External Direct Product Identity

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:
 * $e_S$ be the identity for $\left({S, \circ_1}\right)$

and:
 * $e_T$ be the identity for $\left({T, \circ_2}\right)$.

Then $\left({e_S, e_T}\right)$ is the identity for $\left({S \times T, \circ}\right)$.

Proof
Thus $\left({e_S, e_T}\right)$ is the identity of $\left({S \times T, \circ}\right)$.