<?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>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">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>let </b><font color="Maroon" title="c5">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E49">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> holds <br/>(  ex <font color="Olive" title="b2">t</font> being  <a href="helly.html#NM2" title="HELLY:NM.2">_Subtree</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Olive" title="b2">t</font> &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) or ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) or ( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">Pab</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Pac</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c8">Pbc</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c9">Pca</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c10">VPab</font> = <span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> ;<br/>
<b>set </b><font color="Maroon" title="c11">VPac</font> = <span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> ;<br/>
<b>set </b><font color="Maroon" title="c12">VPbc</font> = <span class="p1">(<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="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> ;<br/>
<b>set </b><font color="Maroon" title="c13">VPca</font> = <span class="p1">(<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">c</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> ;<br/>
<a NAME="E2:63"/><i><font color="Green" title="E50">A2</font></i>: 
<span class="p1">(<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">c</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="helly.html#T31" target="_self" title="HELLY:th.31">Th31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">m</font> =  <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/>
<a NAME="E3:63"/>
<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="c2">a</font>,<font color="Maroon" title="c3">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="c3">b</font>,<font color="Maroon" title="c4">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="c4">c</font>,<font color="Maroon" title="c2">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="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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="E4:63"/><b>then </b><i><font color="Green" title="E51">A3</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">a</font>,<font color="Maroon" title="c3">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="c3">b</font>,<font color="Maroon" title="c4">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="c4">c</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:63"/><b>then </b><i><font color="Green" title="E52">A4</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">a</font>,<font color="Maroon" title="c3">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="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="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:63"/><b>then </b><i><font color="Green" title="E53">A5</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:63"/><i><font color="Green" title="E54">A6</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E2:63"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="helly.html#E4:63"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:63"/><i><font color="Green" title="E55">A7</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E5:63"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E56">A8</font></i>: 
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c16">t</font> being   <a href="helly.html#NM2" title="HELLY:NM.2">_Subtree</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green" title="E57">A9</font></i>: 
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">t</font>
 <b>by </b><i><a class="txt" href="helly.html#E1:63"><i><font color="Green" title="E49">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1"/>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> ) or ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> ) or ( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> ) )
 </span><b>by </b><i><a class="txt" href="helly.html#E1:63"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="helly.html#E1:63_1"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> )
 ;<br/><div class="add"><a NAME="E2:63_1_1_1"/><b>then </b>
<span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E3:63_1"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="helly.html#T33" target="_self" title="HELLY:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_1"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E6:63"><i><font color="Green" title="E53">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_2"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> )
 ;<br/><div class="add"><a NAME="E2:63_1_1_2"/><b>then </b>
<span class="p1">(<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="c2">a</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E3:63_1"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="helly.html#T33" target="_self" title="HELLY:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_2"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E7:63"><i><font color="Green" title="E54">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_3"><b>suppose </b></a><a NAME="E1:63_1_1_3"/>
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font> )
 ;<br/><div class="add"><a NAME="E2:63_1_1_3"/><b>then </b>
<span class="p1">(<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="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E3:63_1"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="helly.html#T33" target="_self" title="HELLY:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_3"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">s</font>
 <b>by </b><i><a class="txt" href="helly.html#E8:63"><i><font color="Green" title="E55">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div>
