intros.
inversion H.
unfold included in |- *.
unfold included in H0.
intros.
apply H0.
simpl in H1.
apply H1.
Qed.
End indexed_int.
(* The following is 'predefined' as "image_map"; I prefer this name: *)
Let range (A B : Setoid) (f : MAP A B) := image f (full A).
Section Compatibility_of_indexed_intersections.
Variable A : Setoid.
Variable I : Setoid.

(** - If $(F_i)_i$ (indexed by $f$) and $(G_j)_j$ (indexed by $g$) denote the same set of subsets of $A$, then the indexed intersections of the $F_i$ and $G_j$ are equal *)
Lemma indexed_intersection_comp :
forall f g : MAP I (part_set A),
Equal f g -> Equal (indexed_intersection f) (indexed_intersection g).