intros A H'; try assumption.
unfold included in |- *; auto with algebra.
intros x H'0; try assumption.
apply union_part_prop_rev with (A := A); auto with algebra.
Qed.

(* Specifications: *)
Lemma union_part_upper_bound :
forall Y : part_set E,
(forall A : part_set E, in_part A P -> included A Y) ->
included union_part Y.