intro.
red in |- *.
intros.
unfold indexed_intersection in H.
simpl in H.
apply H.
Qed.

(* if a \in indexed_intersection then a \in A_i (\forall i \in I) *)
Lemma in_indexed_intersection_then_in_subsets :
forall a : A,
in_part a indexed_intersection -> forall i : I, in_part a (f i).