I have three vector subspaces of a vector space $H$ , I need to prove the following :
$$ E \cap (F + (E \cap G)) = (E \cap F) + (E \cap G)$$
I've already proved that $E + ( F \cap G) \subset (E+F) \cap (E + G)$ , The first thing I thought of is to prove the double inclusion, i.e :
$$ E \cap (F + (E \cap G)) \subset (E \cap F) + (E \cap G)$$
and
$$ (E \cap F) + (E \cap G) \subset E \cap (F + (E \cap G)) $$
But I have no idea how to do that .