∀ {A : Type u_5} [inst : CommSemiring A] {R : Type u_3} [inst_1 : CommSemiring R] [inst_2 : Algebra A R]
  {M : Submonoid R} (S : Type u_2) [inst_3 : CommSemiring S] [inst_4 : Algebra R S] [inst_5 : IsLocalization M S]
  {P : Type u_4} [inst_6 : CommSemiring P] [inst_7 : Algebra A P] {T : Submonoid P} (Q : Type u_1)
  [inst_8 : CommSemiring Q] [inst_9 : Algebra P Q] [inst_10 : IsLocalization T Q] (h : R ≃ₐ[A] P)
  (H : Submonoid.map h M = T) (x y : S),
  (IsLocalization.ringEquivOfRingEquiv S Q h.toRingEquiv H).toFun (x + y) =
    (IsLocalization.ringEquivOfRingEquiv S Q h.toRingEquiv H).toFun x +
      (IsLocalization.ringEquivOfRingEquiv S Q h.toRingEquiv H).toFun y