfFilter_notempty base). Qed.
imports: UniMath.Foundations.Preamble UniMath.MoreFoundations.Tactics UniMath.Topology.Prelim UniMath.MoreFoundations.PartA