TITLE: tensor products and intersections of subspaces
QUESTION [5 upvotes]: 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 but no proof is given.

REPLY [8 votes]: 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.