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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="glib_001.html#NM4">Trail</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:223"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a>  &amp; <font color="Maroon">v</font> is <a href="glib_000.html#V11">endvertex</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:223"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:223"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K5" target="_self">.vertexAt</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:223"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="glib_000.html#D53">GLIB_000:def 53</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="223_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:223_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 ;<br/><a NAME="E2:223_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a></i>;<br/><a NAME="E3:223_1"/><i><font color="Green">E46</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E4:223_1"/><b>then </b><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">naa1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:223_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:223_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E8:223_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a></i>;<br/><a NAME="E9:223_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E10:223_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <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">W</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:223_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E12:223_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">naa1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a>, </i>;<br/><a NAME="E13:223_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, </i>;<br/><a NAME="E14:223_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E15:223_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E16:223_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:223"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 ;<br/>


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