 to verify that ${\widehat}B_{0,0}, \dots, {\widehat}B_{0, \ell_0}$ is an operational sequence of [bracket system]{}s for $(W, {\Delta})$ which corresponds to an analogue ${\widehat}\Omega_0$ of the sequence of [elementary operation]{}s $\Omega_0$ for $(W_0, {\Delta}_0)$ so that if a bracket $b \in B