-- 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
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
toIsLocalizationMap :
  ∀
    (y :
      ↥(Ideal.primeCompl
          (PrimeSpectrum.asIdeal
            ((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)))),
    IsUnit (M := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
            (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          (↑y : A⁰_ f) :
        AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
toIsLocalizationMap₁ :
  ∀ (z : AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal),
    ∃ (x_1 :
      (A⁰_ f) ×
        ↥(Ideal.primeCompl
            (PrimeSpectrum.asIdeal
              ((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)))),
      HMul.hMul (β := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) z
          ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
                (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
              (↑x_1.2 : A⁰_ f) :
            AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) =
        (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
            (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          x_1.1
⊢ Submonoid.IsLocalizationMap.{u_1, u_1} (M := A⁰_ f) (N :=
    AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
    (Ideal.primeCompl
      (PrimeSpectrum.asIdeal
        ((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)))
    (⇑(algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)) :
      (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
have toIsLocalizationMap₂ : ∀ {x_1 y : A⁰_ f},
  Eq (α := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
            (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          x_1 :
        AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
            (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          y :
        AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) →
    ∃ (c :
      ↥(Ideal.primeCompl
          (PrimeSpectrum.asIdeal
            ((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)))),
      (↑c : A⁰_ f) * x_1 = (↑c : A⁰_ f) * y
Tactic: constructor