Inductive Power_set (A : Ensemble U) : Ensemble (Ensemble U) :=
       Definition_of_Power_set :
         forall X : Ensemble U,
         Included U X A -> In (Ensemble U) (Power_set A) X.Inductive Power_set (A : Ensemble U) : Ensemble (Ensemble U) :=
       Definition_of_Power_set :
         forall X : Ensemble U,
         Included U X A -> In (Ensemble U) (Power_set A) X.

(* Specifications: *)
Theorem Empty_set_minimal : forall X : Ensemble U, Included U (Empty_set U) X.
Theorem Power_set_non_empty : forall A : Ensemble U, Non_empty (Ensemble U) (Power_set A).
Theorem Inclusion_is_an_order : Order (Ensemble U) (Included U).
Theorem Inclusion_is_transitive : Transitive (Ensemble U) (Included U).
Theorem Same_set_equivalence : Equivalence (Ensemble U) (Same_set U).
Theorem Same_set_reflexive : Reflexive (Ensemble U) (Same_set U).
Theorem Power_set_PO : PO (Ensemble 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 Empty_set_is_Bottom : Bottom (Ensemble U) Power_set_PO (Empty_set U).
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).
Theorem Singleton_is_finite : forall x : U, Finite U (Singleton U x).
Theorem Union_of_finite_is_finite : forall X Y : Ensemble U, Finite U X -> Finite U Y -> Finite U (Union U X Y).