818 

   818 

   819 \begin{proof}

   819 \begin{proof}

   820 After a small perturbation, we may assume that $q$ is simultaneously transverse to all the splittings in $P$, and

   820 After a small perturbation, we may assume that $q$ is simultaneously transverse to all the splittings in $P$, and

   821 (by Axiom \ref{axiom:splittings}) that $c$ splits along $q$.

   821 (by Axiom \ref{axiom:splittings}) that $c$ splits along $q$.

   822 We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$.

   822 We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$.

   824 \end{proof}

   824 \end{proof}

   825 

   825 

   826 

   826 

   827 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   827 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   828 

   828