Order of Subset Product with Singleton/Proof 1

Proof
From Regular Representations of Subset Product, we have that the left regular representation of $\struct {S, \circ}$ with respect to $a$ is:
 * $\lambda_x \sqbrk S = \set x \circ S = x \circ S$

The result then follows directly from Regular Representation of Invertible Element is Permutation.