∀ {ι : Type u_2} {A : ι → Type u_1} [inst : (i : ι) → Add (A i)] (x : (i : ι) → A i),
  (∀ (i : ι),
      (∀ (a : A i), x i + a = a + x i) ∧
        (∀ (b c : A i), x i + (b + c) = x i + b + c) ∧ ∀ (a b : A i), a + b + x i = a + (b + x i)) →
    (∀ (a : (i : ι) → A i) (x_1 : ι), x x_1 + a x_1 = a x_1 + x x_1) ∧
      (∀ (b c : (i : ι) → A i) (x_1 : ι), x x_1 + (b x_1 + c x_1) = x x_1 + b x_1 + c x_1) ∧
        ∀ (a b : (i : ι) → A i) (x_1 : ι), a x_1 + b x_1 + x x_1 = a x_1 + (b x_1 + x x_1)