∀ {R : Type u_2} {A : Type u_1} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : Semiring A]
  [inst_3 : StarRing A] [inst_4 : Algebra R A] [inst_5 : StarModule R A] (S : StarSubalgebra R A) {a b : A},
  a ∈ S.carrier → b ∈ S.carrier → a + b ∈ S.carrier