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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>  holds not <font color="Olive">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_002.html#D2" target="_self">GLIB_002:def 2</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:8_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; <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> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">e</font> )
 ;<br/><b>set </b><font color="Maroon">v1</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> <font color="Maroon">e</font>;<br/><b>set </b><font color="Maroon">v2</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> <font color="Maroon">e</font>;<br/><b>reconsider </b><font color="Maroon">v1</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> <font color="Maroon">e</font>, <font color="Maroon">v2</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> <font color="Maroon">e</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:8_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v1</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<font color="Maroon">v1</font>;<br/><a NAME="E4:8_1_1_1"/>
 <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> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/><a NAME="E5:8_1_1_1"/><b>then </b>
not <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<font color="Maroon">v1</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/><a NAME="E6:8_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<font color="Maroon">v1</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:8_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> is <a href="glib_000.html#V3">loopless</a>
 
<b>by </b><i><a class="ref" href="glib_000.html#D20">GLIB_000:def 20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">e1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> )
 ;<br/><a NAME="E2:8_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E3:8_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T21">GLIB_000:21</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font>
 ;<br/><b>set </b><font color="Maroon">W1</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font>;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font>;<br/><b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> ;<br/><a NAME="E3:8_1_1_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/><a NAME="E4:8_1_1_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">W1</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">W1</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><a NAME="E5:8_1_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W1</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">v1</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><a NAME="E6:8_1_1_2_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T109">GLIB_001:109</a></i>;<br/><a NAME="E7:8_1_1_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
not <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:8_1_1_2_1"/>
( <font color="Maroon">W1</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">W1</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><a NAME="E9:8_1_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
not <font color="Maroon">W1</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="E10:8_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E11:8_1_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</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="E12:8_1_1_2_1"/><i><font color="Green">E70</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:8_1_1_2_1_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> )
 ;<br/><a NAME="E2:8_1_1_2_1_1"/><b>then </b>
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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:8_1_1_2_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:8_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:8_1_1_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:8_1_1_2_1_1"/>
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D5">GLIB_001:def 5</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_2_1_1"/>
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:8_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/><a NAME="E15:8_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E16:8_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_2"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E5:8_1_1"/><b>then </b>
<font color="Maroon">G</font> is <a href="glib_000.html#V5">non-multi</a>
 
<b>by </b><i><a class="ref" href="glib_000.html#D22">GLIB_000:def 22</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1_1"/>
<font color="Maroon">G</font> is <a href="glib_000.html#V7">simple</a>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D24">GLIB_000:def 24</a></i>;<br/>


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