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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="chord.html#V6">chordal</a> <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a> <a href="glib_003.html#NM3">VGraph</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:179"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> is <a href="lexbfs.html#V21" target="_self">with_property_L3</a>
 <b>and </b><br/><a NAME="E2:179"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">V</font> be    <a href="chord.html#M3">VertexScheme</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E3:179"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">V</font> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>
 ;<br/>

<b>assume </b><a NAME="E4:179"/>
not <font color="Maroon">V</font> is <a href="chord.html#V7">perfect</a>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:179"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E7:179"/><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">H</font> being  <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font></span>)</span> ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">H</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; not <font color="Olive">v</font> is <a href="chord.html#V3">simplicial</a> )
 <b>by </b><i><a class="ref" href="chord.html#D13">CHORD:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">HH</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font></span>)</span>, <font color="Maroon">hv</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">HH</font><b> such that </b><br/><a NAME="E9:179"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">hv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E10:179"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">hv</font> is <a href="chord.html#V3">simplicial</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E29:179"><i><font color="Green">E103</font></i></a></i>;<br/>
<a NAME="E11:179"/><i><font color="Green">E46</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">VLG</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>;<br/>
<a NAME="E12:179"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">V</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="chord.html#D12">CHORD:def 12</a></i>;<br/>
<a NAME="E13:179"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E27:179"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<a NAME="E14:179"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="chord.html#D12">CHORD:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">ha</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">HH</font>, <font color="Maroon">hb</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">HH</font><b> such that </b><br/><a NAME="E16:179"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">ha</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hb</font> &amp; <font color="Maroon">hv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ha</font> &amp; <font color="Maroon">hv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hb</font> )
 <b>and </b><br/><a NAME="E17:179"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">hv</font>,<font color="Maroon">ha</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E18:179"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">hv</font>,<font color="Maroon">hb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E19:179"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">ha</font>,<font color="Maroon">hb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T69">CHORD:69</a></i>;<br/>
<a NAME="E20:179"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E28:179"><i><font color="Green">E102</font></i></a>, <a class="ref" href="chord.html#T107">CHORD:107</a></i>;<br/>
<a NAME="E21:179"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">HH</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E22:179"/>
( <font color="Maroon">hv</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">HH</font> &amp; <font color="Maroon">ha</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">HH</font> &amp; <font color="Maroon">hb</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">HH</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">hv</font>, <font color="Maroon">aa</font> = <font color="Maroon">ha</font>, <font color="Maroon">bb</font> = <font color="Maroon">hb</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E24:179"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><br/> for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/>( <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> iff <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:179_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E2:179_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E3:179_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, <a class="txt" href="lexbfs.html#E33:179"><i><font color="Green">E118</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E4:179_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, , <a class="txt" href="lexbfs.html#E27:179"><i><font color="Green">E101</font></i></a>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<b>thus </b><a NAME="E5:179_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> implies <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, , <a class="txt" href="lexbfs.html#E27:179"><i><font color="Green">E101</font></i></a>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a>, </i>;<br/>

<b>thus </b><a NAME="E6:179_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">y</font> implies <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, , <a class="txt" href="lexbfs.html#E27:179"><i><font color="Green">E101</font></i></a>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_2"><b>now </b></a><div class="add"><a NAME="E1:179_2"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_2_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font> or <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1_1"><b>suppose </b></a><a NAME="E1:179_2_1_1"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">aa</font> = <font color="Maroon">aa</font>;<br/><b>take </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font>;<br/><b>thus </b><a NAME="E2:179_2_1_1"/>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:179_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E33:179"><i><font color="Green">E118</font></i></a></i>;<br/><b>thus </b><a NAME="E4:179_2_1_1"/>
<font color="Maroon">v</font>,<font color="Maroon">aa</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E5:179_2_1_1"/>
<font color="Maroon">v</font>,<font color="Maroon">bb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E6:179_2_1_1"/>
not <font color="Maroon">aa</font>,<font color="Maroon">bb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1_2"><b>suppose </b></a><a NAME="E1:179_2_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font>;<br/><b>take </b><font color="Maroon">aa</font> = <font color="Maroon">aa</font>;<br/><b>thus </b><a NAME="E2:179_2_1_2"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:179_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">aa</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E33:179"><i><font color="Green">E118</font></i></a></i>;<br/><b>thus </b><a NAME="E4:179_2_1_2"/>
<font color="Maroon">v</font>,<font color="Maroon">bb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E5:179_2_1_2"/>
<font color="Maroon">v</font>,<font color="Maroon">aa</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E6:179_2_1_2"/>
not <font color="Maroon">bb</font>,<font color="Maroon">aa</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T45">CHORD:45</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">a</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">bb</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E27:179"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4">.followSet</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E28:179"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bb</font>
 <b>and </b><br/><a NAME="E29:179"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E30:179"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">bb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E31:179"/><i><font color="Green">E116</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">bb</font> <a href="chord.html#R1">are_adjacent</a> 
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">v</font>,<font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="chord.html#R1">are_adjacent</a>  );<br/>
<a NAME="E32:179"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:179"/><i><font color="Green">E118</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">mbb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:179_3"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">mbb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">bb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">mbb</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mbb</font> = <font color="Maroon">mbb</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">mbb</font>
;<br/>

