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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>thus </b><a NAME="E1:80"/>
( <font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> implies ( <font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self" title="GLIB_002:def.3">Def3</a>, <a class="ref" href="glib_002.html#T46" target="_self" title="GLIB_002:th.46">Th46</a></i>;<br/>

<b>assume </b><a NAME="E2:80"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1"/>
not <font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>
 ;<br/><b>then consider </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><br/><a NAME="E3:80_1"/><i><font color="Green" title="E48">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>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">Def2</a></i>;<br/><b>set </b><font color="Maroon" title="c3">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span>;<br/><b>consider </b><font color="Maroon" title="c4">G2</font> being   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span></span>)</span>;<br/><a NAME="E5:80_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#E3:80_1"><i><font color="Green" title="E48">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="E6:80_1"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<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/><a NAME="E7:80_1"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <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> 
 ;<br/><a NAME="E8:80_1"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c4">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:80"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="glib_002.html#E3:80_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="glib_002.html#E6:80_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="glib_002.html#T5" target="_self" title="GLIB_002:th.5">Th5</a></i>;<br/><a NAME="E9:80_1"/>
( <font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a>  )
 <b>by </b><i><a class="txt" href="glib_002.html#E7:80_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="glib_000.html#T55" title="GLIB_000:th.55">GLIB_000:55</a></i>;<br/><a NAME="E10:80_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E2:80"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="glib_002.html#E8:80_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="glib_002.html#T40" target="_self" title="GLIB_002:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E11:80_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
<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#E2:80"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="glib_002.html#D3" target="_self" title="GLIB_002:def.3">Def3</a></i>;<br/>


</div>
