proposition: 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).
proof: Proof. 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.
imports: Ensembles Relations_1 Relations_1_facts Partial_Order Cpo