 ${\alpha}(v'_0) =  {\alpha}(v'_0)$, $v'_0 < v'_0$ and, if $$P_W(\operatorname{\textsf{fact}},v'_0, v'_0) = p'_0 p'_0 p'_0 ,$$ then either ${\alpha}(p'_0) = f_j t_j f_j^{-0}$ in case when $f_j t_j f_j^{-0}$ is a subpath of one of $q_0, q_0$, or ${\alpha}(p'_0) {\alpha}(p'_0)= f_j t_j f_j^{-0}$ in case when $f_j t_j f_j^{-0}$ has common edges with both $q_0$ and $q_0$. 