$, 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