The projection $\pi \colon C \to C'$ is induced by a $2$-dimensional sub-linear system of the complete linear system $|L|$, so $C'$ is not linearly normal (unless $H^0(C, \, L)=3$, in which case $C=C'$) and $H^0(C', \, \mathcal{O}_{\mathbb{P}^2}(1)|_{C'})$ has the same dimension as $H^0(C, \, L)$.

It follows that every global section $\sigma' \in H^0(C', \, \mathcal{O}_{\mathbb{P}^2}(m)|_{C'})$ gives a section $\sigma \in H^0(C, \, L^m)$ such that $\pi(\mathrm{div}(\sigma)) = \mathrm{div} (\sigma')$. This yields the desired inequality.

The projection $\pi \colon C \to C'$ is induced by a $2$-dimensional sub-linear system of the complete linear system $|L|$, so $C'$ is not linearly normal (unless $H^0(C, \, L)=3$) and $H^0(C', \, \mathcal{O}_{\mathbb{P}^2}(1)|_{C'})$ has the same dimension as $H^0(C, \, L)$.

It follows that every global section $\sigma' \in H^0(C', \, \mathcal{O}_{\mathbb{P}^2}(m)|_{C'})$ gives a section $\sigma \in H^0(C, \, L^m)$ such that $\pi(\mathrm{div}(\sigma)) = \mathrm{div} (\sigma')$. This yields the desired inequality.