Definition: Let $K/F$ be a field extension and let $p(x)\in F[x]$, we say that $K$ is splitting field of $p$ over $F$ if $p$ splits in $K$ and $K$ is generated by $p$'s roots; i.e. if $a_{0},...,a_{n}\in K$ are the roots of $p$ then $K=F(a_{0},...a_{n})$.
What I am trying to understand is this: in my lecture notes it is written that if $K/E$,$E/F$ are field extensions then $K$ is splitting field of $p$ over $F$ iff $K$ is splitting field of $p$ over $E$.
If I assume $K$ is splitting field of $p$ over $F$ then $\begin{align*}K=F(a_{0,}...,a_{n})\subset E(a_{0,}...,a_{n})\subset K &\implies F(a_{0,}...,a_{n})=E(a_{0,}...,a_{n})\\ &\implies K=E(a_{0,}...,a_{n}). \end{align*}$
Can someone please help with the other direction ? help is appreciated!