Composition of Right Regular Representations

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

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

Let $\rho_x \circ \rho_y$ be defined as the composition of the mappings $\rho_x$ and $\rho_y$.

Then $\forall x, y \in S$:
 * $\rho_x \circ \rho_y = \rho_{y * x}$

Proof
Let $z \in S$.