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.

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.