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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T24">GLIB_000:24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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="E4:53"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E39</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>then </b><i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K4">.last()</a>  = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>
;<br/><a NAME="E3:53_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E4:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E2:53"><i><font color="Green">E37</font></i></a>, <a class="txt" href="glib_002.html#E2:53_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E5:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:53"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:53"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E8:53"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:53"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="glib_001.html#T9">GLIB_001:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:53"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E36"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>thus </b><a NAME="E13:53"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="glib_000.html#V11">endvertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E12:53"><i><font color="Green">E41</font></i></a></i>;<br/>


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