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.