Operation Induced on Power Set on Set of Singleton Subsets preserves Associativity

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

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

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

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

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

The result follows from Isomorphism Preserves Associativity.