(****************************************************************************)
(** Characterization of a relation over two sets                            *)

Inductive Rel (U : Type) (A B : Ensemble U) (R : Relation U) : Prop :=
    Rel_intro :
      (forall x y : U, R x y -> In U A x) ->
      (forall x y : U, R x y -> In U B y) -> Rel U A B R.(****************************************************************************)
(** Characterization of a relation over two sets                            *)

Inductive Rel (U : Type) (A B : Ensemble U) (R : Relation U) : Prop :=
    Rel_intro :
      (forall x y : U, R x y -> In U A x) ->
      (forall x y : U, R x y -> In U B y) -> Rel U A B R.Inductive Im (R : Relation U) (A : Ensemble U) (y : U) : Prop :=
    Im_intro : forall x : U, R x y -> In U A x -> Im R A y.Inductive Im (R : Relation U) (A : Ensemble U) (y : U) : Prop :=
    Im_intro : forall x : U, R x y -> In U A x -> Im R A y.

(* Specifications: *)
Lemma Im_stable_par_incl : forall (R : Relation U) (A1 A2 : Ensemble U), Included U A1 A2 -> Included U (Im R A1) (Im R A2).