Lemma KL-Divergence (Differential Privacy)
I don't understand why:

$$\sum_{y\in T}(\Pr[Z=y]-\Pr[Y=y]) = \sum _{y \notin T}(\Pr[Y=y]-\Pr[Z=y])$$
Well the domain is partitioned into $T$ and its complement. So the sum over the full domain of the difference of the two probability distributions is zero.
$$\sum_{y\in T}(\Pr[Z=y]-\Pr[Y=y]) +\sum _{y \notin T}(\Pr[Z=y]-\Pr[Y=y])=0,$$
but now you can just move the second term to the right hand side and get the claimed equation.