from $|p|<M$ that $-p_{i_{\ell-1}}+p+p_{i_\ell} \neq 0$, and $v_\ell$ is again a nontrivial power of $\sigma$. Thus, in all subcases, $v_\ell$ is again a non-empty word ending with $\sigma^{\pm 1}$ as desired.

This concludes the proof that, if $\varepsilon_\ell=+1$, then $v_\ell$ is a non-empty word ending with $\sigma^{\pm 1}$. The analogous arguments show that, if $\varepsilon_\ell=+1$, then $v_{\ell+1}$ is a non-empty word beginning with $\tau^{\pm 1}$. The symmetric arguments show that, if $\varepsilon_\ell=-1$, then $v_\ell$ is a nonempty word ending with $\tau^{\pm1}$ and $v_{\ell+1}$ is a non-empty word beginning with $\sigma^{\pm1}$.

\[nontrivial\] If the last letter of $v_{s+1}$ is $\sigma^{\pm1}$ and $y \in \{x\} \cup U_\tau \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma^{\varepsilon}_i(U_\tau)$, or if the last letter of $v_{s+1}$ is $\tau^{\pm1}$ and $y \in \{x\} \cup U_\sigma \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma^{\varepsilon}_i(U_\sigma)$, then $w(y) \in U_\sigma \cup U_\tau$.

We will prove the claim by induction on $s$. If $s=0$, $w=v_1$ is merely a reduced word in $\sigma^{\pm1}$ and $\tau^{\pm1}$. In this case, $w(y) \in U_\sigma \cup U_\tau$ follows from the classical ping-pong argument as in the proof of Free Subgroup Criterion.

Now, as the induction hypothesis, suppose that the claim is true for $s-1 \geq 0$, and let $w=v_1 \gamma_{i_1}^{\varepsilon_1} v_2 \gamma_{i_2}^{\varepsilon_2} \cdots \gamma_{i_s}^{\varepsilon_s} 