to $c$ or $v_{\beta\gamma}$ and let $w^{-1}$ and $w^{+1}$ denote the vertices directly before and after $w$ in $[c, v_{ \beta \gamma}]_S$. We want to show that the link connecting $w^{-1}$ to $w^{+1}$ in $S\backslash w$ is at most $5M$. Assume $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$. Consider the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha \beta}]_S \cup [v_{\alpha \beta}, c]_S \cup [c, w^{-1}]_S$, where geodesics are taken to be on $g_*$ where appropriate. The Bounded Geodesic Image Theorem, and our assumption that $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$, implies that $w$ must be somewhere on the path. $w$ cannot be in $[w^{+1}, v_{\beta\gamma}]_S$, $[v_{ \beta \gamma}, \beta_0]_S$, or $[c, w^{-1}]_S$ since that would contradict the fact that they are geodesics or the definition of how we chose $c$ and $v_{\beta\gamma}$. Therefore, $w$ is in $[\beta_0, v_{\alpha \beta}]_S$ or $[v_{\alpha \beta}, c]_S$. Without loss of generality assume $w \in [\beta_0, v_{\alpha \beta}]_S$. We can apply the same logic to the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \gamma_0]_S \cup [\gamma_0, v_{\alpha \gamma}]_S \cup [v_{\alpha \gamma}, c]_S \cup [c, w^{-1}]_S$. Now $w$ has to be in $[v_{\alpha \gamma}, c]_S$ so that it doesn’t contradict the 