∀ {p : ℕ} [inst : Fact (Nat.Prime p)] {R : Type u_1} [inst_1 : NormedRing R] [inst_2 : Algebra ℤ_[p] R]
  [inst_3 : IsBoundedSMul ℤ_[p] R] [inst_4 : IsUltrametricDist R] [inst_5 : CompleteSpace R] [inst_6 : NormMulClass R]
  {r : R} (hr : ‖r‖ < 1),
  ↑((PadicInt.continuousAddCharEquiv_of_norm_mul p R).symm ⟨r, hr⟩) = PadicInt.addChar_of_value_at_one r ⋯