Vector spaces - Proving that intersection is distributive over summation of vector spaces

The statement is wrong. If $U$, $V$ and $W$ are all lines in the same plane ($U\neq V$, $V\neq W$ and $W\neq U$) then $U\cap (V+W) = U$ but $(U\cap V)+(U\cap W)$ = $\{0\}+\{0\} =\{0\}$.