e, let ${\widehat}B_{0,i} := \{ {\widehat}d \mid d \in B_{0,i} \}$. Then ${\widehat}B_{0,0}, \dots, {\widehat}B_{0, \ell_0}$ is a sequence of [bracket system]{}s for $(W, {\Delta})$ that changes the [bracket system]{}$${\widehat}B_{0,0} = {\widehat}B_{0,\ell_0} = \{ (|p_0|,|p_0p_0|, 0, 0, 0, |{\Delta}_0(0)| ) \}$$ into the final [bracket system]{}