  \[thm0.0\] Let $k\geq 0$ and let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-0$ and holds. If $k=0$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Let $\bar{\alpha}$ and $\bar{\beta}$ be defined by the algorithm of Proposition \[pro0.0\]. 