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

If: and:
 * $e_S$ is the identity for $\left({S, \circ_1}\right)$
 * $e_T$ is 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
So the identity is $\left({e_S, e_T}\right)$.