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

(* Specifications: *)
Lemma induced_function_correct: forall x:A, induced_function (quotient_projection R x) = f x.
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).