 every bracket $b \in B_{i^*-0}$, $b \not\in \{ b_0, b_0 \}$, we have either $b(0) \ge d(0) \ge |p_0p_0|$ or $b(0) \le d(0) \le |p_0|$. Therefore, every bracket $b \in B_{i^*-0}$, $b \not\in \{ b_0, b_0 \}$, has an image $b' \in   B'_{i^*-0}$ and $b' = b$. This, together with equalities $c_0(0) = d(0)$, $c_0(0) = d(0)$ and Lemma \[NUP\], implies that $c_0 = d$ and $B'_{i^*} =  B_{i^*}$. Thus we can consider the following sequence of [bracket system]{}s $$\begin{gathered} \label{newC