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 ].
Definition equiv_classes : Ensemble (Ensemble A) :=
Im Full_set equiv_class.
Definition quotient : Type :=
{ S:Ensemble A | In equiv_classes S }.
Definition quotient_projection (x:A) : quotient :=
exist _ (equiv_class x) (equiv_class_in_quotient x).

Lemma quotient_projection_collapses_R: forall x1 x2:A,
R x1 x2 -> quotient_projection x1 = quotient_projection x2.