Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$
This is a theorem. The proof is as follows.
\begin{align} (x,y)\in(A\times B)\cap(C\times D)&\iff (x,y)\in(A\times B)\land (x,y)\in(C\times D) \\ &\iff (x\in A \land y\in B) \land (x\in C \land y\in D) \\ &\iff (x\in A \land x\in C) \land (y\in B \land y\in D) \\ &\iff (x\in A \cap C) \land (y\in B \cap D) \\ & \iff (x,y) \in (A \cap C) \times (B \cap D) \end{align}