External Direct Product Inverses

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:
 * $s^{-1}$ be an inverse of $s \in \struct {S, \circ_1}$

and:
 * $t^{-1}$ be an inverse of $t \in \struct {T, \circ_2}$.

Then $\tuple {s^{-1}, t^{-1} }$ is an inverse of $\tuple {s, t} \in \struct {S \times T, \circ}$.

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

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

Also let:
 * $s^{-1}$ be the inverse of $s \in \struct {S, \circ_1}$

and
 * $t^{-1}$ be the inverse of $t \in \struct {T, \circ_2}$.

Then:

Thus the inverse of $\tuple {s, t}$ is $\tuple {s^{-1}, t^{-1} }$.