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-<!doctype html>
-<html>
-  <head>
-    <title>CodeMirror 2: sTeX mode</title>
-    <link rel="stylesheet" href="../../lib/codemirror.css">
-    <script src="../../lib/codemirror.js"></script>
-    <script src="stex.js"></script>
-    <link rel="stylesheet" href="stex.css">
-    <style>.CodeMirror {background: #f8f8f8;}</style>
-    <link rel="stylesheet" href="../../css/docs.css">
-  </head>
-  <body>
-    <h1>CodeMirror 2: sTeX mode</h1>
-     <form><textarea id="code" name="code">
-\begin{module}[id=bbt-size]
-\importmodule[balanced-binary-trees]{balanced-binary-trees}
-\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
-
-\begin{frame}
-  \frametitle{Size Lemma for Balanced Trees}
-  \begin{itemize}
-  \item
-    \begin{assertion}[id=size-lemma,type=lemma] 
-    Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree} 
-    of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
-     $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
-    \termref[cd=graphs-intro,name=node]{nodes} at 
-    \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
-    \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
-   \end{assertion}
-  \item
-    \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
-      \begin{spfcases}{We have to consider two cases}
-        \begin{spfcase}{$i=0$}
-          \begin{spfstep}[display=flow]
-            then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
-            $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
-          \end{spfstep}
-        \end{spfcase}
-        \begin{spfcase}{$i>0$}
-          \begin{spfstep}[display=flow]
-           then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes 
-           \begin{justification}[method=byIH](IH)\end{justification}
-          \end{spfstep}
-          \begin{spfstep}
-           By the \begin{justification}[method=byDef]definition of a binary
-              tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
-            two children that are at depth $i$.
-          \end{spfstep}
-          \begin{spfstep}
-           As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ 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=fbbt-pf]{}
-        \begin{spfstep}
-          Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
-        \end{spfstep}
-        \begin{spfstep}
-          Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
-        \end{spfstep}
-      \end{sproof}
-    \end{itemize}
-  \end{frame}
-\begin{note}
-  \begin{omtext}[type=conclusion,for=binary-tree]
-    This shows that balanced binary trees grow in breadth very quickly, a consequence of
-    this is that they are very shallow (and this compute very fast), which is the essence of
-    the next result.
-  \end{omtext}
-\end{note}
-\end{module}
-
-%%% Local Variables: 
-%%% mode: LaTeX
-%%% TeX-master: "all"
-%%% End: \end{document}
-</textarea></form>
-    <script>
-      var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
-    </script>
-
-    <p><strong>MIME types defined:</strong> <code>text/stex</code>.</p>
-
-  </body>
-</html>