-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic (goal)
Goal:
σ : Type u_1
A : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
i_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)
hi :
  Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
    (Opposite.unop (Opposite.op (basicOpen 𝒜 x)) : TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u)) i_1
n : ℕ
a : ↥(𝒜 n)
i : ℕ
hb' : (fun (x : ℕ) ↦ f ^ x) i ∈ 𝒜 n
⊢ Eq (α := Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
    (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑((CommRingCat.Hom.hom (awayToSection 𝒜 x) : (a : Away 𝒜 x) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
              ((CommRingCat.Hom.hom (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) : (a : Away 𝒜 f) → Away 𝒜 x)
                (mk
                  { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ }))) :
          (x_1 : ↥(basicOpen 𝒜 x)) →
            AtPrime 𝒜 (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal)
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
          i_1 hi)))
    (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑((CommRingCat.Hom.hom ((Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
                (a : (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
              ((CommRingCat.Hom.hom (awayToSection 𝒜 f) : (a : Away 𝒜 f) → (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u))
                (mk
                  { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ }))) :
          (x_1 : ↥(basicOpen 𝒜 x)) →
            AtPrime 𝒜 (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal)
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
          i_1 hi)))
have hf : Eq (α := Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
  ((IsLocalization.map (M := Submonoid.powers x) (T :=
          (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                      Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                        (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                      i_1 hi) :
                  (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
          (Localization (M := A)
            (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                        Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                          (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                        i_1 hi) :
                    (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
          (RingHom.id A) sorry :
        (a : Localization (Submonoid.powers x)) →
          Localization (↑⟨i_1, hi⟩ : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
      (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
        ((CommRingCat.Hom.hom (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) : (a : Away 𝒜 f) → Away 𝒜 x)
          (mk { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := sorry }))) :
    Localization (M := A)
      (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                  Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                    (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                  i_1 hi) :
              (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
  (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑((CommRingCat.Hom.hom ((Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE sorry))) :
                (a : (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
              ((CommRingCat.Hom.hom (awayToSection 𝒜 f) : (a : Away 𝒜 f) → (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u))
                (mk
                  { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := sorry }))) :
          (x_1 : ↥(basicOpen 𝒜 x)) →
            AtPrime 𝒜 (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal)
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
          i_1 hi)) :
    Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
Tactic: erw [ProjectiveSpectrum.Proj.awayToSection_apply]