-- Module: Mathlib.AlgebraicGeometry.Modules.Tilde
Goal:
R : CommRingCat
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
hf :
  ∀ (x : (↑((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) : Type u)),
    Eq (α :=
      (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
            (op (PrimeSpectrum.basicOpen 1))) :
        Type u))
      ((ModuleCat.Hom.hom
              (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen 1) ≫
                ModuleCat.ofHom
                  (IsLocalizedModule.lift (Submonoid.powers 1)
                    (ModuleCat.Hom.hom
                      (tilde.toOpen
                        ((Functor.obj modulesSpecToSheaf M).obj (P :=
                              CategoryTheory.Presheaf.IsSheaf
                                (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                          (op ⊤))
                        (PrimeSpectrum.basicOpen 1)))
                    (ModuleCat.Hom.hom
                      ((Functor.obj modulesSpecToSheaf M).obj (P :=
                            CategoryTheory.Presheaf.IsSheaf
                              (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                        (Quiver.Hom.op (homOfLE ⋯))))
                    ⋯)) :
            (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
              (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen 1))) : Type u))
          x :
        (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op (PrimeSpectrum.basicOpen 1))) :
          Type u))
      ((ModuleCat.Hom.hom
              ((Functor.obj modulesSpecToSheaf M).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
                (Quiver.Hom.op (homOfLE ⋯))) :
            (a : (↑((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)) : Type u)) →
              (↑((modulesSpecToSheaf.obj M).obj.obj (op (PrimeSpectrum.basicOpen 1))) : Type u))
          x :
        (↑((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op (PrimeSpectrum.basicOpen 1))) :
          Type u))
⊢ Eq (α :=
    (Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤) ⟶
      (Functor.obj modulesSpecToSheaf M).obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
        (op (PrimeSpectrum.basicOpen 1)))
    (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen 1) ≫
      ModuleCat.ofHom
        (IsLocalizedModule.lift (Submonoid.powers 1)
          (ModuleCat.Hom.hom
            (tilde.toOpen
              ((Functor.obj modulesSpecToSheaf M).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
                (op ⊤))
              (PrimeSpectrum.basicOpen 1)))
          (ModuleCat.Hom.hom
            ((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
              (Quiver.Hom.op (homOfLE ⋯))))
          ⋯))
    ((Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
      (Quiver.Hom.op (homOfLE ⋯)))

Tactic:
ext x