How did they get from:
$$\Pr[Z_n > t -s + \tau~\mid~Z_n > t-s]$$
to:
$$\Pr[Z_n > t -s + \tau~\mid~Z_n > t-s] = \frac{\Pr[Z_n > t-s +\tau]}{\Pr[Z_n > t -s]}$$
It looks almost like baye's law, except you need a joint probability in the numerator to apply bayes law: $\Pr[Z_n > t -s + \tau~,~Z_n > t-s]$
Here's a scan of the problem from the workbook:
It comes from noticing that the events $A = \left\{Z_n >t-s + \tau\right\}\cap\left\{Z_n >t-s\right\}$ and $B = \left\{Z_n >t-s + \tau\right\}$ are equivalent. If $z_n\in A$ then $z_n > t-s + \tau$ and so $z_n\in B$. Similarly if $z_n\in B$ then $z_n > t-s + \tau$, so $z_n \in A$. Thus $A = B$. Since both sets define the same elements in the sample space we have $\mathbb{P}\left(A\right) = \mathbb{P}\left(B\right)$.