Subset Product within Semigroup is Associative

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

Then the operation $\circ_\mathcal P$ induced on the power set of $S$ is also associative.

Proof
Let $X, Y, Z \in \powerset S$.

Then:

demonstrating that $\circ_\mathcal P$ is associative on $\powerset S$.

Also see

 * Subset Product within Commutative Structure is Commutative