Why if $e$ is independent to $\hat y$, then $\hat\beta $ will be independent of $S^2$.

Question

$\bullet$ $X$ is a matrix $n\times p$,

$\bullet$ $y=X\beta +\varepsilon$ where $\varepsilon\sim\mathcal N(0,\sigma ^2I_{n\times n})$

$\bullet$ $\hat \beta=(X^TX)^{-1}X^Ty$

$\bullet$ $\hat y=X\hat\beta =Hy$

$\bullet$ $H=X(X^TX)^{-1}X^T$

$\bullet$ $S^2=\frac{e^Te}{n-p}$

$\bullet$ $e=y-\hat y$.

In a proof of my course, it's written : "Since $e$ and $\hat y$ are independents, then $S^2$ and $\hat \beta $ are also independents".

So, why do we have this implication ?