-- 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 :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
⊢ awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.Γgerm
        x =
    CategoryStruct.comp (Y :=
      CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl))
      (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯))
      (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1)))) ≫
        Iso.inv (Proj.restrictStalkIso ⋯ x))
have rw : Eq (α :=
  CommRingCat.of (A⁰_ f) ⟶
    CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1))
  (awayToSection 𝒜 f ≫
    CategoryStruct.comp (Y :=
      LocallyRingedSpace.Γ.obj
        (op (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) sorry)))
      (CategoryTheory.Functor.map
          (ObjectProperty.FullSubcategory.obj (P :=
            (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
              (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
            (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)))
          (Quiver.Hom.op (homOfLE sorry)) :
        Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
                (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
              (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
            (op (pbo f) : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
          Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
                (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
              (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
            (Functor.obj (C :=
                (Opens (↑(Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) : Type u_1))ᵒᵖ)
                (Functor.op (Topology.IsOpenEmbedding.functor sorry)) (op ⊤) :
              (Opens (↑Proj.T : Type u_1))ᵒᵖ))
      ((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  sorry).presheaf.germ
            ⊤ x True.intro ≫
          Iso.hom (Proj.restrictStalkIso sorry x) :
        LocallyRingedSpace.Γ.obj
            (op
              (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) sorry)) ⟶
          CommRingCat.of
            (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1)))
  (CategoryStruct.comp (Y :=
    CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl))
    (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 sorry))
    (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1))))))
Tactic: rw [awayToΓ, Category.assoc, ← Category.assoc _ (Iso.inv _),
    Iso.eq_comp_inv, Category.assoc, Category.assoc, Presheaf.Γgerm]