Inductive sequence of words in a biprefix code
Your paper has a typo. It should be $u_0 = v_{j-i}$ rather than $u_0 = v_{j-1}$.
Indeed, consider the following example: the alphabet is $A = \{0,1\}$, and the two codes are $X_1 = A^1$ and $X_2 = A^2$. Take $u = 01$ and $v = 10$. Then $u_0 = 01$, $v_0 = 10$, and $v_1 = 01$. If $i = 0$ and $j = 1$ then $u_i = v_j$, yet $u_0 \neq v_{j-1}$; but $u_0 = v_{j-i}$.
You already know that $u_i = v_j$ implies $u_{i-1} = v_{j-1}$. Applying this step $i$ times, you immediately conclude that $u_0 = v_{j-i}$.