-- Lean 4 proof step from Mathlib.Order.Category.PartOrdEmb (goal)
Goal:
J : Type u
inst : SmallCategory J
inst_1 : IsFiltered J
F : J ⥤ PartOrdEmb
c :
  Cocone (F ⋙ forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier))
hc : IsColimit c
j : J
x y : (↑(F.1 j) : Type u)
k : J
x' y' : (↑(F.obj k) : Type u)
hx :
  Eq (α := (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app k) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
            (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj k))
        x' :
      (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
    (({
            toFun :=
              (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j)) :
                (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj j)) →
                  (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)),
            inj' := ⋯ } :
          (a : (↑(F.1 j) : Type u)) → CoconePt hc)
        x :
      CoconePt hc)
hy :
  Eq (α := (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app k) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
            (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj k))
        y' :
      (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
    (({
            toFun :=
              (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j)) :
                (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj j)) →
                  (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)),
            inj' := ⋯ } :
          (a : (↑(F.1 j) : Type u)) → CoconePt hc)
        y :
      CoconePt hc)
h : x' ≤ y'
l₁ : J
a₁ : k ⟶ l₁
b₁ : j ⟶ l₁
hl₁ :
  Eq (α :=
    (fun (X : Type u) ↦ X)
      ((F ⋙
            forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
              carrier)).obj
        l₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((F ⋙
                  forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier)).map
              a₁) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
            (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj l₁))
        x' :
      (fun (X : Type u) ↦ X)
        ((F ⋙
              forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                carrier)).obj
          l₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((F ⋙
                  forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier)).map
              b₁) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj j)) →
            (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj l₁))
        x :
      (fun (X : Type u) ↦ X)
        ((F ⋙
              forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                carrier)).obj
          l₁))
⊢ x ≤ y
have obtain : ∀ (l₂ : J) (a₂ : k ⟶ l₂) (b₂ : j ⟶ l₂)
  (hl₂ :
    Eq (α :=
      (fun (X : Type u) ↦ X)
        ((F ⋙
              forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                carrier)).obj
          l₂))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((F ⋙
                    forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                      carrier)).map
                a₂) :
            (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
              (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj l₂))
          y' :
        (fun (X : Type u) ↦ X)
          ((F ⋙
                forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                  carrier)).obj
            l₂))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((F ⋙
                    forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                      carrier)).map
                b₂) :
            (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj j)) →
              (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj l₂))
          y :
        (fun (X : Type u) ↦ X)
          ((F ⋙
                forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                  carrier)).obj
            l₂))),
  x ≤ y
Tactic: obtain ⟨l₂, a₂, b₂, hl₂⟩ := (Types.FilteredColimit.isColimit_eq_iff _ hc).1 hy