∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
  [inst_3 : p.IsPrime] (Rₚ : Type u_3) [inst_4 : CommRing Rₚ] [inst_5 : Algebra R Rₚ]
  [inst_6 : IsLocalization.AtPrime Rₚ p] [inst_7 : IsLocalRing Rₚ] (Sₚ : Type u_4) [inst_8 : CommRing Sₚ]
  [inst_9 : Algebra S Sₚ] [inst_10 : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ]
  [inst_11 : Algebra Rₚ Sₚ] [inst_12 : Algebra R Sₚ] [inst_13 : IsScalarTower R S Sₚ] [inst_14 : IsScalarTower R Rₚ Sₚ]
  [inst_15 : p.IsMaximal] (x : S),
  (IsLocalization.AtPrime.equivQuotientMapMaximalIdeal S p Rₚ Sₚ)
      ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) p)) x) =
    (Ideal.Quotient.mk (Ideal.map (algebraMap Rₚ Sₚ) (IsLocalRing.maximalIdeal Rₚ))) ((algebraMap S Sₚ) x)