817 

   817 

   818 \begin{proof}

   818 \begin{proof}

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

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

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

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

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

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

   823 \end{proof}

   823 \end{proof}

   824 

   824 

   825 

   825 

   826 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   826 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   827 

   827