Let $Pr_{(X,Y)}$ be a probability measure of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define $$ \mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq 3\} $$ Is there a way to express $Pr_{(X,Y)}(\mathcal{A})$ as $$ \sum_{k=1}^K F(a_k, b_k)\times c_k $$ for some finite $K$ and $\{a_k, b_k, c_k\}_{k=1}^K$?

Note: if $(X,Y)$ had a uniform distribution, then we could write $$ Pr_{(X,Y)}(\mathcal{A})=F((-\infty,2]\times [-1,\infty))+\frac{1}{2}F((-\infty,2]\times (-\infty,-1]) $$ However, I wonder whether we could do something similar for the general case without uniformity.

Let $Pr_{(X,Y)}$ be a probability distribution of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define $$ \mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq 3\} $$ Is there a way to express $Pr_{(X,Y)}(\mathcal{A})$ as $$ (*)\quad \sum_{k=1}^K F(a_k, b_k)\times c_k $$ for some finite $K$ and $\{a_k, b_k, c_k\}_{k=1}^K$?

Note: if $(X,Y)$ had a uniform distribution, then we could write $$ Pr_{(X,Y)}(\mathcal{A})=Pr_{(X,Y)}((-\infty,2]\times [-1,\infty))+\frac{1}{2}Pr_{(X,Y)}((-\infty,2]\times (-\infty,-1]) $$ which can be rewritten as $(*)$. However, I wonder whether we could do something similar for the general case without uniformity.