∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
  [inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : StarModule R A] [inst_6 : IsScalarTower R A A]
  [inst_7 : SMulCommClass R A A] {S₁ : NonUnitalSubalgebra R A} {S₂ : NonUnitalStarSubalgebra R A},
  S₁.starClosure ≤ S₂ ↔ S₁ ≤ S₂.toNonUnitalSubalgebra