Section Advanced_lemmas.
Variable U : Type.
Variable D : PO U.
Hint Resolve Lub_definition Upper_Bound_Couple_intro.

Lemma Lub_monotonic :
forall A B : Ensemble U,
Included U A B ->
Included U B (Carrier_of U D) ->
forall a b : U, Lub U D A a -> Lub U D B b -> Rel_of U D a b.