∀ {R : Type u_2} [inst : CommSemiring R] {A : Type u_1} [inst_1 : CommSemiring A] [inst_2 : Algebra R A]
  (I J : Submodule R A) {a b : A},
  a ∈ {x | ∀ y ∈ J, x * y ∈ I} → b ∈ {x | ∀ y ∈ J, x * y ∈ I} → ∀ y ∈ J, (a + b) * y ∈ I