intros.
simpl in H.
apply (not_all_ex_not I (fun i : I => in_part a (f i))).
assumption.
Qed.

(* Specifications: *)
(* if one of the A_i is contained in some subset B, then so is the intersection *)
Lemma subset_included_then_indexed_intersection_included :
forall B : part_set A,
(exists i : I, included (f i) B) -> included indexed_intersection B.