Definition induced1 (a:A) : quotient S -> C :=
induced_function (f a) equivS (slices_well_defined a).
Definition eq_fn (f g:quotient S->C) :=
forall b:quotient S, f b = g b.

(* Specifications: *)
Lemma eq_fn_equiv: equivalence eq_fn.
Lemma well_defined_induced1: forall a1 a2:A, R a1 a2 ->
eq_fn (induced1 a1) (induced1 a2).