$p \geq k+1$

Since $b \not\sim x_{p}$, x_p must be somewhere in the set $\{y_{1} \ldots y_{k}\}$. But this is equal to $\{x_{1} \ldots x_{k}\}$, from the condition on c. From $p \geq k+1$, x_p cannot be in this set. So this case is not possible, the path P'₃ does not correspond to an alias for P₂.