Corollary

— Let E and F be algebraically disjoint over K. Let E' be the relative algebraic closure of E in L and F' that of F (V, p. 19). Then E' and F' are algebraically disjoint over K.

Let B be a transcendence basis of E over K; this is also a transcendence basis of E' over K. Since E is algebraically disjoint from F over K, B is algebraically free over F, hence over F' (V, p. 108, Prop. 6); now we can apply Prop. 12.