case step =>
  intro (n' : ℕ) (ih : abs (p^n') ≃ (abs p)^n')
  show abs (p^(step n')) ≃ (abs p)^(step n')
  calc
    _ ≃ abs (p^(step n'))  := eqv_refl
    _ ≃ abs (p^n' * p)     := abs_subst pow_step
    _ ≃ abs (p^n') * abs p := abs_compat_mul
    _ ≃ (abs p)^n' * abs p := mul_substL ih
    _ ≃ (abs p)^(step n')  := eqv_symm pow_step
ℕ ℤ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
ℚ : Type
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Multiplication ℚ
inst✝⁷ : Negation ℚ
inst✝⁶ : Subtraction ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : Metric ℚ
inst✝ : Natural.Exponentiation ℕ fun x x_1 => x * x_1
p : ℚ
n : ℕ
⊢ ∀ (m : ℕ), abs (p ^ m) ≃ abs p ^ m → abs (p ^ step m) ≃ abs p ^ step m
no goals