Section InducedFunction2arg.
(* well-defined function A x B -> C induces a function (A/R) x (B/S) -> C *)
Variable A B C:Type.
Variable R:relation A.
Variable S:relation B.
Variable f:A->B->C.
Hypothesis equivR:equivalence R.
Hypothesis equivS:equivalence S.
Hypothesis well_defined_2arg: forall (a1 a2:A) (b1 b2:B),
R a1 a2 -> S b1 b2 -> f a1 b1 = f a2 b2.
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.

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