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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>thus </b><a NAME="E1:80"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V3" target="_self">Tree-like</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_002.html#T46" target="_self">Th46</a></i>;<br/>

<b>assume </b><a NAME="E2:80"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</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">c<sub>1</sub></font> is <a href="glib_002.html#V2" target="_self">acyclic</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14">.edges()</a> </span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="glib_000.html#NM8">removeEdge</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14">.edges()</a> </span>)</span></span>)</span>;<br/><a NAME="E5:80_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:80_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E6:80_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></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/><a NAME="E7:80_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><a NAME="E8:80_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:80"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_002.html#E3:80_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E6:80_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_002.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:80_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a>  )
 <b>by </b><i><a class="txt" href="glib_002.html#E7:80_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#T55">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">c<sub>4</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E2:80"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_002.html#E8:80_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_002.html#T40" target="_self">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">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V3" target="_self">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:80"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a></i>;<br/>


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