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 $\left({T, \circ_2}\right)$.

Let $\circ_1$ and $\circ_2$ be associative.

Then $\circ$ is also associative.

Proof
Let $\circ_1$ and $\circ_2$ be associative.

Then:

Thus $\circ$ is associative.