Qed.
Hint Resolve Inclusion_is_transitive.

(* Specifications: *)
Theorem Same_set_equivalence : Equivalence (Ensemble U) (Same_set U).
Theorem Same_set_reflexive : Reflexive (Ensemble U) (Same_set U).
Hint Resolve Same_set_reflexive.
Theorem Power_set_PO : PO (Ensemble U).
Hint Resolve Union_minimal.
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.