Order of Subset Product with Singleton

Theorem
Let $\struct {G, \circ}$ be a group.

Let $X, Y \subseteq \struct {G, \circ}$ such that $X$ is a singleton:
 * $X = \set x$

Then:
 * $\order {X \circ Y} = \order Y = \order {Y \circ X}$

where $\order S$ is defined as the order of $S$.