Structure Induced by Associative Operation is Associative

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

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

If $$\oplus$$ is associative, then the operation induced on $$T^S$$ by $$\oplus$$ is also associative.

Proof
Let $$f, g, h \in T^S$$. Let $$\left({T, \oplus}\right)$$ be an associative algebraic structure. Then:

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