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.

Theorem Anyone_is_Upper_Bound_of_Empty_set :
forall x : U, In U (Carrier_of U D) x -> Upper_Bound U D (Empty_set U) x.