have : q ^ m - 1 = q^(m - k)*(q ^ k - 1) + (q^(m - k) - 1) := by
  zify
  simp [one_le_pow m q hq', one_le_pow k q hq', one_le_pow (m - k) q hq']
  push_cast
  rw [mul_sub, mul_one]
  ring_nf
  simp only [ge_iff_le, add_right_inj]
  exact (pow_sub_mul_pow (q : ℤ) hkm).symm
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
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
this : q ^ m - 1 = q ^ (m - k) * (q ^ k - 1) + (q ^ (m - k) - 1)
⊢ k ∣ m