Definition induced_function2: quotient R -> quotient S :=
  induced_function projf equivR projf_well_defined.

(* Specifications: *)
Lemma induced_function2_correct: forall a:A, induced_function2 (quotient_projection R a) = quotient_projection S (f a).
Lemma slices_well_defined: forall (a:A) (b1 b2:B), S b1 b2 -> f a b1 = f a b2.