For a symmetric function $C(x,y)$ and $a,b,c,d \in [0,1]$ with $b\ge a$, $d \ge c$ and further, C(x,y) is non-decreasing in $x,y$. Then, does it hold that:
$$ C(b,d) -C(a,d) -C(b,c) + C(a,c) \ge 0 $$
We clearly have that $$ C(b,d)-C(a,d) \ge 0 $$ but then, $$ C(a,c) - C(b,c) \le 0 $$ does the symmetry allow us to say that these differences cancel each other out