-- 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
hsple :
  ∀ {x : 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})
        x),
    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)))
          x :
        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))
a :
  HMul.hMul (α := Ideal (Generators.Ring (comp (localizationAway T g) P)))
        (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
          (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker)
        (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
          (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker) ⊔
      HMul.hMul (α := Ideal (Generators.Ring (comp (localizationAway T g) P)))
        (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
          (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker)
        (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}) ≤
    RingHom.ker (F :=
      Generators.Ring (comp (localizationAway T g) P) →ₐ[R]
        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)
⊢ HMul.hMul (α := Ideal (Generators.Ring (comp (localizationAway T g) P)))
          (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
            (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker)
          (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
            (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker) ⊔
        HMul.hMul (α := Ideal (Generators.Ring (comp (localizationAway T g) P)))
          (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
            (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker)
          (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 := 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})
          (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
            (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker) ⊔
        HMul.hMul (α := 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})
          (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})) ≤
    RingHom.ker (F :=
      Generators.Ring (comp (localizationAway T g) P) →ₐ[R]
        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)
have a₁ : HMul.hMul (α := 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})
      (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (comp (localizationAway T g) P))
        (Hom.toAlgHom (toComp (localizationAway T g) P)) P.ker) ⊔
    HMul.hMul (α := 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})
      (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}) ≤
  RingHom.ker (F :=
    Generators.Ring (comp (localizationAway T g) P) →ₐ[R]
      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)
Tactic: apply sup_le