∀ (R : Type u_1) [inst : CommRing R] (n m k : ℕ) (hn : n = m + k) (i : Fin m) (j : Fin n),
  (MvPolynomial.pderiv (Sum.inl i))
      ((MvPolynomial.tensorEquivSum R (Fin m) (Fin k) R)
        ((MvPolynomial.universalFactorizationMap R n m k hn) (MvPolynomial.X j))) =
    if ↑j < ↑i then 0
    else if h : ↑j - ↑i < k then MvPolynomial.X (Sum.inr ⟨↑j - ↑i, h⟩) else if ↑j - ↑i = k then 1 else 0