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<b>let </b><font color="Maroon">G</font> be  non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V1" target="_self">connected</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:62"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T24">GLIB_000:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1"/>
 ex <font color="Olive">u</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:62"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_2"/>
 ex <font color="Olive">u</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">u</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:62"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/>
<b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:62"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:62"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="txt" href="glib_002.html#E7:62"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T89">GLIB_001:89</a></i>;<br/>
<a NAME="E10:62"/>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E5:62"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E8:62"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/>
<b>hence </b><a NAME="E11:62"/>
not <font color="Maroon">v</font> is <a href="glib_000.html#V10">isolated</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T136">GLIB_001:136</a></i>;<br/>


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