This is the question:
Let $F$, $G$ and $H$ subspace of a vector space be $E$. Determine if the following statement is true: $$F\cap (H+(F\cap G))=(F\cap H)+(F\cap G)$$
My answer is: $$u\in (F\cap H)+(F\cap G)\Rightarrow u = v(\in F\cap H)+w(\in F\cap G)\Rightarrow \left\{ \begin{array}{lcc} u\in F \\ \\ u=v(\in H)+w(\in G) \end{array} \right.\Rightarrow \left\{ \begin{array}{lcc} u\in F \\ \\ u\in H+(F\cap G) \end{array} \right.\Rightarrow u\in F\cap (H+(F\cap G))\Rightarrow (F\cap H)+(F\cap G)\subseteq F\cap (H+(F\cap G))$$ The other content I think is not fulfilled. I can not think of any counter example. Can you help me?
Thanks
$u\in F\cap(H+(F\cap G)) \implies \begin{cases}u\in F\\ u\in H+(F\cap G)\end{cases}\\ \implies u=v(\in H) + w(\in F\cap G)\implies u-w=v\in H$
but $u-w\in F$ since $u,w\in F$ and $F$ is a subspace so $v\in F\implies v\in F\cap H$.