∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (s : Subsemiring A)
  (h : ∀ (r : R), (algebraMap R A) r ∈ s.carrier) (x : A), (x ∈ { toSubsemiring := s, algebraMap_mem' := h }) = (x ∈ s)