Let $W_i (i =1,2)$ be subspaces of $V$. $W_1 \cap W_2 =0$. If $U$ is a subspace of $V$, is it true that $U \cap (W_1 \oplus W_2) = (U \cap W_1) \oplus (U \cap W_2)$?
If it is true, is there any way to give a formal proof? If it is not, is there a hint for constrcting a contradiction or giving a counter example?