∀ {R : Type u} {A : Type v} {B : Type w} {C : Type w'} [inst : CommSemiring R] [inst_1 : Semiring A]
  [inst_2 : Algebra R A] [inst_3 : Semiring B] [inst_4 : Algebra R B] [inst_5 : Semiring C] [inst_6 : Algebra R C]
  (S : Subalgebra R A) (g : B →ₐ[R] C) (f : A →ₐ[R] B),
  Subalgebra.map g (Subalgebra.map f S) = Subalgebra.map (g.comp f) S