-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor (goal)
Goal:
A B σ τ : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
inst_3 : CommRing B
inst_4 : SetLike τ B
inst_5 : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst_6 : GradedRing 𝒜
inst_7 : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : LE.le (α := HomogeneousIdeal ℬ) (ℬ₊) (HomogeneousIdeal.map f (𝒜₊))
p : ProjectiveSpectrum ℬ
c :
  NumDenSameDeg 𝒜
    (Ideal.primeCompl
      (toIdeal (σ := σ) (𝒜 := 𝒜)
        (asHomogeneousIdeal
          (((ProjectiveSpectrum.comap f hf : C(ProjectiveSpectrum ℬ, ProjectiveSpectrum 𝒜)) :
              (a : ProjectiveSpectrum ℬ) → ProjectiveSpectrum 𝒜)
            p))))
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have hf₁ : Eq (α := Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
  (Localization.mk ((f : (a : A) → B) (↑c.num : A))
      (Subtype.mk (p := fun (x : B) ↦ x ∈ p.asHomogeneousIdeal.toIdeal.primeCompl) ((f : (a : A) → B) (↑c.den : A)) sorry) :
    Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
  (val (ι := ℕ) (σ := τ) (𝒜 := ℬ)
      ((CommRingCat.Hom.hom
            (CommRingCat.ofHom
              (↑(stalkIso' ℬ p) :
                (↑((structureSheaf ℬ).presheaf.stalk p) : Type u) →+* AtPrime ℬ p.asHomogeneousIdeal.toIdeal)) :
          (a : (↑((Proj ℬ).presheaf.stalk p) : Type u)) → AtPrime ℬ p.asHomogeneousIdeal.toIdeal)
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              (toSheafedSpace ℬ).presheaf.obj
                (Opposite.op
                  (Functor.obj (C := Opens (↑(↑(toSheafedSpace 𝒜).toPresheafedSpace : TopCat) : Type u))
                    (Opens.map (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (sheafedSpaceMap f hf))))
                    (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))))
              (Y := (toSheafedSpace ℬ).presheaf.stalk p)
              ((toSheafedSpace ℬ).presheaf.germ
                (Functor.obj (C := Opens (↑(↑(toSheafedSpace 𝒜).toPresheafedSpace : TopCat) : Type u))
                  (Opens.map (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (sheafedSpaceMap f hf))))
                  (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))
                p sorry) :
            (a :
                (↑((toSheafedSpace ℬ).presheaf.obj
                      (Opposite.op
                        ((Opens.map (sheafedSpaceMap f hf).hom.base).obj
                          (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A))))) :
                  Type u)) →
              (↑((toSheafedSpace ℬ).presheaf.stalk p) : Type u))
          ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (X :=
                  (toSheafedSpace 𝒜).presheaf.obj (Opposite.op (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))) (Y :=
                  (toSheafedSpace ℬ).presheaf.obj
                    (Opposite.op
                      (Functor.obj (C := Opens (↑(↑(toSheafedSpace 𝒜).toPresheafedSpace : TopCat) : Type u))
                        (Opens.map (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (sheafedSpaceMap f hf))))
                        (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))))
                  (NatTrans.app (PresheafedSpace.Hom.c (InducedCategory.Hom.hom (sheafedSpaceMap f hf)))
                    (Opposite.op (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))) :
                (a :
                    (↑((toSheafedSpace 𝒜).presheaf.obj (Opposite.op (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A)))) :
                      Type u)) →
                  (↑((toSheafedSpace ℬ).presheaf.obj
                        (Opposite.op
                          ((Opens.map (sheafedSpaceMap f hf).hom.base).obj
                            (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A))))) :
                    Type u))
              (sectionInBasicOpen 𝒜
                (((ProjectiveSpectrum.comap f hf : C(ProjectiveSpectrum ℬ, ProjectiveSpectrum 𝒜)) :
                    (a : ProjectiveSpectrum ℬ) → ProjectiveSpectrum 𝒜)
                  p)
                c) :
            (↑((toSheafedSpace ℬ).presheaf.obj
                  (Opposite.op
                    (Functor.obj (C := Opens (↑(↑(toSheafedSpace 𝒜).toPresheafedSpace : TopCat) : Type u))
                      (Opens.map (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (sheafedSpaceMap f hf))))
                      (ProjectiveSpectrum.basicOpen 𝒜 (↑c.den : A))))) :
              Type u)))) :
    Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
Tactic: erw [PresheafedSpace.stalkMap_germ_apply]