-- Lean 4 proof step from Mathlib.RingTheory.Extension.Cotangent.LocalizationAway (goal)
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : CommRing T
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
g : S
inst_7 : IsLocalization.Away g T
P : Generators R S ι
a : Generators.Ring (comp (localizationAway T g) P)
hx :
  Membership.mem (γ := Ideal (Generators.Ring (comp (localizationAway T g) P)))
    (Ideal.span
      {HMul.hMul (α := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R) (β := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
            ((rename Sum.inr : (a : MvPolynomial ι R) → MvPolynomial (Unit ⊕ ι) R) (P.σ g) :
              MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
            (X (Sum.inl ())) -
          1})
    ((HMul.hMul (α := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R) (β := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
          ((rename Sum.inr : (a : MvPolynomial ι R) → MvPolynomial (Unit ⊕ ι) R) (P.σ g) :
            MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
          (X (Sum.inl ())) -
        1) *
      a)
rw :
  HMul.hMul (α :=
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      (β :=
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      ((compLocalizationAwayAlgHom T g P :
            (a : ((localizationAway T g).comp P).Ring) →
              Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g)))
          (HMul.hMul (α := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R) (β := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
              ((rename Sum.inr : (a : MvPolynomial ι R) → MvPolynomial (Unit ⊕ ι) R) (P.σ g) :
                MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
              (X (Sum.inl ())) -
            1) :
        Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
          ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
            P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      ((compLocalizationAwayAlgHom T g P :
            (a : ((localizationAway T g).comp P).Ring) →
              Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g)))
          a :
        Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
          ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
            P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)) =
    0
rw₁ :
  HMul.hMul (α :=
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      (β :=
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      (0 :
        Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
          ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
            P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
      ((compLocalizationAwayAlgHom T g P :
            (a : ((localizationAway T g).comp P).Ring) →
              Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g)))
          a :
        Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
          ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
            P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)) =
    0
⊢ Eq (α :=
    Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
      ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
        P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
    ((compLocalizationAwayAlgHom T g P :
          (a : ((localizationAway T g).comp P).Ring) →
            Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g)))
        ((HMul.hMul (α := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R) (β := MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
              ((rename Sum.inr : (a : MvPolynomial ι R) → MvPolynomial (Unit ⊕ ι) R) (P.σ g) :
                MvPolynomial.{u_4, u_1} (Unit ⊕ ι) R)
              (X (Sum.inl ())) -
            1) *
          a) :
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
    (0 :
      Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
        ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
          P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
have rw₂ : Eq (α :=
  Localization.Away (M := P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2)
    ((Ideal.Quotient.mk (HPow.hPow (β := ℕ) P.ker 2) : (a : P.Ring) → P.Ring ⧸ P.ker ^ 2) (P.σ g) :
      P.Ring ⧸ HPow.hPow (β := ℕ) P.ker 2))
  0 0
Tactic: rw [map_mul, compLocalizationAwayAlgHom_relation_eq_zero, zero_mul]