have hkm : k ≤ m := by
  have : 0 < q ^ m - 1 := by
    have : 1 < q ^ m := by
      calc
        _ < q := hq
        _ = q^1 := (Nat.pow_one q).symm
        _ ≤ q ^ m := (pow_le_pow_iff_right hq).mpr hm
    exact Nat.sub_pos_of_lt this
  have : q ^ k - 1 ≤ q ^ m - 1 := Nat.le_of_dvd this H
  have :  q ^ k ≤ q ^ m  := by
    zify at this
    simp at this
    simpa [Nat.sub_add_cancel <| one_le_pow m q hq'] using this
  exact (pow_le_pow_iff_right hq).mp this
case neg
R : Type ?u.38021
inst✝¹ : DecidableEq R
inst✝ : DivisionRing R
k n q : ℕ
hq : 1 < q
hk : 0 < k
m : ℕ
h : ∀ m_1 < m, 0 < m_1 → q ^ k - 1 ∣ q ^ m_1 - 1 → k ∣ m_1
hm : 0 < m
H : q ^ k - 1 ∣ q ^ m - 1
hq' : 0 < q
hkeqm : ¬k = m
⊢ k ∣ m
case neg
R : Type ?u.38021
inst✝¹ : DecidableEq R
inst✝ : DivisionRing R
k n q : ℕ
hq : 1 < q
hk : 0 < k
m : ℕ
h : ∀ m_1 < m, 0 < m_1 → q ^ k - 1 ∣ q ^ m_1 - 1 → k ∣ m_1
hm : 0 < m
H : q ^ k - 1 ∣ q ^ m - 1
hq' : 0 < q
hkeqm : ¬k = m
hkm : k ≤ m
⊢ k ∣ m