Prop. Let $X$ have a gamma distribution with parameter $\alpha$ and $\beta$. Then for any $x > 0$, the cdf of $X$ is given by
\[P(X\leqslant x)=F(x;\alpha,\beta)=F(\frac{x}{\beta};\alpha)\]
where $F(\cdot;\alpha)$ is the incomplete gamma function.
1
\[
F(x;\alpha,\beta)=\int_{0}^{x}\frac{1}{\beta^{\alpha}\Gamma(\alpha)}t^{\alpha-1}e^{-\frac{t}{\beta}}dt
\]
Let $u=\frac{t}{\beta}$, then $dt=\beta du$. Then the integral above becomes
\[
\begin{split}
F(x;\alpha,\beta)&=\int_{0}^{\frac{x}{\beta}}\frac{t^{\alpha-1}}{\beta^{\alpha-1}\Gamma(\alpha)}\cdot\frac{1}{\beta}\cdot e^{-u}\cdot\beta du\\
&=\int_{0}^{\frac{x}{\beta}}\frac{1}{\Gamma(\alpha)}\cdot u^{\alpha-1}\cdot e^{-u}du\\
&=F(\frac{x}{\beta};\alpha)
\end{split}
\]