in Stichtenoth's book "Algebraic function fields and codes" there is the following proposition (p. 13, Proposition 1.3.3):
Let $F/K$ be a function field, $x\in F$, and let $P_1,\ldots, P_r$ be zeros of $x$. Then$$ \sum_{i=1}^r v_{P_i}(x)\cdot\deg (P_i) ≤ [F : K(x)]$$
In his proof, Stichtenoth choose elements $s_{ij}$ such that $(s_{ij}(P_i))_{1\leq j\leq \deg(P_i)}$ forms a basis of the residue class field $F_{P_i}=\mathcal{O}_{P_i}/P_i$. With the help of the weak approximation theorems, he find elements $z_{ij}$ such that $$ v_{P_i}(z_{ij}-{s_{ij}})>0\text{ and }v_{P_k}(z_{ij})\geq v_i(x). $$
In his final argument, he said that the elements $(z_{ij}(P_i))_{1\leq j\leq \deg(P_i)}$ form a basis of $F_{P_i}=\mathcal{O}_{P_i}/P_i$.
My question is: why?
I can show that $z_{ij}(P_i)\neq 0$ in $F_{P_i}$, but why do they form a basis? What argument do I miss to conclude that?
Sincerely, Hypertrooper