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.
Lemma description_of_fbar: forall xbar:quotient R, exists! y:B, exists x:A, quotient_projection R x = xbar /\ f x = y.