Proof involving the expression $(H_1 \setminus Z_1) \times (H_2 \setminus Z_2)$

Question

Prove that $(H_1 \setminus Z_1) \times (H_2 \setminus Z_2) = \left(H_1 \times H_2 \right) \setminus \left(\left(H_1 \times Z_2 \right) \cup \left(H_2 \times Z_1 \right)\right)$, given that $Z_1 \subseteq H_1$ and $Z_2 \subseteq H_2$.

I can see the intuition behind it graphically by plotting on the plane, but have difficulty using set theoretic notation.

Answer 1

$$(h_1,h_2)\in (H_1\backslash Z_1)\times(H_2\backslash Z_2)$$ $$\iff h_i\in H_i \text{ and } h_i \not \in Z_i \text{ for } i \in \{1,2\}$$ $$\iff (h_1,h_2) \in (H_1,H_2) \text{ and } (h_1,h_2) \not \in (H_1\times Z_2) \text{ and } (h_1,h_2) \not \in (Z_1\times H_2)$$ $$\iff (h_1,h_2)\in(H_1\times H_2)\backslash(H_1\times Z_2\cup Z_1\times H_2)$$

This proves the set equality.