Lemma 19.9.22B103C3B777A

Proof. Sections with distinct valuations are linearly independent: in a nontrivial finite linear combination of them, the smallest valuation occurs in exactly one summand and cannot cancel. Hence there are at most $\dim V$ values.

For the reverse inequality, we argue by induction on $\dim V$, the case $\dim V=0$ being immediate. The nonempty set of values has a smallest element $a$ in the lexicographic order. The Taylor coefficient of the monomial $z^ a$ is a nonzero linear functional on $V$. Its kernel has codimension one, and every nonzero element of the kernel has valuation strictly larger than $a$. By the induction hypothesis the kernel contributes $\dim V-1$ distinct values, all different from $a$. $\square$