Theorem Lub_of_Union : Conditionally_complete U D -> forall A B : Ensemble U, Included U (Union U A B) (Carrier_of U D) -> forall x : U, In U (Carrier_of U D) x -> Upper_Bound U D (Union U A B) x -> forall a b : U, Lub U D A a -> Lub U D B b -> exists bsup : U, Lub U D (Union U A B) bsup /\ Lub U D (Couple U a b) bsup.