intros A a b H' H'0.
apply Lub_definition; simpl.
- apply Upper_Bound_definition; simpl; auto with sets.
intros y H'1; elim H'1; auto with sets.
- intros y H'1; elim H'1; simpl; auto with sets.
Qed.

Theorem Intersection_is_Glb :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Glb (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b)
(Intersection U a b).