Partial Gamma Function expressed as Integral/Lemma

Theorem
Let $m \in \Z_{\ge 1}$.

Then:
 * $\displaystyle (1): \int_0^m \left({1 - \frac t m}\right)^m t^{x - 1} \, \mathrm d t = m^x \int_0^1 \left({1 - t}\right)^m t^{x - 1} \, \mathrm d t$

for $x > 0$.

Proof
Let:

Recalculating the limits:

Hence:

Thus $(1)$ can be written