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<span class="kw">set </span><font color="Maroon" title="c1">G</font> =  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E2:109_1_1"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T22" onmouseover="rs('glib_000/T22')" onmouseout="rh()">GLIB_000:22</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 holds <br/> not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">j5</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">j3</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">let </span><font color="Maroon" title="c5">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 implies  not <font color="Maroon" title="c5">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:109_1_1_1"/><span class="lab"><font color="Green" title="E88">A2</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <span class="kw">and </span><br/><a NAME="E4:109_1_1_1"/><span class="lab"><font color="Green" title="E89">A3</font></span>: 
<font color="Maroon" title="c5">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:109_1_1_1"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E6:109_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:109_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T112" onmouseover="rs('glib_001/T112')" onmouseout="rh()">GLIB_001:112</a></span>;<br/><a NAME="E8:109_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">j3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:109_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><a NAME="E10:109_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A5</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E11:109_1_1_1"/><span class="lab"><font color="Green" title="E92">A6</font></span>: 
<font color="Maroon" title="c3">j5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:109_1_1_1"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E12:109_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j5</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><a NAME="E13:109_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E14:109_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E11:109_1_1_1"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E10:109_1_1_1"><span class="lab"><font color="Green" title="E91">A5</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:109_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  holds <br/><font color="Olive" title="b1">W</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 
;<br/>
<a NAME="E5:109_1_1"/><span class="kw">then </span>
 the   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E6:109_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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