# An alternative description of an holomorphic map associated to a complete linear system

I need an help with an exercise in Miranda's book "Algebraic curves and Riemann surfaces". More precisely is the exercise in Problems V.4 I.

Given a Riemann surface $X$ and a divisor $D$ on $X$ such that $|D|$ is base point free, let $S\colon \mathbb P(L(D))\to |D| \quad [f]\mapsto div(f)+D$ be the standard isomorphism between these two spaces.

Now consider the set $\{E\in|D|: E\geq p \}$, where $p\in X$. It is easy to see that, under the bijection $S$, $$\{E\in|D|: E\geq p \}=\{div(g)+D:g\in L(D-p)\} =S(\mathbb P(L(D-p))).$$ Since $|D|$ is base point free, we have that $L(D-p)$ is a codimension one subspace of $L(D)$, so $\{E\in|D|: E\geq p \}$ is a hyperplane in $|D|$ and hence is an element of $|D|^*$ (if $V$ is a vector space, with $(\mathbb P V)^*$ we mean the codimension one subset of $V$).

Now Miranda says that the map $\phi \colon X\to |D|^*$ sending $p$ to $\{E\in|D|: E\geq p \}$ is the map $\phi_D\colon X \to \mathbb P^n$, where $\phi_D$ is the holomorphic map associated to the complete linear system $|D|$. The problem is that I don't know how to prove this last part. I know that $\phi_D$ is given by a basis $\{f_0,\dots, f_n\}$ of $L(D)$, so $\phi_D(p)=[f_0(p):\dots:f_n(p)]$. Moreover I can choose functions such that $f_0\in L(D)-L(D-p)$ and $f_1,\dots,f_n\in L(D-p)$ so that $\phi_D(p)=[1:0:\dots:0]$ but now I don't know how to conclude.