Definition quotient_projection (x:A) : quotient :=
exist _ (equiv_class x) (equiv_class_in_quotient x).

(* Specifications: *)
Lemma quotient_projection_correct: forall x:A,
proj1_sig (quotient_projection x) = equiv_class x.
Lemma quotient_projection_surjective: forall xbar:quotient,
exists x:A, quotient_projection x = xbar.
Lemma quotient_projection_collapses_R: forall x1 x2:A,
R x1 x2 -> quotient_projection x1 = quotient_projection x2.
Lemma quotient_projection_minimally_collapses_R: forall x1 x2:A,
quotient_projection x1 = quotient_projection x2 -> R x1 x2.