Conversely, let D be a derivation of X. So it is a (r, l)-derivatin of X. Then
D(x ∗ y) = (x ∗ D(y)) ∧ (D(x) ∗ y)
= (D(x) ∗ y) ∗ ((D(x) ∗ y) ∗ (x ∗ D(y)))
= x ∗ D(y) = (y ∗ D(x)) ∗ ((y ∗ D(x)) ∗ (x ∗ D(y)))
= (x ∗ D(y)) ∧ (y ∗ D(x)).
Hence, D is a left derivation of X.