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_Inhabited : forall X:Ensemble U, Inhabited (Ensemble U) (Power_set X).
Theorem Inclusion_is_an_order : Order (Ensemble U) (Included U).
Theorem Inclusion_is_transitive : Transitive (Ensemble U) (Included U).