External Direct Product Identity

Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of the two monoids $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.

Let:
 * $e_S$ be the identity for $\struct {S, \circ_1}$

and:
 * $e_T$ be the identity for $\struct {T, \circ_2}$.

Then $\tuple {e_S, e_T}$ is the identity for $\struct {S \times T, \circ}$.

Proof
Thus $\tuple {e_S, e_T}$ is the identity of $\struct {S \times T, \circ}$.