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.

Theorem Singleton_has_lub :
forall x : U, In U (Carrier_of U D) x -> Lub U D (Singleton U x) x.