generalize Rel_of_transitive; intro RoT; red in RoT.
intros A B H' a b bsup H'0 H'1 H'2; try assumption.
elim H'0; elim H'1; elim H'2.
intros H'3 H'4 H'5 H'6 H'7 H'8; try assumption.
elim H'3; elim H'5; elim H'7.
intros H'9 H'10 H'11 H'12 H'13 H'14; try assumption.
apply Lub_definition; auto 8 with sets.
intros y H'15; apply H'4.
apply Upper_Bound_definition; auto with sets.
elim H'15; auto with sets.
elim (Upper_Bound_Couple_inv U D a b y);
[ intros H'22 H'23; elim H'23; intros H'24 H'25; try exact H'24; clear H'23
| idtac ]; auto with sets.
intros y0 H'16; elim H'16.
intros x H'17; apply RoT with (y := a); auto with sets.
intros x H'17; apply RoT with (y := b); auto with sets.
Qed.

(* Specifications: *)
Lemma LUaux2 :
forall A B : Ensemble U,
Included U (Union U A B) (Carrier_of U D) ->
forall a b bsup : U,
Lub U D A a ->
Lub U D B b -> Lub U D (Couple U a b) bsup -> Lub U D (Union U A B) bsup.