J : Type u₁
CategoryTheory.Category.{v₁, u₁} J
⊢ ∀ {X Y Z : CategoryTheory.Bicone J} (f : X ⟶ Y) (g : Y ⟶ Z),
  CategoryTheory.CategoryStruct.comp f g =
    CategoryTheory.BiconeHom.casesOn (motive := fun a a_1 x => X = a → Y = a_1 → HEq f x → (X ⟶ Z)) f
      (fun h =>
        Eq.ndrec (motive := fun {X} =>
          (f : X ⟶ Y) → Y = CategoryTheory.Bicone.left → HEq f CategoryTheory.BiconeHom.left_id → (X ⟶ Z))
          (fun f h =>
            Eq.ndrec (motive := fun {Y} =>
              (Y ⟶ Z) →
                (f : CategoryTheory.Bicone.left ⟶ Y) →
                  HEq f CategoryTheory.BiconeHom.left_id → (CategoryTheory.Bicone.left ⟶ Z))
              (fun g f h => g) ⋯ g f)
          ⋯ f)
      (fun h =>
        Eq.ndrec (motive := fun {X} =>
          (f : X ⟶ Y) → Y = CategoryTheory.Bicone.right → HEq f CategoryTheory.BiconeHom.right_id → (X ⟶ Z))
          (fun f h =>
            Eq.ndrec (motive := fun {Y} =>
              (Y ⟶ Z) →
                (f : CategoryTheory.Bicone.right ⟶ Y) →
                  HEq f CategoryTheory.BiconeHom.right_id → (CategoryTheory.Bicone.right ⟶ Z))
              (fun g f h => g) ⋯ g f)
          ⋯ f)
      (fun j h =>
        Eq.ndrec (motive := fun {X} =>
          (f : X ⟶ Y) → Y = CategoryTheory.Bicone.diagram j → HEq f (CategoryTheory.BiconeHom.left j) → (X ⟶ Z))
          (fun f h =>
            Eq.ndrec (motive := fun {Y} =>
              (Y ⟶ Z) →
                (f : CategoryTheory.Bicone.left ⟶ Y) →
                  HEq f (CategoryTheory.BiconeHom.left j) → (CategoryTheory.Bicone.left ⟶ Z))
              (fun g f h =>
                CategoryTheory.BiconeHom.casesOn (motive := fun a a_1 t =>
                  CategoryTheory.Bicone.diagram j = a → Z = a_1 → HEq g t → (CategoryTheory.Bicone.left ⟶ Z)) g
                  (fun h => CategoryTheory.Bicone.noConfusion h) (fun h => CategoryTheory.Bicone.noConfusion h)
                  (fun j_1 h => CategoryTheory.Bicone.noConfusion h) (fun j_1 h => CategoryTheory.Bicone.noConfusion h)
                  (fun {j_1 k} f h =>
                    CategoryTheory.Bicone.noConfusion h fun val_eq =>
                      Eq.ndrec (motive := fun {j_2} =>
                        (f : j_2 ⟶ k) →
                          Z = CategoryTheory.Bicone.diagram k →
                            HEq g (CategoryTheory.BiconeHom.diagram f) → (CategoryTheory.Bicone.left ⟶ Z))
                        (fun f h =>
                          Eq.ndrec (motive := fun {Z} =>
                            (g : CategoryTheory.Bicone.diagram j ⟶ Z) →
                              HEq g (CategoryTheory.BiconeHom.diagram f) → (CategoryTheory.Bicone.left ⟶ Z))
                            (fun g h => CategoryTheory.BiconeHom.left k) ⋯ g)
                        val_eq f)
                  ⋯ ⋯ ⋯)
              ⋯ g f)
          ⋯ f)
      (fun j h =>
        Eq.ndrec (motive := fun {X} =>
          (f : X ⟶ Y) → Y = CategoryTheory.Bicone.diagram j → HEq f (CategoryTheory.BiconeHom.right j) → (X ⟶ Z))
          (fun f h =>
            Eq.ndrec (motive := fun {Y} =>
              (Y ⟶ Z) →
                (f : CategoryTheory.Bicone.right ⟶ Y) →
