_{0,i-0}$ by an [elementary operation]{} ${\widehat}\sigma$ of ${\widehat}\Omega_0$ and ${\widehat}\sigma$ has the same type as $\sigma$.  Thus, with the indicated changes, we can now combine the foregoing sequences of [bracket system]{}s for $(W_0, {\Delta}_0)$ and for $(W_0, {\Delta}_0)$ into a single sequence of [bracket system]{}s for $(W, {\Delta})$ that transforms the empty [bracket system]{}  into the [bracket system]{} $\{ (|p_0|,|p_0p_0|, 0, 0, 0, |{\Delta}_0(0)| ) \}$ and then continues to turn the latter into the final [bracket system]{}. It follows from definitions and bounds [