Let $f(x)$ denote the pdf of a $\chi^2$-distribution with $n\in\mathbb{N}$ degrees of freedom given by $f(x) = \frac{2^{-n/2}}{\Gamma(n/2)}\cdot x^{n/2-1}\cdot\mathrm e^{-1/2x}\cdot\textbf{1}_{[0,\infty)}(x),$ where $\textbf{1}_A(x)=\begin{cases}1,&x\in A,\\0,&\text{else.}\end{cases}$
Furthermore we define $\Gamma(1/2)=\sqrt{\pi},\;\Gamma(1)=1$ and $\Gamma(r+1)=r\cdot\Gamma(r)$.
Assume we have two independant random variables $X_1,X_2\sim\mathcal{N}(\mu,\sigma^2)$ with unknown $\mu$ and unknown $\sigma$ and their sampling variance $S_X^2=\frac{1}{n-1}\sum\limits_{i=1}^n(X_i-\overline{X})^2$ with $\overline{X}=\frac{1}{n}\sum\limits_{i=1}^nX_i$.
Show that $\frac{1}{\sigma^2}S_X^2=\frac{(X_1-\overline{X})^2+(X_2-\overline{X})^2}{\sigma^2}$ is $\chi^2$ distributed with one degree of freedom.
To be honest, i have no idea at all how to start because this huge amount of information intimidates me. Can anyone explain me an appropriate ansatz to proove this?