\] Let there be a sequence $\Omega$ of [elementary operation]{}s that transforms the empty [bracket system]{}  for the pair $(W, {\Delta})$ into the final [bracket system]{}  and has size bounded by $(k_0, k_0)$ and let $c$ be a starting bracket for $(W, {\Delta})$. Then there is also a sequence of [elementary operation]{}s that converts the [bracket system]{}  $\{ c \}$ into the final [bracket system]{}  and has size bounded by $(k_0+0, k_0+0)$.  