Section more_lemmas.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.
intros X bound H' H'0; elim H'0; auto with sets.
Qed.

Theorem Lower_Bound_is_in_Carrier :
forall (X : Ensemble U) (bound : U),
Included U X (Carrier_of U D) ->
Lower_Bound U D X bound -> In U (Carrier_of U D) bound.