-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (hypothesis)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    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 (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
⊢ ∃ (ι : Type u) (x : Fintype ι) (a : ι → M) (b : ι → R) (ibU : ∀ (i : ι), basicOpen (b i) ≤ U),
    U ≤ ⨆ (i : ι), basicOpen (b i) ∧
      (∀ (i j : ι), b j • a i = b i • a j) ∧
        ∀ (i : ι),
          Eq (α :=
            (fun (X : Type u) ↦ X)
              (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                (Sheaf.presheaf (structureSheafInType R M)) (op (basicOpen (b i)))))
            ((ConcreteCategory.hom (FC := TypeCat.Fun)
                    (CategoryTheory.Functor.map (Sheaf.presheaf (structureSheafInType R M)) ⋯.hom.op) :
                  (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).presheaf.obj (op U))) →
                    (fun (X : Type u) ↦ X) ((structureSheafInType R M).presheaf.obj (op (basicOpen (b i)))))
                s :
              (fun (X : Type u) ↦ X)
                (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                  (Sheaf.presheaf (structureSheafInType R M)) (op (basicOpen (b i)))))
            (const (a i) (b i) (basicOpen (b i)) ⋯ :
              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 (b i))))
have this : ∀ (i j : ↥U),
  Eq (α := LocalizedModule (Submonoid.powers (g i * g j)) M)
    (LocalizedModule.mk (g i • f j)
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
        sorry))
    (LocalizedModule.mk (g j • f i)
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
        sorry))
Tactic: have (i j : _) : LocalizedModule.mk (g i • f j) ⟨g i * g j, Submonoid.mem_powers _⟩ =
      LocalizedModule.mk (g j • f i) ⟨g i * g j, Submonoid.mem_powers _⟩ := by
    refine toBasicOpenₗ_injective (g i * g j) ?_
    simp only [toBasicOpenₗ_mk]
    have := H i
    trans (structureSheafInType R M).obj.map (homOfLE ?_).op s
    · refine .trans (Subtype.ext <| funext fun a ↦ ?_) congr((structureSheafInType R M).obj.map
        (homOfLE ((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right)).op $(H j))
      exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]; ring_nf⟩
    · refine congr((structureSheafInType R M).obj.map (homOfLE ((PrimeSpectrum.basicOpen_mul (g i)
        (g j)).trans_le inf_le_left)).op $(H i)).symm.trans (Subtype.ext <| funext fun a ↦ ?_)
      exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]⟩
    · exact ((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right).trans (igU _)