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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:52"/><i><font color="Green">E41</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">W</font> being  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E4:52_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K11">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#T155">GLIB_001:155</a></i>;<br/><a NAME="E5:52_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K11">.vertexSeq()</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:52_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#D14">GLIB_001:def 14</a></i>;<br/><a NAME="E7:52_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E8:52_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">src</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">tar</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:52"/><i><font color="Green">E44</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:52"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i>
<b>consider </b><font color="Maroon">W</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:52"/><i><font color="Green">E62</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E10:52"/><i><font color="Green">E64</font></i>: 
3 <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="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E11:52"/><b>then </b><i><font color="Green">E65</font></i>: 
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 ;<br/><a NAME="E2:52_2"/><b>then </b>
<font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><a NAME="E3:52_2"/><b>then </b>
<font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:52"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E4:52_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E13:52"/><b>then </b><i><font color="Green">E67</font></i>: 
not <font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:52_3"/><i><font color="Green">E69</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; not <font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a> )
 ;<br/><a NAME="E2:52_3"/><b>then </b><i><font color="Green">E70</font></i>: 
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E3:52_3"/>
2 <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="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:52_3"/><b>then </b>
2 <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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenW2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:52_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_3_1"/>
not <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  is <a href="glib_000.html#V11">endvertex</a>
 ;<br/><a NAME="E2:52_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K36">.degree()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_000.html#D54">GLIB_000:def 54</a></i>;<br/><a NAME="E3:52_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_000.html#T22">GLIB_000:22</a></i>;<br/><a NAME="E4:52_3_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89">GLIB_001:89</a></i>;<br/><a NAME="E5:52_3_1"/><b>then </b>
not <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  is <a href="glib_000.html#V10">isolated</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T136">GLIB_001:136</a></i>;<br/><a NAME="E6:52_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K36">.degree()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="glib_000.html#D52">GLIB_000:def 52</a></i>;<br/><a NAME="E7:52_3_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="glib_000.html#T22">GLIB_000:22</a></i>;<br/><a NAME="E8:52_3_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E9:52_3_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:52_3_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">e1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:52_3_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; <font color="Maroon">e1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_1_1"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e1</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">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b><font color="Maroon">e2</font> = <font color="Maroon">e2</font>;<br/><b>thus </b><a NAME="E2:52_3_1_1_1_1"/>
( <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; <font color="Maroon">e2</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">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_1_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">e1</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">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b><font color="Maroon">e1</font> = <font color="Maroon">e1</font>;<br/><b>thus </b><a NAME="E2:52_3_1_1_1_2"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; <font color="Maroon">e1</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">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:52_3_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; <font color="Maroon">e</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">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E17:52_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T67">GLIB_000:67</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:52_3_1_2_1_1"/><i><font color="Green">E118</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><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font>;<br/><a NAME="E4:52_3_1_2_1_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <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> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( for <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font> ) )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D22">GLIB_001:def 22</a></i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>;<br/><a NAME="E5:52_3_1_2_1_1"/><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E6:52_3_1_2_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E7:52_3_1_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T65">GLIB_000:65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1_1_1_1"/>
(  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52"><i><font color="Green">E59</font></i></a>, </i>;<br/><a NAME="E3:52_3_1_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E4:52_3_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><a NAME="E5:52_3_1_2_1_1_1_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T133">GLIB_001:133</a></i>;<br/><a NAME="E6:52_3_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E5:52"><i><font color="Green">E60</font></i></a>, <a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E7:52_3_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19">.rfind</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T107">GLIB_001:107</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:52_3_1_2_1_1_1_1_2"/><i><font color="Green">E163</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E5:52"><i><font color="Green">E60</font></i></a>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E5:52_3_1_2_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:52_3_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:52_3_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:52_3_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:52_3_1_2_1_1_1_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:52_3_1_2_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E12:52_3_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:52"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> ) or ( <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> ) )
 </span><b>by </b><i>, <a class="txt" href="glib_002.html#E10:52"><i><font color="Green">E64</font></i></a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><b>hence </b><a NAME="E3:52_3_1_2_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1_2"/>
( <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> )
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E3:52_3_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E9:52"><i><font color="Green">E62</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:52_3_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 2 <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="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:52_3_1_2_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="glib_002.html#E11:52"><i><font color="Green">E65</font></i></a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:52_3_1_2_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:52_3_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_2"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T152">GLIB_001:152</a></i>;<br/><a NAME="E3:52_3_1_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E4:52_3_1_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:52_3_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:52_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:52_3"/>
<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  is <a href="glib_000.html#V11">endvertex</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E15:52"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  is <a href="glib_000.html#V11">endvertex</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E16:52"/><i><font color="Green">E171</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E2:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/>
<a NAME="E17:52"/>
not <font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a>  is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T121">GLIB_001:121</a></i>;<br/>
<a NAME="E18:52"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4">.last()</a>  is <a href="glib_000.html#V11">endvertex</a>
 
<b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E19:52"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  is <a href="glib_000.html#V11">endvertex</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T23">GLIB_001:23</a></i>;<br/>
<a NAME="E20:52"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E21:52"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E22:52"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">v1</font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Olive">v2</font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Olive">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Olive">v1</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
