Hint Resolve Empty_set_has_lub.

(* Specifications: *)
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.
Theorem Conditionally_complete_has_a_bottom :
Conditionally_complete U D -> exists bot : U, Bottom U D bot.