G(I^{k-1})$ and $\tilde{v}=x_ix_j$ for some $j$. In the further discussions we will distinguish four cases.

-   $\deg_{x_i}u\neq 0$ and $\deg_{x_j}u\neq 0$,

-   $\deg_{x_i}u\neq 0$ and $\deg_{x_j}u= 0$,

-   $\deg_{x_i}u= 0$ and $\deg_{x_j}u\neq 0$,

-   $\deg_{x_i}u= 0$ and $\deg_{x_j}u= 0$.

We now first consider the cases (i), (ii) and (iii) and construct in these cases $w\in V(G^{(u,v)}_{I^{k}})$ such that following conditions hold:

-   $\deg (\lcm(u,w)) =2k+1$;

-   $w/\tilde{v}\in G({I^{k-1}})$.

Condition ($\alpha$) implies that $\{u,w\}\in E(G^{(u,v)}_{I^{k}})$. In the case that $\deg(\lcm(w,v))\leq 2k+1$, $w$ is connected to $v$ in $G^{(u,v)}_{I^{k}}$, and so $u$ and $v$ are connected in $G^{(u,v)}_{I^{k}}$. If $\deg(\lcm(w,v))>2k+1$, then condition ($\beta$) allows us to use induction on $k$ as before, and to conclude that $w$ is connected to $v$ by a path in $G^{(u,v)}_{I^{k}}$, and hence $u$ and $v$ are connected in $G^{(u,v)}_{I^{k}}$. Thus ($\alpha$) together with ($\beta$) implies that there is a path in $G^{(u,v)}_{I^{k}}$ connecting $u$ and $v$.

There exists a factor $\tilde{u}\in G(I)$ of $u$ such that $u/\tilde{u}\in G(I^{k-1})$ which in the cases (i) and (iii) is of the form $x_jx_{i_1}$ for some $i_1$ and in case (ii) is of the form $x_ix_{i_2}$ for some $i_2$. It is seen that $\tilde{v}\neq \tilde{u}$, since otherwise $u/\tilde{v}, v/\tilde{v}\in G(I^{k-1})$, 