I have essentially the same question found here, except on that post the question was not answered fully. Or at least not in a way I can follow. I'm able to follow everything in Brin-Stuck's proof except for the inequality
$$ |\phi_{n+1}(z)-\psi_{n+1}(z)|\geq \frac{\text{length}(f_n(c^u))}{1+2L}-L(1+L)\text{length}(f_n(c^u)). $$
I'm trying to use $|\phi_{n+1}(z)-\psi_{n+1}(z)|\geq |A|-|B|$, where $|A|$ is the length of a geodesic from $f_n(y,\psi_n'(y))$ to $f_n(y,\phi_n'(y))$, and $|B|$ is the length of a geodesic from $f_n(y,\phi_n'(y))$ to $(z,\phi_{n+1}(z))$.
For $|B|$, if I let $y'$ be the projection of $f_n(y,\phi_n'(y))$ onto $E^s(n+1)$, then $f_n(y,\phi_n'(y))=(y',\phi_{n+1}(y'))$ and
\begin{align*} |B| &= |(y',\phi_{n+1}(y'))-(z,\phi_{n+1}(z))|=|(y'-z,\phi_{n+1}(y')-\phi_{n+1}(z))| \newline &=|y'-z| + |\phi_{n+1}(y')-\phi_{n+1}(z)|\newline &\leq |y'-z|(1+L) \end{align*}
and then because we are in $K^u_L(n+1)$, $|y'-z|<L\cdot\text{length}(f_n(c^u))$. So then
$$ -|B| \geq -L(1+L)\text{length}(f_n(c^u)).$$
However I still cannot get the other part of the inequality and would appreciate help.