Power Structure Operation on Set of Singleton Subsets is Closed

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 the algebraic structure $\struct {S', \circ_\PP}$ is closed.

Proof
Let $A, B \in S'$.

Then:
 * $\exists a, b \in S: A = \set a, B = \set b$

Hence:

Hence the result.