intro H'; elim H'.
intro h; elim h; intros bot E; clear h.
intro H'0; exists bot.
apply Lub_definition.
apply Upper_Bound_definition.
elim E; auto.
intros y H'1; elim H'1; auto.
intros y H'1; elim H'1; auto.
elim E; auto.
Qed.
Hint Resolve Empty_set_has_lub.

Theorem Upper_downward_stable :
forall (A B : Ensemble U) (maj : U),
Included U A (Carrier_of U D) ->
Included U B (Carrier_of U D) ->
Included U A B -> Upper_Bound U D B maj -> Upper_Bound U D A maj.