Integer Less One divides Power Less One/Corollary

Corollary to Integer Less One divides Power Less One
Let $m, n, q \in \Z_{>0}$.

Let:
 * $m \mathop \backslash n$

where $\backslash$ denotes divisibility.

Then:
 * $\left({q^m - 1}\right) \mathop \backslash \left({q^n - 1}\right)$

Proof
By hypothesis:
 * $m \mathop \backslash n$

By definition of divisibility:
 * $\exists k \in \Z: k m = n$

Thus:
 * $q^n = q^{k m} = \left({q^m}\right)^k$

Then by Integer Less One divides Power Less One:
 * $\left({q^m - 1}\right) \mathop \backslash \left({\left({q^m}\right)^k - 1}\right)$

Hence the result.