External Direct Product Associativity

Theorem
Let $$\left({S \times T, \circ}\right)$$ be the external direct product of the two algebraic structures $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$.

If $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$ are associative, then $$\left({S \times T, \circ}\right)$$ is also associative.

Generalized Result
Let $$\left({S, \circ}\right) = \prod_{k=1}^n S_k$$ be the external direct product of the algebraic structures $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$$.

If $$\circ_1, \ldots, \circ_n$$ are all associative, then so is $$\circ$$.

Proof
Let $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$ be associative.

and we see that $$\left({S \times T, \circ}\right)$$ is associative.

Proof of Generalized Result
This follows from above and an application of the Principle of Mathematical Induction.