I have a fairly simple question about the projection from a linear subnspace in $\mathbb{P}^n$. Following Shafarevich, let $E = \{L_1 = ... = L_{n-r} = 0\}$ be a $r$ - dimensional linear subspace in $\mathbb{P}^n$ with $L_1,...,L_{n-r}$ linearly independent linear functionals. The the projection with center $E$ is defined as $\pi_L : \mathbb{P}^n-L\rightarrow \mathbb{P}^{n-r-1}$, $x\mapsto (L_1(x):...:L_{n-r}(x))$.
The geometric interpretation is supposed to be as follows: Given $E$ and a $n-r-1$ - dimensional subspace $H$ disjoint from $E$ and $x\in \mathbb{P}^n-E$, then there is a unique $r+1$ - dimensional subspace $L_x = \langle E,x \rangle$ through $E$ and $x$ that intersects $H$ in exactly one point, which is given by $\pi_L(x)$.
Here is my problem: I can see why the intersection $L_q\cap H$ is a one dimensional affine subspace, i.e. a point in $\mathbb{P}^n$. But I don't see why it equals $\pi_L(x)= (L_1(x):...:L_{n-r}(x))$.