Corollary 4

— Let $E'$ be an extension of K and E, $K'$ two subextensions of $E'$. Suppose that E is quasi-Galois over K and $E' = K'(E)$; then $E'$ is quasi-Galois over $K'$.

Let $(f_i)_{i \in I}$ be a family of non-constant polynomials in $K[X]$ whose splitting field over $K$ is $E$. Then it is clear that $E'$ is the splitting field of the family $(f_i)_{i \in I}$ over $K'$, so it is quasi-Galois over $K'$.