Definition Power_set_PO : Ensemble U -> PO (Ensemble U).

(* Specifications: *)
Theorem Strict_Rel_is_Strict_Included : same_relation (Ensemble U) (Strict_Included U) (Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U))).
Lemma Strict_inclusion_is_transitive_with_inclusion : forall x y z:Ensemble U, Strict_Included U x y -> Included U y z -> Strict_Included U x z.
Lemma Strict_inclusion_is_transitive_with_inclusion_left : forall x y z:Ensemble U, Included U x y -> Strict_Included U y z -> Strict_Included U x z.
Lemma Strict_inclusion_is_transitive : Transitive (Ensemble U) (Strict_Included U).
Theorem Empty_set_is_Bottom : forall A:Ensemble U, Bottom (Ensemble U) (Power_set_PO A) (Empty_set U).
Theorem Union_minimal : forall a b X:Ensemble U, Included U a X -> Included U b X -> Included U (Union U a b) X.
Theorem Intersection_maximal : forall a b X:Ensemble U, Included U X a -> Included U X b -> Included U X (Intersection U a b).
Theorem Union_increases_l : forall a b:Ensemble U, Included U a (Union U a b).
Theorem Union_increases_r : forall a b:Ensemble U, Included U b (Union U a b).
Theorem Intersection_decreases_l : forall a b:Ensemble U, Included U (Intersection U a b) a.
Theorem Intersection_decreases_r : forall a b:Ensemble U, Included U (Intersection U a b) b.
Theorem Union_is_Lub : forall A a b:Ensemble U, Included U a A -> Included U b A -> Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b).
Theorem Intersection_is_Glb : forall A a b:Ensemble U, Included U a A -> Included U b A -> Glb (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Intersection U a b).