Corollary 3

— Let N be a quasi-Galois extension of K contained in $\Omega$, and E a subextension of N. Let $u$ be a K-homomorphism of E into $\Omega$; then $u(E) \subset N$ and there exists a K-automorphism $v$ of N which induces $u$ on E.

Let $w$ be a K-automorphism of $\Omega$ extending $u$ (V, p. 52, Prop. 1); since N is quasi-Galois over K, we have $w(N) = N$, whence $w(E) \subset N$ and so $w$ induces a K-automorphism $v$ of N.