Structure Induced by Associative Operation is Associative

Theorem
Let $\left({T, \circ}\right)$ be an algebraic structure, and let $S$ be a set.

Let $\left({T^S, \oplus}\right)$ be the structure on $T^S$ induced by $\circ$.

Let $\circ$ be associative.

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

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

Let $\left({T, \circ}\right)$ be an associative algebraic structure.

Then: