Section Advanced_lemmas.
Variable U : Type.
Variable D : PO U.
Hint Resolve Lub_definition Upper_Bound_Couple_intro.
intros A B H' H'0 a b H'1 H'2; try assumption.
elim H'1; auto with sets.
intros H'3 H'4; apply H'4.
apply Upper_downward_stable with (B := B); auto with sets.
elim H'2; auto with sets.
Qed.

Lemma Compatible_lubs :
forall A B C : Ensemble U,
Included U A C ->
Included U B C ->
Included U C (Carrier_of U D) ->
forall a b c : U,
Lub U D A a -> Lub U D B b -> Lub U D C c -> Compatible U D a b.