-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ Functor.obj (Functor.const I) S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : LocallyOfFinitePresentation f
inst_3 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
inst_5 : ∀ (i : I), QuasiSeparatedSpace.{u} ↥(D.obj i)
a : c.pt ⟶ X
ha : c.π ≫ t = CategoryTheory.Functor.map (Functor.const I) (a ≫ f)
𝒰 :
  IsOpenCover (X := ↥c.pt)
    fun
      (V :
        { x :
          TopologicalSpace.Opens ↥c.pt ×
            { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
              x.1 ∈ X.affineOpens ∧
                x.1 ≤
                  (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                    (↑x.2 : TopologicalSpace.Opens ↥S) } //
          Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt)) c.pt.affineOpens x.1 ∧
            x.1 ≤
              (Opens.comap (TopCat.Hom.hom a.base) : (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }) ↦
    (↑V :
        TopologicalSpace.Opens ↥c.pt ×
          { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
            x.1 ∈ X.affineOpens ∧
              x.1 ≤
                (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                  (↑x.2 : TopologicalSpace.Opens ↥S) }).1
obtain :
  ∀ (i : I)
    (s :
      Finset
        { x :
          TopologicalSpace.Opens ↥c.pt ×
            { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
              x.1 ∈ X.affineOpens ∧
                x.1 ≤
                  (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                    (↑x.2 : TopologicalSpace.Opens ↥S) } //
          Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt)) c.pt.affineOpens x.1 ∧
            x.1 ≤
              (Opens.comap (TopCat.Hom.hom a.base) : (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 })
    (𝒱 : ↥s → (D.obj i).Opens) (h𝒱 : IsOpenCover 𝒱)
    (h𝒱𝒰 :
      ∀ (j : ↥s),
        IsAffineOpen (𝒱 j) ∧
          (↑(↑j :
                    { x :
                      TopologicalSpace.Opens ↥c.pt ×
                        { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                          x.1 ∈ X.affineOpens ∧
                            x.1 ≤
                              (Opens.comap (TopCat.Hom.hom f.base) :
                                  (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                                (↑x.2 : TopologicalSpace.Opens ↥S) } //
                      x.1 ∈ c.pt.affineOpens ∧
                        x.1 ≤
                          (Opens.comap (TopCat.Hom.hom a.base) :
                              (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                            (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }) :
                TopologicalSpace.Opens ↥c.pt ×
                  { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                    x.1 ∈ X.affineOpens ∧
                      x.1 ≤
                        (Opens.comap (TopCat.Hom.hom f.base) :
                            (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                          (↑x.2 : TopologicalSpace.Opens ↥S) }).1 =
            c.π.app i ⁻¹ᵁ 𝒱 j),
    ∃ (i : I) (g : D.obj i ⟶ X), c.π.app i ≫ g = a ∧ g ≫ f = t.app i
⊢ ∃ (i : I) (g : D.obj i ⟶ X), c.π.app i ≫ g = a ∧ g ≫ f = t.app i

Tactic:
obtain ⟨i, s, 𝒱, h𝒱, h𝒱𝒰⟩ := Scheme.exists_isOpenCover_and_isAffine D c hc _ 𝒰 fun U ↦ U.2.1

Premises used: And.left, AlgebraicGeometry.Scheme.exists_isOpenCover_and_isAffine, Subtype.property