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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c1">G</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> &amp; <font color="Maroon" title="c1">G</font> is <a href="glib_000.html#V3" title="GLIB_000:attr.3">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" title="c2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E1:7_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 ;<br/><a NAME="E2:7_1_1_1"/>
not <font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:7_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><a NAME="E3:7_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <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="glib_001.html#T137" title="GLIB_001:th.137">GLIB_001:137</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:7_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E6:7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:7_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="glib_002.html#E5:7_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="glib_000.html#T26" title="GLIB_000:th.26">GLIB_000:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:7_1_1"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>
 
<b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">G'</font> = <font color="Maroon" title="c1">G</font> as  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <b>by </b><i><a class="txt" href="glib_002.html#E1:7_1_1"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<a NAME="E5:7_1_1"/>
<font color="Maroon" title="c2">G'</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
;<br/>
<b>hence </b><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:7_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="glib_002.html#D3" target="_self" title="GLIB_002:def.3">Def3</a></i>;<br/>


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