Require Export Ensembles.
Require Export Image.
Require Export Relation_Definitions.
Require Export EnsemblesSpec.
Set Implicit Arguments.
Section Quotient.
Variable A:Type.
Variable R:relation A.
Hypothesis equivR:equivalence R.
Definition equiv_class (x:A) : Ensemble A :=
[ y:A | R x y ].

(* Specifications: *)
Lemma R_impl_equality_of_equiv_class:
forall x y:A, R x y -> equiv_class x = equiv_class y.
Lemma equality_of_equiv_class_impl_R:
forall x y:A, equiv_class x = equiv_class y -> R x y.