∀ {f : AbsoluteValue ℚ ℝ} (q : ℕ)
  (motive : (∃ (h : Fact (Nat.Prime q)), ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = (Rat.AbsoluteValue.padic q) ↑n) → Prop)
  (x : ∃ (h : Fact (Nat.Prime q)), ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = (Rat.AbsoluteValue.padic q) ↑n),
  (∀ (hq_prime : Fact (Nat.Prime q)) (h_equiv : ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = (Rat.AbsoluteValue.padic q) ↑n),
      motive ⋯) →
    motive x