rw [Algebra.trace_eq_of_equiv_equiv (equivQuotMaximalIdealOfIsLocalization p Rₚ)
  (quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)]
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (Algebra.trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ))
        ((quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ) ((Ideal.Quotient.mk pS) x))) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

case he
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdealOfIsLocalization p Rₚ) =
    (↑(quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))