-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
D : GlueData
i : D.toLocallyRingedSpaceGlueData.J
y : (↑(D.toLocallyRingedSpaceGlueData.U i).toTopCat : Type u)
j : PreZeroHypercover.I₀ (S := D.U i) (Precoverage.ZeroHypercover.toPreZeroHypercover (D.U i).affineCover)
z : ↥(PreZeroHypercover.X (Precoverage.ZeroHypercover.toPreZeroHypercover (D.U i).affineCover) j)
hz : (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (D.U i).affineCover) j) z = y
⊢ ∃ (R : CommRingCat.{u}) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ D.toLocallyRingedSpaceGlueData.glued),
    Membership.mem (α := (↑D.toLocallyRingedSpaceGlueData.glued.toTopCat : Type u)) (γ :=
        Set (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u))
        (Set.range (ι := (↑(↑(Spec.toLocallyRingedSpace.obj (op R)).toPresheafedSpace : TopCat) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                TopCat.carrier) f.base) :
            (a : (↑(↑(Spec.toLocallyRingedSpace.obj (op R)).toPresheafedSpace : TopCat) : Type u)) →
              (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u)))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                (Y := (↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat))
                (Hom.base (LocallyRingedSpace.Hom.toHom (D.toLocallyRingedSpaceGlueData.ι i))) :
              (a : (↑(↑(D.toLocallyRingedSpaceGlueData.U i).toPresheafedSpace : TopCat) : Type u)) →
                (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u))
            y :
          (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u)) ∧
      LocallyRingedSpace.IsOpenImmersion f
have refine : Membership.mem (α := (↑D.toLocallyRingedSpaceGlueData.glued.toTopCat : Type u)) (γ :=
    Set (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u))
    (Set.range (ι :=
      (↑(↑(Spec.toLocallyRingedSpace.obj
                  (op
                    (Classical.indefiniteDescription
                        (fun (R : CommRingCat.{u}) ↦
                          Nonempty.{u + 1}
                            ((D.U i).restrict (U :=
                                Functor.obj (C := TopologicalSpace.Opens (↑(D.U i).toTopCat : Type u))
                                  (Opens.toTopCat (D.U i).toTopCat)
                                  (↑((fun (x : (↑(D.U i).toTopCat : Type u)) ↦
                                          Classical.choose (p := fun (U : OpenNhds x) ↦
                                            ∃ (R : CommRingCat.{u}),
                                              Nonempty.{u + 1}
                                                ((D.U i).restrict sorry ≅ Spec.toLocallyRingedSpace.obj (op R)))
                                            sorry)
                                        j) :
                                    TopologicalSpace.Opens (↑(D.U i).toTopCat : Type u)))
                                (f :=
                                (Opens.inclusion'
                                    (↑((fun (x : (↑(D.U i).toTopCat : Type u)) ↦
                                            Classical.choose (p := fun (U : OpenNhds x) ↦
                                              ∃ (R : CommRingCat.{u}),
                                                Nonempty.{u + 1}
                                                  ((D.U i).restrict sorry ≅ Spec.toLocallyRingedSpace.obj (op R)))
                                              sorry)
                                          j) :
                                      TopologicalSpace.Opens (↑(D.U i).toTopCat : Type u)) :
                                  Functor.obj (C := TopologicalSpace.Opens (↑(D.U i).toTopCat : Type u))
                                      (Opens.toTopCat (D.U i).toTopCat)
                                      (↑((fun (x : (↑(D.U i).toTopCat : Type u)) ↦
                                              Classical.choose (p := fun (U : OpenNhds x) ↦
                                                ∃ (R : CommRingCat.{u}),
                                                  Nonempty.{u + 1}
                                                    ((D.U i).restrict sorry ≅ Spec.toLocallyRingedSpace.obj (op R)))
                                                sorry)
                                            j) :
                                        TopologicalSpace.Opens (↑(D.U i).toTopCat : Type u)) ⟶
                                    (D.U i).toTopCat))
                                sorry ≅
                              Spec.toLocallyRingedSpace.obj (op R)))
                        sorry).1)).toPresheafedSpace :
            TopCat) :
        Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            (Y := (↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat))
            (Hom.base
              (LocallyRingedSpace.Hom.toHom
                (Hom.toLRSHom
                    (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (D.U i).affineCover) j) ≫
                  D.toLocallyRingedSpaceGlueData.ι i)))) :
        (a :
            (↑(↑(Spec.toLocallyRingedSpace.obj
                        (op
                          (Classical.indefiniteDescription
                              (fun (R : CommRingCat) ↦
                                Nonempty ((D.U i).restrict sorry ≅ Spec.toLocallyRingedSpace.obj (op R)))
                              sorry).1)).toPresheafedSpace :
                  TopCat) :
              Type u)) →
          (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) (Y :=
            (↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat))
            (Hom.base (LocallyRingedSpace.Hom.toHom (D.toLocallyRingedSpaceGlueData.ι i))) :
          (a : (↑(↑(D.toLocallyRingedSpaceGlueData.U i).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u))
        y :
      (↑(↑D.toLocallyRingedSpaceGlueData.glued.toPresheafedSpace : TopCat) : Type u)) ∧
  LocallyRingedSpace.IsOpenImmersion (Y := D.toLocallyRingedSpaceGlueData.glued)
    (Hom.toLRSHom (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (D.U i).affineCover) j) ≫
      D.toLocallyRingedSpaceGlueData.ι i)
Tactic: refine ⟨_, ((D.U i).affineCover.f j).toLRSHom ≫
    D.toLocallyRingedSpaceGlueData.toGlueData.ι i, ?_⟩