I am having trouble understanding the difference between a proposition and theorem in chapter 14 (Galois theory) of Dummit and Foote.
Prop. 5 (p.562). Let $E$ be a splitting field over $F$ of the polynomial $f(x)\in F[x]$. Then $|\operatorname{Aut}(E/F)|\leq [E:F]$.
Thm. 9 (p.570) Let $G=\{\sigma_{1}=1,\sigma_{2},\cdots,\sigma_{n} \}$ be a subgroup of automorphisms of a field $K$ and let $F$ be the fixed field. Then $[K:F]=n=|G|$.
My question: to me it seems that $G=Aut(K/F)$, then by Prop. 5 it seems we get the inequality $G=\operatorname{Aut}(K/F)\leq [K:F]$. If this is the case, I do not understand why we get equality in Thm. 9 and not in Prop. 5.