                  HEq f (CategoryTheory.BiconeHom.right j) → (CategoryTheory.Bicone.right ⟶ Z))
              (fun g f h =>
                CategoryTheory.BiconeHom.casesOn (motive := fun a a_1 t =>
                  CategoryTheory.Bicone.diagram j = a → Z = a_1 → HEq g t → (CategoryTheory.Bicone.right ⟶ Z)) g
                  (fun h => CategoryTheory.Bicone.noConfusion h) (fun h => CategoryTheory.Bicone.noConfusion h)
                  (fun j_1 h => CategoryTheory.Bicone.noConfusion h) (fun j_1 h => CategoryTheory.Bicone.noConfusion h)
                  (fun {j_1 k} f h =>
                    CategoryTheory.Bicone.noConfusion h fun val_eq =>
                      Eq.ndrec (motive := fun {j_2} =>
                        (f : j_2 ⟶ k) →
                          Z = CategoryTheory.Bicone.diagram k →
                            HEq g (CategoryTheory.BiconeHom.diagram f) → (CategoryTheory.Bicone.right ⟶ Z))
                        (fun f h =>
                          Eq.ndrec (motive := fun {Z} =>
                            (g : CategoryTheory.Bicone.diagram j ⟶ Z) →
                              HEq g (CategoryTheory.BiconeHom.diagram f) → (CategoryTheory.Bicone.right ⟶ Z))
                            (fun g h => CategoryTheory.BiconeHom.right k) ⋯ g)
                        val_eq f)
                  ⋯ ⋯ ⋯)
              ⋯ g f)
          ⋯ f)
      (fun {j k} f_1 h =>
        Eq.ndrec (motive := fun {X} =>
          (f : X ⟶ Y) → Y = CategoryTheory.Bicone.diagram k → HEq f (CategoryTheory.BiconeHom.diagram f_1) → (X ⟶ Z))
          (fun f h =>
            Eq.ndrec (motive := fun {Y} =>
              (Y ⟶ Z) →
                (f : CategoryTheory.Bicone.diagram j ⟶ Y) →
                  HEq f (CategoryTheory.BiconeHom.diagram f_1) → (CategoryTheory.Bicone.diagram j ⟶ Z))
              (fun g f h =>
                CategoryTheory.BiconeHom.casesOn (motive := fun a a_1 x =>
                  CategoryTheory.Bicone.diagram k = a → Z = a_1 → HEq g x → (CategoryTheory.Bicone.diagram j ⟶ Z)) g
                  (fun h => CategoryTheory.Bicone.noConfusion h) (fun h => CategoryTheory.Bicone.noConfusion h)
                  (fun j_1 h => CategoryTheory.Bicone.noConfusion h) (fun j_1 h => CategoryTheory.Bicone.noConfusion h)
                  (fun {j_1 k_1} g_1 h =>
                    CategoryTheory.Bicone.noConfusion h fun val_eq =>
                      Eq.ndrec (motive := fun {j_2} =>
                        (g_2 : j_2 ⟶ k_1) →
                          Z = CategoryTheory.Bicone.diagram k_1 →
                            HEq g (CategoryTheory.BiconeHom.diagram g_2) → (CategoryTheory.Bicone.diagram j ⟶ Z))
                        (fun g_2 h =>
                          Eq.ndrec (motive := fun {Z} =>
                            (g : CategoryTheory.Bicone.diagram k ⟶ Z) →
                              HEq g (CategoryTheory.BiconeHom.diagram g_2) → (CategoryTheory.Bicone.diagram j ⟶ Z))
                            (fun g h => CategoryTheory.BiconeHom.diagram (CategoryTheory.CategoryStruct.comp f_1 g_2)) ⋯
                            g)
                        val_eq g_1)
                  ⋯ ⋯ ⋯)
              ⋯ g f)
          ⋯ f)
      ⋯ ⋯ ⋯