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

<b>set </b><font color="Maroon">W</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:111_1_1_1_1"/>
( <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font> or  not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:111_1_1_1_1_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:111_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:111_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:111_1_1_1_1_2"/>
not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:111_1_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:111_1_1_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:111_1_1"/>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 ;<br/>


</div>
