Cancellable iff Regular Representations Injective

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Then $a \in S$ is cancellable iff:
 * the left regular representation $\lambda_a \left({x}\right)$ is injective

and
 * the right regular representation $\rho_a \left({x}\right)$ is injective.