have pow_lift {x y : ℕ} {z : ℤ} : z ≃ x - y → p^x / p^y ≃ p^z := by
  intro (_ : z ≃ x - y)
  calc
    _ = p^x / p^y     := rfl
    _ ≃ p^((x:ℤ) - y) := eqv_symm pow_diff
    _ ≃ p^z           := pow_substR (Rel.symm ‹z ≃ x - y›)
ℕ ℤ : Type
inst✝¹¹ : Natural ℕ
inst✝¹⁰ : Integer ℤ
ℚ : Type
inst✝⁹ : Core ℚ
inst✝⁸ : Addition ℚ
inst✝⁷ : Multiplication ℚ
inst✝⁶ : Negation ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Natural.Exponentiation ℕ fun x x_1 => x * x_1
inst✝¹ : Exponentiation ℚ
p : ℚ
inst✝ : AP (p ≄ 0)
a b : ℤ
n m : ℕ
h✝¹ : a ≃ Coe.coe n - Coe.coe m
k j : ℕ
h✝ : b ≃ Coe.coe k - Coe.coe j
this : a + b ≃ Coe.coe (n + k) - Coe.coe (m + j)
⊢ p ^ (a + b) ≃ p ^ a * p ^ b
ℕ ℤ : Type
inst✝¹¹ : Natural ℕ
inst✝¹⁰ : Integer ℤ
ℚ : Type
inst✝⁹ : Core ℚ
inst✝⁸ : Addition ℚ
inst✝⁷ : Multiplication ℚ
inst✝⁶ : Negation ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Natural.Exponentiation ℕ fun x x_1 => x * x_1
inst✝¹ : Exponentiation ℚ
p : ℚ
inst✝ : AP (p ≄ 0)
a b : ℤ
n m : ℕ
h✝¹ : a ≃ Coe.coe n - Coe.coe m
k j : ℕ
h✝ : b ≃ Coe.coe k - Coe.coe j
this : a + b ≃ Coe.coe (n + k) - Coe.coe (m + j)
pow_lift : ∀ {x y : ℕ} {z : ℤ}, z ≃ Coe.coe x - Coe.coe y → p ^ x / p ^ y ≃ p ^ z
⊢ p ^ (a + b) ≃ p ^ a * p ^ b