<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">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="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b2">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="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> &amp; not <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/> <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>  st <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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <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="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> &amp; not <font color="Olive" title="b2">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>  &amp; not <font color="Olive" title="b3">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>  holds <br/> <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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></span><br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <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="c4">a</font>,<font color="Maroon" title="c6">c</font> &amp; not <font color="Maroon" title="c5">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>  &amp; not <font color="Maroon" title="c6">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>  implies  <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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> )</span><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="c4">a</font>,<font color="Maroon" title="c5">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="c4">a</font>,<font color="Maroon" title="c6">c</font>
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green" title="E50">A3</font></i>: 
not <font color="Maroon" title="c5">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="E4:62"/><i><font color="Green" title="E51">A4</font></i>: 
not <font color="Maroon" title="c6">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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></span><br/>

<a NAME="E5:62"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E2:62"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E3:62"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="helly.html#T37" target="_self" title="HELLY:th.37">Th36</a></i>;<br/>
<a NAME="E6:62"/><i><font color="Green" title="E53">A6</font></i>: 
not <font color="Maroon" title="c3">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E2:62"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E4:62"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="helly.html#T37" target="_self" title="HELLY:th.37">Th36</a></i>;<br/>
<a NAME="E7:62"/><i><font color="Green" title="E54">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="helly.html#D3" target="_self" title="HELLY:def.3">Def3</a></i>;<br/>
<a NAME="E8:62"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<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></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E2:62"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E5:62"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="#E6:62"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="helly.html#E44"><i><font color="Green" title="E39">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:62"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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="#E7:62"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
