-- 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
a : (↑(CommRingCat.of (Away 𝒜 f)) : Type u)
⊢ Eq.{u + 1} (α :=
    (x_1 :
        ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)) :
            TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))) →
      (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
        (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)))
    (↑((CommRingCat.Hom.hom (CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection 𝒜 x) :
            (a : (↑(CommRingCat.of (Away 𝒜 f)) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
          a) :
      (x_1 : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
        (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
          (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)))
    (↑((CommRingCat.Hom.hom (awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
            (a : (↑(CommRingCat.of (Away 𝒜 f)) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
          a) :
      (x_1 : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
        (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
          (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)))
have hf : ∀ (i : (↑(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),
  Eq (α :=
    Localization (M := A)
      (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                  Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                    (Opposite.unop (Opposite.op (basicOpen 𝒜 x)) :
                      TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                    x_1)
                  i 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) ≫ awayToSection 𝒜 x) :
                (a : (↑(CommRingCat.of (Away 𝒜 f)) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
              a) :
          (x_1 : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)))
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            (Opposite.unop (Opposite.op (basicOpen 𝒜 x)) :
              TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            x_1)
          i hi)))
    (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑((CommRingCat.Hom.hom (awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE sorry))) :
                (a : (↑(CommRingCat.of (Away 𝒜 f)) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
              a) :
          (x_1 : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)))
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            (Opposite.unop (Opposite.op (basicOpen 𝒜 x)) :
              TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
            x_1)
          i hi)))
Tactic: ext ⟨i, hi⟩