Identity of Group Direct Product/Proof 2
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Theorem
Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.
Let $e_G$ be the identity for $\struct {G, \circ_1}$.
Let $e_H$ be the identity for $\struct {H, \circ_2}$.
Then $\tuple {e_G, e_H}$ is the identity for $\struct {G \times H, \circ}$.
Proof
A specific instance of External Direct Product Identity, where the algebraic structures in question are groups.
$\blacksquare$