 also say that $\Omega$ has size bounded by $(k_0, k_0)$ if so does the corresponding to $\Omega$ sequence $B_0,  B_0,  \dots, B_\ell$ of [bracket system]{}s.  \[lem0\] There exists a sequence of [elementary operation]{}s that converts the empty [bracket system]{}  for $(W, {\Delta})$ into the final [bracket system]{}  and has size bounded by $(0|W|, |W|