$\livar{V}{i-1}$ cannot contain
            leaves.
          \end{spfstep}
          \begin{spfstep}[type=conclusion]
           Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
          \end{spfstep}
        \end{spfcase}
      \end{spfcases}
    \end{sproof}
  \item 
    \begin{assertion}[id=fbbt,type=corollary]	
      A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
    \end{assertion}
  \item
      \begin{sproof}[for=fbbt,id