If $L$ is the splitting field of a polynomial $f$ over $K$, $r \in M \supset L$ any element. Is $L[r]$ the splitting field of $f$ over $K[r]$?

Question

I was studying the proof of the equivalence: "A field extension L : K is normal and finite if and only if L is a splitting field for some polynomial over K" from Ian Stewart's book, and then, he implicitly used the following steatment:

If $L$ is the splitting field of some polynomial $f$ over $K$ and $r_1$ is any element in some extension $M \supset L$. Then $L[r_1]$ is the splitting field of $f \in K[r_1]$.

I was trying to see why this is true. But nothing came to my mind. I know, it seems kind of obvious but I was not able to justify it. My guess is it has something to do with the extension degree in $L : K$. Could someone please help me to understand why?

Thanks.