In algebraic geometric coding theory, I encountered this theorem : 
In the above proof they've used a Corollary that is given as follows : "We have $l(D)=0$ whenever $\text{deg}(D)<0$ for any divisor $D$." Here $\mathcal{L}(G)$ is the Riemann-Roch space over the divisor $G$ and $l(G)$ is its dimension. I have understood upto the part that "$f\in\mathcal{L}(G-P_{i_1}-\dots-P_{i_z})$", but after how they have proved the dimension of the code $k$ equals $l(G)$ is not clear to me using the Corollary I mentioned above. Please clarify the proof for me. Any help is appreciated.
Edit : $N(F)$ is the number of rational places of $F$ and $\psi$ is the $\mathbb{F}_q$-linear map $\psi:\mathcal{L}(G)\to\mathbb{F}_q^n$ given by $$\psi(f)=(f(P_1),\dots,f(P_n)) \ \ \ \ \text{for all} \ \ \ f\in\mathcal{L}(G)$$