intros X bound H' H'0; elim H'0; auto with sets.
Qed.

(* Specifications: *)
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.
Theorem Lub_is_unique :
Conditionally_complete U D ->
forall (C : Ensemble U) (bsup1 bsup2 : U),
Lub U D C bsup1 -> Lub U D C bsup2 -> bsup1 = bsup2.