I'm looking at theorem $4.1$ about separable degree of S.Lang on page $240$.
Let $E\supset F\supset k$ be a tower, then $$[E:k]_S=[E:F]_S\cdot [F:k]_S$$Furthermore if $E$ is finite over $k$ then $[E:k]_S$ is finite and $$[E:k]_S\leq [E:k]~~~~~~~~~~~~~(1)$$
I could prove the first part but for $(1)$ I have some problems. Therefore I looked at the prove of S.Lang and he uses Proposition $2.7$. We haven't discussed this proposition, therefore I wanted to ask if there is a way to prove $(1)$ without using the proposition $2.7$? It would be very nice if someone could help me.
Thanks for your help.