Structure Induced by Commutative Operation is Commutative

Theorem
Let $\struct {T, \circ}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Let $\circ$ be a commutative operation.

Then the operation $\oplus$ induced on $T^S$ by $\circ$ is also commutative.

Proof
Let $\struct {T, \circ}$ be a commutative algebraic structure.

Let $f, g \in T^S$.

Then: