∀ (R : Type u_1) [inst : CommRing R] (m k : ℕ) (i i_1 : Fin m),
  (if i_1 < i then 0
    else if h : ↑i_1 - ↑i < k then MvPolynomial.X (Sum.inr ⟨↑i_1 - ↑i, ⋯⟩) else if ↑i_1 - ↑i = k then 1 else 0) =
    if i ≤ i_1 ∧ ↑i_1 ≤ ↑i + k then
      if h : ↑i_1 - ↑i < k then MvPolynomial.X (Sum.inr ⟨↑i_1 - ↑i, ⋯⟩) else if ↑i_1 - ↑i = k then 1 else 0
    else 0