Identity of Group Direct Product

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 $\struct {G \times H, \circ}$.