External Direct Product Associativity/Sufficient Condition

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

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

Then $\circ$ is also associative.

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

Thus $\circ$ is associative.

Also see

 * External Direct Product Commutativity
 * External Direct Product Identity
 * External Direct Product Inverses