Partial Gamma Function expressed as Integral/Lemma

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

Then:
 * $(1): \quad \displaystyle \int_0^m \paren {1 - \frac t m}^m t^{x - 1} \rd t = m^x \int_0^1 \paren {1 - t}^m t^{x - 1} \rd t$

for $x > 0$.

Proof
Let:

Recalculating the limits:

Hence:

Thus $(1)$ can be written: