-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (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
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ Eq (α :=
    (Spec A⁰_ f).presheaf.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).toPresheafedSpace :
                TopCat))
              (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
            (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          x) ⟶
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.stalk
        x)
    (LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x)
    (Iso.hom (specStalkEquiv 𝒜 f x f_deg hm) ≫
      Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : ProjectiveSpectrum 𝒜))) ≫
        Iso.inv (Proj.restrictStalkIso ⋯ x))
have refine : Eq (α :=
  (A⁰_ f) →+*
    (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                sorry).presheaf.stalk
          x) :
      Type u_1))
  (RingHom.comp (β :=
    (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (X :=
                (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                        sorry).toPresheafedSpace :
                  TopCat))
                (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
              (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
            x)) :
      Type u_1))
    (CommRingCat.Hom.hom (LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x))
    (algebraMap (A⁰_ f)
      (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  carrier) (X :=
                  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                          sorry).toPresheafedSpace :
                    TopCat))
                  (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                  (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
              x)) :
        Type u_1)))
  (RingHom.comp (β :=
    (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (X :=
                (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                        sorry).toPresheafedSpace :
                  TopCat))
                (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
              (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
            x)) :
      Type u_1))
    (CommRingCat.Hom.hom
      (Iso.hom (specStalkEquiv 𝒜 f x f_deg hm) ≫
        Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : ProjectiveSpectrum 𝒜))) ≫
          Iso.inv (Proj.restrictStalkIso sorry x)))
    (algebraMap (A⁰_ f)
      (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  carrier) (X :=
                  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                          sorry).toPresheafedSpace :
                    TopCat))
                  (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                  (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
              x)) :
        Type u_1)))
Tactic: refine CommRingCat.hom_ext <|
    IsLocalization.ringHom_ext (R := A⁰_ f) ((toSpec 𝒜 f).base x).asIdeal.primeCompl
      (S := (Spec.structureSheaf (A⁰_ f)).presheaf.stalk ((toSpec 𝒜 f).base x)) <| ?_