Section Build_from_interior.
Variable X:Type.
Variable int : Ensemble X -> Ensemble X.
Hypothesis int_deflationary: forall S:Ensemble X,
Included (int S) S.
Hypothesis int_respects_intersection: forall S1 S2:Ensemble X,
int (Intersection S1 S2) = Intersection (int S1) (int S2).
Hypothesis int_full: int Full_set = Full_set.
Hypothesis int_idempotent: forall S:Ensemble X,
int (int S) = int S.

Lemma intersection_family_union : forall (S:Ensemble X) (F:Ensemble (Ensemble X)),
In F S -> Intersection S (FamilyUnion F) = S.