deletions | additions       diff --git a/subsubsection_Heap_Comparisons_Assume_we__.tex b/subsubsection_Heap_Comparisons_Assume_we__.tex  index 095b602..99ba690 100644  --- a/subsubsection_Heap_Comparisons_Assume_we__.tex  +++ b/subsubsection_Heap_Comparisons_Assume_we__.tex 
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\begin{proof}  This follows immediately from the continuity of our intervals and the given non-intersection.  To see why assume the opposite. Say we had two intervals $A$ and $B$ as above with $\|b_0\| \leq \|a_\alpha\|$. If Assume  the ordering changed at by  scan-angle $\phi$, that is $\|a_\phi\| \leq \|b_\phi\|$. If we define the function $f(\theta):=\|a_\theta\| - \|b_\theta\|$, then by rewriting the conditions above: \begin{align*}  f(\alpha) \geq 0 \\