Order of Group Element equals Order of Coprime Power/Proof 2

Proof
Let $\left|{g}\right| = n$.

Necessary Condition
Let $m \perp n$.

Then by Bézout's Lemma:
 * $\exists x, y \in \Z: x m + y n = 1$

Let $h = g^m$.

Then:
 * $h^x = g^{m x} = g^{1 - y n} = g g^{- y n} = g e = g$

and so $g$ is also a power of $h$.

Hence from Order of Group Element not less than Order of Power:
 * $\left|{g}\right| \le \left|{h}\right| \le \left|{g}\right|$

and it follows that: $\left|{g}\right| = \left|{h}\right|$