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 $\displaystyle \left({S, \circ}\right) = \prod_{k=1}^n S_k$ 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
Follows directly from the above.