-- Module: Mathlib.AlgebraicGeometry.Modules.Tilde
Goal:
R : CommRingCat
M_1 : ModuleCat (↑R : Type u)
M N : (Spec R).Modules
f : Functor.obj modulesSpecToSheaf M ⟶ Functor.obj modulesSpecToSheaf N
U : (Opens ↥(Spec R))ᵒᵖ
t :
  (↑(Functor.obj
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
        U) :
    Type u)
m : (↑(M.val.1 U) : Type u)
x : ↥(Spec R)
hxU : x ∈ unop U
r : (↑R : Type u)
hxr : x ∈ (↑(basicOpen r) : Set (PrimeSpectrum (↑R : Type u)))
hrU : basicOpen r ≤ unop U
refine :
  Eq (α :=
    (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
          (op (basicOpen r))) :
      Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
            LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
            ModuleCat.carrier)
            ((Functor.obj modulesSpecToSheaf N).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
              (homOfLE hrU).op) :
          (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
            (↑((modulesSpecToSheaf.obj N).obj.obj (op (basicOpen r))) : Type u))
        ((ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m)) :
      (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
            (op (basicOpen r))) :
        Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
            LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
            ModuleCat.carrier)
            ((Functor.obj modulesSpecToSheaf N).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
              (homOfLE hrU).op) :
          (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
            (↑((modulesSpecToSheaf.obj N).obj.obj (op (basicOpen r))) : Type u))
        (HSMul.hSMul (α :=
          (↑(Functor.obj
                (ObjectProperty.FullSubcategory.obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                U) :
            Type u))
          ((RingHom.id
                  (↑(Functor.obj
                        (ObjectProperty.FullSubcategory.obj (P :=
                          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                        U) :
                    Type u) :
                (a : (↑((Spec R).ringCatSheaf.obj.obj U) : Type u)) → (↑((Spec R).ringCatSheaf.obj.obj U) : Type u))
              t :
            (↑(Functor.obj
                  (ObjectProperty.FullSubcategory.obj (P :=
                    CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                  U) :
              Type u))
          ((ModuleCat.Hom.hom (f.hom.app U)).toFun m)) :
      (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
            (op (basicOpen r))) :
        Type u))
⊢ ∃ (V : Opens ↥(Spec R)) (hV : V ≤ unop U),
    x ∈ V ∧
      Eq (α :=
        (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
              (op V)) :
          Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
                LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
                ModuleCat.carrier)
                ((Functor.obj modulesSpecToSheaf N).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
                  (homOfLE hV).op) :
              (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
                (↑((modulesSpecToSheaf.obj N).obj.obj (op V)) : Type u))
            ((ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m)) :
          (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
                (op V)) :
            Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
                LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
                ModuleCat.carrier)
                ((Functor.obj modulesSpecToSheaf N).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).map
                  (homOfLE hV).op) :
              (a : (↑((modulesSpecToSheaf.obj N).obj.obj (op (unop U))) : Type u)) →
                (↑((modulesSpecToSheaf.obj N).obj.obj (op V)) : Type u))
            (HSMul.hSMul (α :=
              (↑(Functor.obj
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                    U) :
                Type u))
              ((RingHom.id
                      (↑(Functor.obj
                            (ObjectProperty.FullSubcategory.obj (P :=
                              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R)))
                              (Spec R).ringCatSheaf)
                            U) :
                        Type u) :
                    (a : (↑((Spec R).ringCatSheaf.obj.obj U) : Type u)) → (↑((Spec R).ringCatSheaf.obj.obj U) : Type u))
                  t :
                (↑(Functor.obj
                      (ObjectProperty.FullSubcategory.obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))) (Spec R).ringCatSheaf)
                      U) :
                  Type u))
              ((ModuleCat.Hom.hom (f.hom.app U)).toFun m)) :
          (↑((Functor.obj modulesSpecToSheaf N).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec R))).obj
                (op V)) :
            Type u))

Tactic:
refine ⟨_, hrU, hxr, ?_⟩