have h : ∀ k : ℕ, P k → ∃ m : ℕ, m < k ∧ P m :=
  λ k h ↦ Exists.elim (bad_exists_descent hn h)
    (λ c h0 ↦ ⟨c, h0.2.1, h0.1, k, h0.2.1, h0.2.2⟩)
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), P k → ∃ m < k, P m
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b