Order of Group Element equals Order of Coprime Power/Proof 1

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

Then from Order of Power of Group Element:
 * $\forall m \in \Z: \left|{g^m}\right| = \dfrac n {\gcd \left\{{m, n}\right\}}$

where $\gcd \left\{{m, n}\right\}$ denotes the greatest common divisor of $m$ and $n$.

Thus:
 * $\left|{g^m}\right| = \left|{g^n}\right| \iff \gcd \left\{{m, n}\right\} = 1$

The result follows by definition of coprime integers.