Power Structure Operation on Set of Singleton Subsets preserves Commutativity

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

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Let $S'$ denote the set of singleton elements of $\powerset S$.

Then $\circ_\PP$ is commutative $\circ$ is commutative.

Proof
From Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets:
 * $\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$

The result follows from Isomorphism Preserves Commutativity.