Hint Resolve Empty_set_is_Bottom.

(* Specifications: *)
Theorem Union_is_Lub :
forall a b : Ensemble U,
Included U a A ->
Included U b A ->
Lub (Ensemble U) Power_set_PO (Couple (Ensemble U) a b) (Union U a b).
Theorem Intersection_is_Glb :
forall a b : Ensemble U,
Included U a A ->
Included U b A ->
Glb (Ensemble U) Power_set_PO (Couple (Ensemble U) a b)
(Intersection U a b).
Theorem Empty_set_zero :
forall X : Ensemble U, Union U (Empty_set U) X = X.
Theorem Union_commutative :
forall A B : Ensemble U, Union U A B = Union U B A.
Theorem Union_associative :
forall A B C : Ensemble U,
Union U (Union U A B) C = Union U A (Union U B C).
Theorem Non_disjoint_union :
forall (X : Ensemble U) (x : U),
In U X x -> Union U (Singleton U x) X = X.
Theorem Finite_plus_one_is_finite :
forall (X : Ensemble U) (x : U),
Finite U X -> Finite U (Union U (Singleton U x) X).