intros a b; red.
intros x H'; elim H'; auto with sets.
Qed.
#[local]
Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l
Intersection_decreases_r : core.

(* Specifications: *)
Theorem Union_is_Lub :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b).