Composition of Left Regular Representation with Right

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

Let $\lambda_x, \rho_y$ be the left and right regular representations of $\left({S, *}\right)$ with respect to $x$ and $y$ respectively.

Let $\lambda_x \circ \rho_y$, $\rho_y \circ \lambda_x$ etc. be defined as the composition of the mappings $\lambda_x$ and $\rho_y$.

Then $\forall x, y \in S$:
 * $\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$

Proof
Let $z \in S$.