∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (s : Subsemiring L)
  (hK : ∀ (x : K), (algebraMap K L) x ∈ s)
  (hi :
    ∀ x ∈ { toSubsemiring := s, algebraMap_mem' := hK }.carrier,
      x⁻¹ ∈ { toSubsemiring := s, algebraMap_mem' := hK }.carrier)
  (x : L), x ∈ { toSubsemiring := s, algebraMap_mem' := hK, inv_mem' := hi } ↔ x ∈ s