I am reading Sheldon Ross: A First Course in Probability and have a problem proving some property. I didn't find anything in the internet so will be grateful if somebody points me in the right direction. Here is the property:
$P(a_1 \lt X \le a_2, b_1 \lt Y \le b_2) = F(a_1, b_1) + F(a_2, b_2) - F(a_1, b_2) -F(a_2, b_1)$
Here is my start:
$P(a_1 \lt X \le a_2, b_1 \lt Y \le b_2) =P(\{a_1 \lt X \le a_2\} \cap \{b_1 \lt Y \le b_2\})$
$...=P(\{a_1 \lt X \le a_2\}) + P(\{b_1 \lt Y \le b_2\})-P(\{a_1 \lt X \le a_2\} \cup \{b_1 \lt Y \le b_2\})$
What are the next steps?
This answer can only be useful if you are familiar with expectations already.
$$\mathbf{1}_{a_{1}<X\leq a_{2}}\mathbf{1}_{b_{1}<Y\leq b_{2}}=$$$$\left(\mathbf{1}_{X\leq a_{2}}-\mathbf{1}_{X\leq a_{1}}\right)\left(\mathbf{1}_{Y\leq b_{2}}-\mathbf{1}_{Y\leq b_{1}}\right)=$$$$\mathbf{1}_{X\leq a_{2}}\mathbf{1}_{Y\leq b_{2}}-\mathbf{1}_{X\leq a_{1}}\mathbf{1}_{Y\leq b_{2}}-\mathbf{1}_{X\leq a_{2}}\mathbf{1}_{Y\leq b_{1}}+\mathbf{1}_{X\leq a_{1}}\mathbf{1}_{Y\leq b_{1}}=$$$$\mathbf{1}_{X\leq a_{2},Y\leq b_{2}}-\mathbf{1}_{X\leq a_{1},Y\leq b_{2}}-\mathbf{1}_{X\leq a_{2},Y\leq b_{1}}+\mathbf{1}_{X\leq a_{1},Y\leq b_{1}}$$
Taking the expectations on both sides results:$$P\left(a_{1}<X\leq a_{2},b_{1}<Y\leq b_{2}\right)=F\left(a_{2},b_{2}\right)-F\left(a_{1},b_{2}\right)-F\left(a_{2},b_{1}\right)+F\left(a_{1},b_{1}\right)$$