Opposite Group is Group

Theorem
Let $\struct {G, \circ}$ be a group.

Let $\struct {G, *}$ be the opposite group to $G$.

Then $\struct {G, *}$ is a group.

$G \, 0$: Closure
$\struct {G, *}$ is closed:
 * $b \circ a \in G \implies a * b \in G$

$G \, 1$: Associativity
$*$ is associative on $G$:

$G \, 2$: Identity
Let $e$ be the identity of $\struct {G, \circ}$:

Thus $e$ is the identity of $\struct {G, *}$.

$G \, 3$: Inverses
Let the inverse of $a \in \struct {G, \circ}$ be $a^{-1}$:

Thus $a^{-1}$ is the inverse of $a \in \struct {G, *}$

So all the group axioms are satisfied, and $\struct {G, *}$ is a group.