Now, we show that D is a (r, l)-derivation of X. Then
D(x ∗ y) = x ∗ D(y)
= (x ∗ 0) ∗ D(y)
= (x ∗ (D(0) ∗ D(0)) ∗ D(y)
= (x ∗ ((x ∗ D(x)) ∗ (D(y) ∗ y))) ∗ D(y)
= (x ∗ ((x ∗ D(y)) ∗ (D(x) ∗ y))) ∗ D(y)
= (x ∗ D(y)) ∗ ((x ∗ D(y)) ∗ (D(x) ∗ y))
= (D(x) ∗ y) ∧ (x ∗ D(y)).
Therefore, D is a derivation of X.