∀ {R : Type u} [inst : Monoid R] {A : Type v} {B : Type w} {C : Type w₁} [inst_1 : NonUnitalNonAssocSemiring A]
  [inst_2 : DistribMulAction R A] [inst_3 : NonUnitalNonAssocSemiring B] [inst_4 : NonUnitalNonAssocSemiring C]
  [inst_5 : DistribMulAction R B] [inst_6 : DistribMulAction R C] (f : A →ₙₐ[R] B) (g : A →ₙₐ[R] C) (i : A),
  (f.prod g) i = Pi.prod (⇑f) (⇑g) i