deletions | additions       diff --git a/begin_claim_The_algorithm_runs__.tex b/begin_claim_The_algorithm_runs__.tex  index fb44dfe..8ff203c 100644  --- a/begin_claim_The_algorithm_runs__.tex  +++ b/begin_claim_The_algorithm_runs__.tex 
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&= O(n\log n) + n^{\log 2}O(1) + \sum_{j=0}^{\log n-1} 2^j (\log n - \log{2^j}) \\   &= O(n\log n) + O(n) + \left(\log n\sum_{j=0}^{\log n-1} 2^j - \sum_{j=0}^{\log n-1} 2^j\log{2^j} \right)\\   &= O(n\log n) + \left(\log n \times 2^{\log n} - C \right) &\text{with $C\geq0$}\\  &= O(n\log n) + \left(n\log n - C \right) &\comment{inner &\tag{inner  bracket $\geq 0$} \\ \end{eqnarray}  \end{proof}