∀ {R : Type u_3} {A : Type u_1} {B : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
  [inst_3 : Semiring B] [inst_4 : Algebra R B] (S : Subalgebra R A) (S₁ : Subalgebra R B) {a b : A × B},
  a ∈ (S.prod S₁.toSubsemiring).carrier →
    b ∈ (S.prod S₁.toSubsemiring).carrier → a * b ∈ (S.prod S₁.toSubsemiring).carrier