Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{p_1,...,p_n\}$ be the singular points of $C$. Assume $\vert D \vert$ be a linear system of divisors; that is it consist of all $D' \in \vert D \vert$ are divisors $D' \subset C$ such that there exist a $f \in K(C)$ with $D' = \operatorname{div}(f) +D$.

We assume that *every* member of $\vert D \vert$ has every singular point $p_i$ the multiplicity $ \ge a_i$ ( where $a_i \in \mathbb{N}$ with $a_i \ge 1$)

That is we can build another linear system $\vert L \vert$ consists of all $L':= D' - \sum_i a_i \cdot (p_i)$ where $D' \in D$.

*Question:* Why and how to see that $\dim_k \vert D \vert= \dim_k \vert L \vert$?

This question is closely related to my other question Linear system on singular plane curve