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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c1">G</font>  st not <font color="Olive" title="b1">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 &amp; not <font color="Olive" title="b1">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Olive" title="b1">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> not <font color="Olive" title="b1">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a></span><br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 &amp; not <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Maroon" title="c2">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )</span><br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green" title="E85">A1</font></i>: 
( not <font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 &amp; not <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a></span><br/>

<b>assume </b><a NAME="E2:106"/>
<font color="Maroon" title="c2">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:106"/><i><font color="Green" title="E86">A2</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</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="c2">W</font> &amp; <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <b>and </b><br/><a NAME="E5:106"/><i><font color="Green" title="E87">A3</font></i>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E6:106"/>
( <font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> ) ) )
 <b>by </b><i><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:106"/><i><font color="Green" title="E88">A4</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E5:106"><i><font color="Green" title="E87">A3</font></i></a></i>;<br/>
<div><i><font color="Green" title="E89">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:106_1"/><b>then </b>
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a></i>;<br/><a NAME="E3:106_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:106_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E5:106_1"/><b>then </b>
3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E6:106_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E4:106"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:106"/><b>then </b>
3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="#E4:106"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/>
<a NAME="E11:106"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E4:106"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E12:106"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E8:106"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="#E9:106"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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