∀ {R : Type u_3} {A : Type u_1} {B : Type u_2} [inst : CommSemiring R] [inst_1 : NonUnitalSemiring A]
  [inst_2 : StarRing A] [inst_3 : Module R A] [inst_4 : NonUnitalSemiring B] [inst_5 : StarRing B] [inst_6 : Module R B]
  (S : NonUnitalStarSubalgebra R A) (S₁ : NonUnitalStarSubalgebra R B) (c : R) {x : A × B},
  x ∈ (S.prod S₁.toNonUnitalSubalgebra).carrier → c • x ∈ (S.prod S₁.toNonUnitalSubalgebra).carrier