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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</a>;<br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font>, <font color="Maroon" title="c4">P3</font>, <font color="Maroon" title="c5">P4</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c5">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E5:62"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E6:62"/><i><font color="Green" title="E53">A6</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E7:62"/><i><font color="Green" title="E54">A7</font></i>: 
not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon" title="c5">P4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P3</font>
 ;<br/><a NAME="E2:62_1"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="helly.html#T13" target="_self" title="HELLY:th.13">Th13</a></i>;<br/><a NAME="E3:62_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E4:62"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></i>;<br/><a NAME="E4:62_1"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E2:62_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:62_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="helly.html#E3:62"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="helly.html#E6:62"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="helly.html#T31" target="_self" title="HELLY:th.31">Th31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:62"/><b>then </b><i><font color="Green" title="E55">A9</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:62"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="helly.html#E4:62"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="helly.html#T36" target="_self" title="HELLY:th.36">Th36</a></i>;<br/>
<a NAME="E10:62"/><i><font color="Green" title="E56">A10</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="helly.html#E2:62"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="helly.html#E5:62"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="helly.html#E6:62"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="helly.html#T50" target="_self" title="HELLY:th.50">Th50</a></i>;<br/>
<a NAME="E11:62"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E3:62"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="helly.html#E4:62"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="helly.html#E7:62"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="helly.html#E9:62"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="helly.html#T50" target="_self" title="HELLY:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E12:62"/>
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E10:62"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="helly.html#T41" target="_self" title="HELLY:th.41">Th41</a></i>;<br/>


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