Set of Finite Subsets under Induced Operation is Closed

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 $\circ$.

Let $T \subseteq \powerset S$ be the set of all finite subsets of $S$.

Then the algebraic structure $\struct {T, \circ_\PP}$ is closed.

Proof
Let $X, Y \in T$.

Then: