Section Lemmas_on_partial_orders.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.
Hint Resolve Rel_of_reflexive.
Hint Resolve Rel_of_antisymmetric.
Hint Resolve Rel_of_transitive.
Hint Resolve Couple_included_in_carrier.
Hint Resolve Singleton_has_lub.
Hint Resolve Empty_set_has_Upper_Bound.
intros x H'; apply Upper_Bound_definition; auto with sets.
intros y H'0; elim H'0; auto with sets.
Qed.
Hint Resolve Anyone_is_Upper_Bound_of_Empty_set.
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.