Structure Induced by Associative Operation is Associative

Theorem
Let $\struct {T, \circ}$ be an algebraic structure.

Let $S$ be a set.

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

Let $\circ$ be associative.

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

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

Let $\struct {T, \circ}$ be an associative algebraic structure.

Then: