∀ {R : Type u_2} [inst : CommRing R] (I : Ideal R) (M : Type u_1) [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  {a b : (n : ℕ) → M ⧸ I ^ n • ⊤},
  a ∈ {f | ∀ {m n : ℕ} (hmn : m ≤ n), (AdicCompletion.transitionMap I M hmn) (f n) = f m} →
    b ∈ {f | ∀ {m n : ℕ} (hmn : m ≤ n), (AdicCompletion.transitionMap I M hmn) (f n) = f m} →
      ∀ (m n : ℕ) (hmn : m ≤ n), (AdicCompletion.transitionMap I M hmn) ((a + b) n) = (a + b) m