-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
z : (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)
⊢ homogeneousLocalizationToStalk 𝒜 x
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
            CommRingCat.carrier) (X := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
            (stalkToFiberRingHom 𝒜 x) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1)) → (↑(CommRingCat.of (at x)) : Type u_1))
        z) =
    z
have obtain : ∀ (U : Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) (hxU : x ∈ U)
  (s :
    ToType (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC := CommRingCat.carrier)
      (Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) (op U))),
  Eq (α := (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
    (homogeneousLocalizationToStalk 𝒜 x
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
            CommRingCat.carrier) (X := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
            (stalkToFiberRingHom 𝒜 x) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1)) → (↑(CommRingCat.of (at x)) : Type u_1))
        (((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                CommRingCat.carrier) (X :=
                Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U))
                (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
                (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                  Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U) ⟶
                    Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
              (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U)) :
                  Type u_1) →+*
                (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
            (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
              (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
          s)))
    (((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
            CommRingCat.carrier) (X :=
            Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
              (op U))
            (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
            (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
              Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U) ⟶
                Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
          (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U)) :
              Type u_1) →+*
            (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
        (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
          (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
      s)
Tactic: obtain ⟨U, hxU, s, rfl⟩ := (Proj.structureSheaf 𝒜).presheaf.exists_germ_eq z
Cleared hypotheses: z