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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:211_1"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 ;<br/><a NAME="E2:211_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a> ;<br/><a NAME="E3:211_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:211_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">W</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Olive">x</font>,<font color="Olive">x</font>
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:211"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>
 ;<br/>

<a NAME="E3:211"/>
( <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:211"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 <b>by </b><i/>;<br/>


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