Subset Product within Semigroup is Associative

Theorem
Let $\struct {S, \circ}$ be a semigroup.

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

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

Then:

demonstrating that $\circ_\PP$ is associative on $\powerset S$.

Also see

 * Subset Product within Commutative Structure is Commutative