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
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
this✝ : 0 < q ^ m - 1
this : q ^ k - 1 ≤ q ^ m - 1
⊢ k ≤ m
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
this✝¹ : 0 < q ^ m - 1
this✝ : q ^ k - 1 ≤ q ^ m - 1
this : q ^ k ≤ q ^ m
⊢ k ≤ m