If $F$ is a radical extension of $K$ and $E$ is an intermediate field, then $F$ is a radical extension of $E$.

Question

I want to prove that if $F$ is a radical extension of $K$ and $E$ is an intermediate field, then $F$ is a radical extension of $E$.

I thought about the proof and wanted to apply this fact: if $F$ is a radical extension field of $K$ and $E$ is an intermediate field, then $\mathrm{Aut}_KE$ is a solvable group. But I don't know how to proceed from here. Am I on a right track?