Corollary 1

— Let E and K' be two extensions of K contained in the same extension of K. Suppose that E is algebraic over K and denote by E, the relative separable algebraic closure of K in E. Then $K'(E_s)$ is the relative separable algebraic closure of K' in $K'(E)$.

For $K'(E_s)$ is a separable algebraic extension of K' by Prop. 10 (V, p. 42); since E is p-radical over E_s, the extension $K'(E)$ of $K'(E_s)$ is p-radical (V, p. 25, Cor.). Now it suffices to apply Prop. 13.