Definition Family := Ensemble (Ensemble T).Inductive FamilyUnion: Ensemble T :=
  | family_union_intro: forall (S:Ensemble T) (x:T),
    In F S -> In S x -> In FamilyUnion x.Inductive FamilyUnion: Ensemble T :=
  | family_union_intro: forall (S:Ensemble T) (x:T),
    In F S -> In S x -> In FamilyUnion x.Inductive FamilyIntersection: Ensemble T :=
  | family_intersection_intro: forall x:T,
    (forall S:Ensemble T, In F S -> In S x) ->
    In FamilyIntersection x.Inductive FamilyIntersection: Ensemble T :=
  | family_intersection_intro: forall x:T,
    (forall S:Ensemble T, In F S -> In S x) ->
    In FamilyIntersection x.

Lemma subfamily_intersection: forall F G:Family T, Included F G -> Included (FamilyIntersection G) (FamilyIntersection F).