∀ {R : Type u_2} {A : Type u_3} {B : Type u_4} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : Semiring A]
  [inst_3 : Algebra R A] [inst_4 : StarRing A] [inst_5 : Semiring B] [inst_6 : Algebra R B] [inst_7 : StarRing B]
  [inst_8 : StarModule R A] [inst_9 : StarModule R B] (f : A →⋆ₐ[R] B) (s : Set A),
  StarSubalgebra.map f (StarAlgebra.adjoin R s) = StarAlgebra.adjoin R (⇑f '' s)