Composition of Left Regular Representations

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

Let $\lambda_x$ be the left regular representation of $\left({S, *}\right)$ with respect to $x$.

Let $\lambda_x \circ \lambda_y$ be defined as the composition of the mappings $\lambda_x$ and $\lambda_y$.

Then $\forall x, y \in S$:
 * $\lambda_x \circ \lambda_y = \lambda_{x * y}$

Proof
Let $z \in S$.