-- Module: Mathlib.AlgebraicGeometry.Gluing
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
H :
  ∀ (l : J) (hli : l ⟶ k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))
    (hlj : l ⟶ k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))) (y : (F ⋙ forget).obj l)
    (hy₁ :
      Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hli) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₁.fst))
            y :
          (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
        ((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
              Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
            x :
          ↥(F.obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))))))
    (hy₂ :
      Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hlj) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₂.fst))
            y :
          (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
        ((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
              Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
            x :
          ↥(F.obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))),
    ∃ (U_1 : (pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)).Opens),
      x ∈ U_1 ∧
        U_1.ι ≫
            pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
              Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
                F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2 =
          U_1.ι ≫
            pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
              Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
                F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2
⊢ ∃ (U_1 : (pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)).Opens),
    x ∈ U_1 ∧
      U_1.ι ≫
          pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
              F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2 =
        U_1.ι ≫
          pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
              F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2

Tactic:
obtain ⟨l, hli, hlj, y, hy₁, hy₂⟩ := (F ⋙ forget).exists_map_eq_of_isLocallyDirected k₁.2.1 k₂.2.1
    ((pullback.fst _ _ ≫ (F.map k₁.2.1).isoOpensRange.inv) x)
    ((pullback.snd _ _ ≫ (F.map k₂.2.1).isoOpensRange.inv) x) (by
      simp only [Functor.comp_obj, forget_obj, Functor.comp_map, forget_map,
        ConcreteCategory.hom_ofHom, Hom.comp_base, TopCat.hom_comp, ContinuousMap.comp_apply,
        TypeCat.Fun.coe_mk]
      simp only [← Hom.comp_apply]
      congr 5
      simpa using congr($(pullback.condition (f := (F.obj i).homOfLE h₁)
        (g := (F.obj i).homOfLE h₂)) ≫ Scheme.Opens.ι _))

Premises used: AlgebraicGeometry.Scheme.Hom.isoOpensRange, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Limits.pullback.snd, AlgebraicGeometry.Scheme.forget, CategoryTheory.Functor.comp, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', CategoryTheory.Functor.map, Prod.fst, CategoryTheory.Functor.exists_map_eq_of_isLocallyDirected, CategoryTheory.Limits.pullback.fst, CategoryTheory.CategoryStruct.comp, AlgebraicGeometry.PresheafedSpace.Hom.base, Sigma.snd, DFunLike.coe, CategoryTheory.Iso.inv