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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">G</font> is <a href="glib_000.html#V4">trivial</a> &amp; <font color="Maroon">G</font> is <a href="glib_000.html#V3">loopless</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>given </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><a NAME="E1:7_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 ;<br/><a NAME="E2:7_1_1_1"/>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E3:7_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T137">GLIB_001:137</a></i>;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:7_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E6:7_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T26">GLIB_000:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:7_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> is <a href="glib_002.html#V2" target="_self">acyclic</a>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as  <a href="glib_000.html#V4">trivial</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i/>;<br/>
<a NAME="E5:7_1_1"/>
<font color="Maroon">G'</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 
;<br/>
<b>hence </b><a NAME="E6:7_1_1"/>
<font color="Maroon">G</font> is <a href="glib_002.html#V3" target="_self">Tree-like</a>
 <b>by </b><i>, </i>;<br/>


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