-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
s : R
f : M
n : ℕ
⊢ Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen s)))
    ((toBasicOpenₗ R M s : (a : LocalizedModule.Away s M) → (structureSheafInType R M).obj.obj (op (basicOpen s)))
        (LocalizedModule.mk f
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers s) x)
            ((fun (x : ℕ) ↦ s ^ x) n) ⋯)) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen s)))
    (const f
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers s) x)
              ((fun (x : ℕ) ↦ s ^ x) n) ⋯) :
          R)
        (basicOpen s) ⋯ :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen s)))
have apply : Eq (α :=
  Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op (basicOpen s)))
  ((HPow.hPow (α :=
          Module.End R
            (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              (op (basicOpen s))))
          ((algebraMap R
                  (Module.End R
                    (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                      (ObjectProperty.FullSubcategory.obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                        (structureSheafInType R M))
                      (op (basicOpen s)))) :
                (a : R) → Module.End R ((structureSheafInType R M).obj.obj (op (basicOpen s))))
              s :
            Module.End R
              (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                  (structureSheafInType R M))
                (op (basicOpen s))))
          n :
        (a : (structureSheafInType R M).obj.obj (op (basicOpen s))) →
          (structureSheafInType R M).obj.obj (op (basicOpen s)))
      ((toBasicOpenₗ R M s : (a : LocalizedModule.Away s M) → (structureSheafInType R M).obj.obj (op (basicOpen s)))
        (LocalizedModule.mk f
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers s) x)
            ((fun (x : ℕ) ↦ s ^ x) n) sorry))) :
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen s)))
  ((HPow.hPow (α :=
          Module.End R
            (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              (op (basicOpen s))))
          ((algebraMap R
                  (Module.End R
                    (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                      (ObjectProperty.FullSubcategory.obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                        (structureSheafInType R M))
                      (op (basicOpen s)))) :
                (a : R) → Module.End R ((structureSheafInType R M).obj.obj (op (basicOpen s))))
              s :
            Module.End R
              (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                  (structureSheafInType R M))
                (op (basicOpen s))))
          n :
        (a : (structureSheafInType R M).obj.obj (op (basicOpen s))) →
          (structureSheafInType R M).obj.obj (op (basicOpen s)))
      (const f
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers s) x)
              ((fun (x : ℕ) ↦ s ^ x) n) sorry) :
          R)
        (basicOpen s) sorry) :
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen s)))
Tactic: apply ((Module.End.isUnit_iff _).mp ((isUnit_basicOpen_end ..).pow n)).1 ?_