-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
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)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
obtain :
  ∀ (t : Finset ↥U)
    (ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))),
    ∃ (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))))
⊢ ∃ (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))))

Tactic:
obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun i ↦ (basicOpen (g i) : Set (PrimeSpectrum R)))
    (fun _ ↦ (basicOpen _).2) (fun x hx ↦ Set.mem_iUnion_of_mem ⟨x, hx⟩ (hxg _))

Premises used: PrimeSpectrum.basicOpen, Subtype.mk, IsCompact.elim_finite_subcover, SetLike.coe, Set.mem_iUnion_of_mem, TopologicalSpace.Opens.is_open'