Identity of Group Direct Product/Proof 2

Theorem
Let $\left({G \times H, \circ}\right)$ be the group direct product of the two groups $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$.

If: and:
 * $e_G$ is the identity for $\left({G, \circ_1}\right)$
 * $e_H$ is the identity for $\left({H, \circ_2}\right)$

then $\left({e_G, e_H}\right)$ is the identity for $\left({G \times H, \circ}\right)$.

Proof
A specific instance of External Direct Product Identity, where the algebraic structures in question are groups.