Given two real vector spaces $V$ and $U$ with subspaces $A, C \subset V$, $A\cap C \neq \{0\}$ and $B, D \subset U$, $B\cap D \neq \{0\}$, is it true that 


$$(A\otimes B)\cap (C\otimes D) = (A\cap C)\otimes (B\cap D)$$


and if so, is there a proof available in a paper or textbook? It is stated to be true in a comment to a previous question [Tensor Products and Intersections](https://mathoverflow.net/questions/49300/tensor-products-and-intersections) but no proof is given. 



tensor products and intersections of subspaces
See e.g. Lemma 1.4.5 in the book 


S. Dascalescu - C. Nastasescu - S. Raianu: "Hopf Algebras. An Introduction", Pure and Applied Mathematics, 2001


There the statement is proved under the assumption that $C\subseteq A$ and $D\subseteq B$: the general case easily follows from this. 


