Equation for linear subspaces Let $Z$ be a vector space over some field $K$ and $U, V, W \subseteq Z$ linear subspaces. I am trying to prove or disprove the statement $$(U+W)\cap V = U\cap V + W\cap V,$$
where $X + Y := \{x+y\,|\, x\in X, \,y \in Y\}$.
Since I could not come up with counter-examples (I tried with vectors from $\mathbb R^3$, maybe there are examples that are more advanced I did not come up with), I started trying to prove it.
I tried to start with the "$\subseteq$"-direction:  
Let $x \in (U + W)\cap V.$ Then $x\in U+W$ and $x\in V$. We want to show that $x\in U\cap V + W\cap V$, i.e. find $u \in U\cap V$ and $w\in W \cap V$ with $x = u + w$. But here I already don't know how to proceed since I don't know how to find those $u, v$. Any help is appreciated.
EDIT: The "$\supseteq $"-dircetion should be true, I already proved that.

Take $Z=\Bbb{R}^2$ and $U$, $V$, $W$ to be three distinct one-dimensional subspaces. Then any one of $U$, $V$, and $W$ is contained in the sum of the other two, so, in particular, $V\cap(U+W)=V$. However, the intersection of any two of these subspaces is $\{0\}$, so $V\cap U+V\cap W=\{0\}+\{0\}=\{0\}$.
