Section InducedFunction3.
(* well-defined function A x B -> C induces a function (A/R) x (B/S) -> C/T *)
Variable A B C:Type.
Variable R:relation A.
Variable S:relation B.
Variable T:relation C.
Variable f:A->B->C.
Hypothesis equivR:equivalence R.
Hypothesis equivS:equivalence S.
Hypothesis equivT:equivalence T.
Hypothesis well_defined3: forall (a1 a2:A) (b1 b2:B),
R a1 a2 -> S b1 b2 -> T (f a1 b1) (f a2 b2).
Definition projf2 (a:A) (b:B) :=
quotient_projection T (f a b).
Definition induced_function3 : quotient R -> quotient S -> quotient T :=
induced_function2arg projf2 equivR equivS projf2_well_defined.

Lemma induced_function3_correct: forall (a:A) (b:B),
induced_function3 (quotient_projection R a)
(quotient_projection S b) =
quotient_projection T (f a b).