-- Lean 4 proof step from Mathlib.RingTheory.Trace.Quotient (goal)
Goal:
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.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst_4 : CommRing Rₚ
inst_5 : CommRing Sₚ
inst_6 : Algebra R Rₚ
inst_7 : IsLocalization.AtPrime Rₚ p
inst_8 : IsLocalRing Rₚ
inst_9 : Algebra S Sₚ
inst_10 : Algebra R Sₚ
inst_11 : Algebra Rₚ Sₚ
inst_12 : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst_13 : IsScalarTower R S Sₚ
inst_14 : IsScalarTower R Rₚ Sₚ
x : S
h :
  Eq (α :=
    RingHom.{u_1, u_4} R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (algebraMap R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (@RingHom.comp _ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ)) _ _ Semiring.toNonAssocSemiring _
      (algebraMap (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (algebraMap Rₚ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))) (algebraMap R Rₚ)))
h₁ :
  Eq (α :=
    RingHom.{u_1, u_4} R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
      (algebraMap Rₚ
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (algebraMap R Rₚ))
    (@RingHom.comp _ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ)) _ _ Semiring.toNonAssocSemiring _
      (algebraMap (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (algebraMap Rₚ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))) (algebraMap R Rₚ)))
h₂ :
  Eq (α :=
    RingHom.{u_1, u_4} R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
      (algebraMap Rₚ
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (algebraMap R Rₚ))
    (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
      (@RingHom.comp _ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ)) _ _ Semiring.toNonAssocSemiring _
        (algebraMap (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))
          (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
        (algebraMap Rₚ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))))
      (algebraMap R Rₚ))
⊢ Eq (α :=
    RingHom.{u_1, u_4} R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (algebraMap R
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (@RingHom.comp.{u_1, u_3, u_4} _ (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ)) _ _
      Semiring.toNonAssocSemiring _
      (algebraMap (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (algebraMap R (HasQuotient.Quotient Rₚ (B := Ideal Rₚ) (maximalIdeal Rₚ))))
have h₃ : Eq (α :=
  RingHom.{u_1, u_4} R
    (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
  (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
    (algebraMap Rₚ
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (algebraMap R Rₚ))
  (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
    (algebraMap Rₚ
      (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
    (algebraMap R Rₚ))
Tactic: rw [IsScalarTower.algebraMap_eq R Rₚ (Rₚ ⧸ _), IsScalarTower.algebraMap_eq R Rₚ (Sₚ ⧸ _),
      ← RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq Rₚ]