Proposition 8

— Let $E$ be an extension of $K$ and $(E_i)_{i \in I}$ a family of subextensions of $E$ such that $E = K \left( \bigcup_{i \in I} E_i \right)$. If each extension $E_i$ is algebraic and separable over $K$, the same is true of $E$.

This follows at once from Prop. 6 (V, p. 39).

$^1$ The reader may convince himself that Th. 1 is used nowhere before the proof of Lemma 1 of V, p. 78.