-- 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)
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
x : (↑(PrimeSpectrum.Top R) : Type u)
hx : x ∈ U
V : Opens (↑(PrimeSpectrum.Top R) : Type u)
hxV :
  (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦ x ∈ unop (op U)) x hx) :
      (↑(PrimeSpectrum.Top R) : Type u)) ∈
    V
iVU : V ⟶ unop (op U)
f : M
g : R
hfg :
  ∀ (x : ↥V),
    ∃ (hs : g ∉ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal),
      Eq (α := Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        ((fun (x : ↥V) ↦
            (↑s : (x : ↥(unop (op U))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
              ((iVU : (a : ↥V) → ↥(unop (op U))) x))
          x)
        (LocalizedModule.mk f
          (Subtype.mk (p := fun (x_1 : R) ↦ x_1 ∈ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) g hs))
g' : R
hxg' :
  (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦ x ∈ unop (op U)) x hx) :
      (↑(PrimeSpectrum.Top R) : Type u)) ∈
    (↑(basicOpen g') : Set (PrimeSpectrum R))
hg'U : (↑(basicOpen g') : Set (PrimeSpectrum R)) ⊆ (↑V : Set (↑(PrimeSpectrum.Top R) : Type u))
refine_1 : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g' * g)) x
refine_2 : basicOpen (g' * g) ≤ U
refine_3 :
  ∀ (x : ↥(unop (op (basicOpen (g' * g))))) (y : (↑(PrimeSpectrum.Top R) : Type u))
    (hy : Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) y),
    Eq (α :=
      Localizations M
        (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
          (↑(PrimeSpectrum.Top R) : Type u)))
      ((↑(const (g' • f) (g' * g) (basicOpen (g' * g)) ⋯) :
            (x : ↥(unop (op (basicOpen (g' * g))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
          (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
            Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
        Localizations M
          (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y
                hy) :
            (↑(PrimeSpectrum.Top R) : Type u)))
      ((↑((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))
                      (Quiver.Hom.op (LE.le.hom ⋯))) :
                  (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' * g)))))
                s) :
            (x : ↥(unop (op (basicOpen (g' * g))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
          (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
            Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
        Localizations M
          (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y
                hy) :
            (↑(PrimeSpectrum.Top R) : Type u)))
⊢ ∃ (g : R) (_ : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen g) x) (i : basicOpen g ≤ U) (f : M),
    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)))
      (const f g (basicOpen g) ⋯)
      ((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))
              i.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))))
        s)

Tactic:
refine ⟨g' * g, ?_, ?_, g' • f, Subtype.ext <| funext fun ⟨y, hy⟩ ↦ ?_⟩