Section InducedFunction2.
(* well-defined function A->B induces a function A/R->B/S *)
Variable A B:Type.
Variable R:relation A.
Variable S:relation B.
Variable f:A->B.
Hypothesis equivR: equivalence R.
Hypothesis equivS: equivalence S.
Hypothesis well_defined2: forall a1 a2:A, R a1 a2 -> S (f a1) (f a2).
Definition projf (a:A) : quotient S :=
quotient_projection S (f a).
Definition induced_function2: quotient R -> quotient S :=
induced_function projf equivR projf_well_defined.

Lemma induced_function2_correct: forall a:A,
induced_function2 (quotient_projection R a) =
quotient_projection S (f a).