-- Module: Mathlib.RingTheory.Localization.AtPrime.Basic
Goal:
R_1 : Type u_1
inst : CommSemiring R_1
S_1 : Type u_2
inst_1 : CommSemiring S_1
inst_2 : Algebra R_1 S_1
P_1 : Type u_3
inst_3 : CommSemiring P_1
p_1 : Ideal R_1
inst_4 : p_1.IsPrime
Rₚ_1 : Type u_4
inst_5 : CommSemiring Rₚ_1
inst_6 : Algebra R_1 Rₚ_1
inst_7 : IsLocalization.AtPrime Rₚ_1 p_1
inst_8 : IsLocalRing Rₚ_1
Sₚ_1 : Type u_5
inst_9 : CommSemiring Sₚ_1
inst_10 : Algebra S_1 Sₚ_1
inst_11 : IsLocalization (algebraMapSubmonoid S_1 p_1.primeCompl) Sₚ_1
inst_12 : Algebra Rₚ_1 Sₚ_1
P : Ideal S_1
S : Type u_6
R : Type u_7
inst_13 : CommRing R
p : Ideal R
inst_14 : p.IsMaximal
Rₚ : Type u_8
inst_15 : CommRing Rₚ
inst_16 : Algebra R Rₚ
inst_17 : IsLocalization.AtPrime Rₚ p
inst_18 : IsLocalRing Rₚ
Sₚ : Type u_9
inst_19 : CommRing S
inst_20 : Algebra R S
inst_21 : CommRing Sₚ
inst_22 : Algebra S Sₚ
inst_23 : Algebra R Sₚ
inst_24 : Algebra Rₚ Sₚ
inst_25 : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst_26 : IsScalarTower R S Sₚ
inst_27 : IsScalarTower R Rₚ Sₚ
h :
  Eq.{u_9 + 1} (α := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))
    (Ideal.map (R := S) (F := S →+* Sₚ) (algebraMap S Sₚ) (Ideal.map (F := R →+* S) (algebraMap R S) p))
x : Sₚ
refine_2 :
  ∀ (x : S) (s : ↥(algebraMapSubmonoid S p.primeCompl)),
    ∃ (a : S),
      Eq.{u_9 + 1} (α :=
        HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
        (((algebraMap S
                  (HasQuotient.Quotient Sₚ (B := Ideal Sₚ)
                    (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))) :
                RingHom.{u_6, u_9} S
                  (HasQuotient.Quotient Sₚ (B := Ideal Sₚ)
                    (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))) :
              (a : S) → Sₚ ⧸ pSₚ)
            a :
          HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
        ((Ideal.Quotient.mk (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)) : (a : Sₚ) → Sₚ ⧸ pSₚ)
            (mk' Sₚ x s) :
          HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
⊢ ∃ (a : S),
    Eq.{u_9 + 1} (α :=
      HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
      (((algebraMap S
                (HasQuotient.Quotient Sₚ (B := Ideal Sₚ)
                  (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))) :
              RingHom.{u_6, u_9} S
                (HasQuotient.Quotient Sₚ (B := Ideal Sₚ)
                  (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))) :
            (a : S) → Sₚ ⧸ pSₚ)
          a :
        HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
      ((Ideal.Quotient.mk (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)) : (a : Sₚ) → Sₚ ⧸ pSₚ) x :
        HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))

Tactic:
obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq
      (Algebra.algebraMapSubmonoid S p.primeCompl) x

Premises used: Ideal.primeCompl, Algebra.algebraMapSubmonoid, rfl, IsLocalization.exists_mk'_eq