I am reading Max Karoubi's "K-Theory" and I think I'm overlooking some trivial fact. We have a vector bundle $E\rightarrow X$ and a morphism $p:E\rightarrow E$ with $p^2=p$. He is showing that $\ker p$ is locally trivial. First he assumes that $E=X\times V$ for a vector space $V$. Here's where I'm stuck:
He defines $f:X\longrightarrow \operatorname{End}(V)$ by
$$ f(x)=1-p_{x_0}-p_x+2p_xp_{x_0}, $$
where $p_x$ is the restriction of $p$ to the fiber over $x$ and $x_0$ is a basepoint. The claim is that $p_{x_0}\circ f(x)=f(x)\circ p_x$. When I compute both sides I get
$$ 2p_{x_0}p_xp_{x_0}-p_{x_0}p_x=2p_{x}p_{x_0}p_{x}-p_{x_0}p_x $$
which says
$$ 2p_{x_0}p_xp_{x_0}=2p_{x}p_{x_0}p_{x}. $$
Why is that true? Thanks.