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.
Definition induced_function (xbar:quotient R) : B :=
proj1_sig (constructive_definite_description _
(description_of_fbar xbar)).

Lemma induced_function_unique: forall fbar:quotient R->B,
(forall x:A, fbar (quotient_projection R x) = f x) ->
(forall xbar:quotient R, fbar xbar = induced_function xbar).