 obtained from $B_{0,j}$ by replacing every bracket $b \in B_{0,j}$ with ${\widehat}b$. Then it is easy to verify that ${\widehat}B_{0,0}, \dots, {\widehat}B_{0, \ell_0}$ is an operational sequence of [bracket system]{}s for $(W, {\Delta})$ that changes the empty [bracket system]{} into $${\widehat}B_{0, \ell_0} = \{  (|p_0|, |p_0|+ |p_0|, 