Power Set of Magma under Induced Operation is Magma

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

Let $\struct {\powerset S, \circ_\PP}$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\powerset S$ by $\circ$.

Then $\struct {\powerset S, \circ_\PP}$ is a magma.

That is, $\circ_\PP$ is closed in $\powerset S$.

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

Let $A, B \subseteq S$.

Thus $\struct {\powerset S, \circ_\PP}$ is a magma.