End Quotient.
Section InducedFunction.
(* well-defined A->B induces a function A/R->B *)
Variable A B:Type.
Variable R:Relation A.
Variable f:A->B.
Hypothesis equiv:equivalence R.
Hypothesis well_defined: forall x y:A, R x y -> f x = f y.

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