Set Difference of Cartesian Products

Theorem

 * $\left({S_1 \times S_2}\right) \setminus \left({T_1 \times T_2}\right) = \left({S_1 \times \left({S_2 \setminus T_2}\right)}\right) \cup \left({\left({S_1 \setminus T_1}\right) \times S_2}\right)$

Proof
Let $\left({x, y}\right) \in \left({S_1 \times S_2}\right) \setminus \left({T_1 \times T_2}\right)$.

Then:

The result follows from the definition of subset and set equality.