case step =>
  intro (n' : ℕ) (ih : x^n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0) (_ : x^(step n') ≃ 0)
  show x ≃ 0 ∧ step n' ≄ 0
  have : x^n' * x ≃ 0 := AA.eqv_substL pow_step ‹x^(step n') ≃ 0›
  have : x^n' ≃ 0 ∨ x ≃ 0 := AA.zero_prod this
  have : x ≃ 0 :=
    match this with
    | Or.inl (_ : x^n' ≃ 0) => (ih ‹x^n' ≃ 0›).left
    | Or.inr (_ : x ≃ 0) => ‹x ≃ 0›
  have : step n' ≄ 0 := step_neqv_zero
  exact And.intro ‹x ≃ 0› ‹step n' ≄ 0›
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ ∀ (m : ℕ), (x ^ m ≃ 0 → x ≃ 0 ∧ m ≄ 0) → x ^ step m ≃ 0 → x ≃ 0 ∧ step m ≄ 0
no goals