<b>thus </b><a NAME="E4:179_3"/>
<font color="Maroon">mbb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:179_3"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mbb</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="lexbfs.html#E33:179"><i><font color="Green">E118</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:179_3"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mbb</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="lexbfs.html#E33:179"><i><font color="Green">E118</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:179_3"/>
<font color="Maroon">v</font>,<font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mbb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green">E120</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:179_3"/>
not <font color="Maroon">a</font>,<font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mbb</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">mb</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E35:179"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">mb</font>]
 <b>and </b><br/><a NAME="E36:179"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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">mb</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="lexbfs.html#E38:179"><i><font color="Green">E122</font></i></a>, );<br/></i>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mb</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E38:179"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E39:179"/><i><font color="Green">E327</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E28:179"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E40:179"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E30:179"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ma</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E42:179"/><i><font color="Green">E331</font></i>: 
( <font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ma</font> = <font color="Maroon">ma</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a></i>;<br/>
<a NAME="E44:179"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><i><font color="Green">E334</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:179_4"/>
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:179_4"/><b>then </b><i><font color="Green">E335</font></i>: 
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, , <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green">E121</font></i></a>, <a class="txt" href="lexbfs.html#E30:179"><i><font color="Green">E104</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a></i>;<br/><a NAME="E3:179_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E4:179_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="chord.html#D12">CHORD:def 12</a></i>;<br/><a NAME="E5:179_4"/><b>then </b>
<font color="Maroon">a</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="lexbfs.html#E32:179"><i><font color="Green">E117</font></i></a>, <a class="ref" href="chord.html#T16">CHORD:16</a></i>;<br/><a NAME="E6:179_4"/><b>then </b>
<font color="Maroon">a</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E7:179_4"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, <a class="ref" href="finseq_4.html#T34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">P</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> ex <font color="Olive">v1</font>, <font color="Olive">v2</font>, <font color="Olive">v3</font>, <font color="Olive">v4</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Olive">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Olive">P</font> is <a href="chord.html#V5">chordal</a> &amp; <font color="Olive">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <font color="Olive">v2</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Olive">v3</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Olive">v4</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="glib_001.html#K3">.first()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v4</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v4</font> &amp; <font color="Olive">x</font>,<font color="Olive">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Olive">v1</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v4</font> ) );<br/>
<a NAME="E46:179"/><i><font color="Green">E336</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green">Lemma127</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, <a class="txt" href="lexbfs.html#E31:179"><i><font color="Green">E116</font></i></a>, </i>;<br/>
<a NAME="E47:179"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[4]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:179_5"/><i><font color="Green">E339</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E3:179_5"/><i><font color="Green">E341</font></i>: 
( <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">c</font>,<font color="Maroon">v</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E4:179_5"/><i><font color="Green">E342</font></i>: 
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Olive">x</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Maroon">v</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <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="E6:179_5"/><i><font color="Green">E343</font></i>: 
( not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 5 &amp; <font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 2 )
 <b>and </b><br/><a NAME="E7:179_5"/><i><font color="Green">E344</font></i>: 
( <font color="Maroon">e1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">b</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">a</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">v</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K8">}</a></span> )
 <b>and </b><br/><a NAME="E8:179_5"/><i><font color="Green">E345</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , , <a class="ref" href="chord.html#T47">CHORD:47</a></i>;<br/>
<a NAME="E9:179_5"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E10:179_5"/><b>then </b><i><font color="Green">E346</font></i>: 
not <font color="Maroon">P</font> is <a href="chord.html#V5">chordal</a>
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="chord.html#T90">CHORD:90</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="E12:179_5"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">c</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">Qr</font> = <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> ;<br/>
<a NAME="E13:179_5"/><i><font color="Green">E348</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E14:179_5"/><i><font color="Green">E349</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<a NAME="E15:179_5"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; 5 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:179_5"/><b>then </b><i><font color="Green">E351</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/>
<a NAME="E17:179_5"/><i><font color="Green">E352</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <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">c</font>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<div><i><font color="Green">E353</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:179_5_1"/><i><font color="Green">E354</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_5_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_1"><b>suppose </b></a><a NAME="E1:179_5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_2"><b>suppose </b></a><a NAME="E1:179_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_3"><b>suppose </b></a><a NAME="E1:179_5_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_3"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:179_5"/><i><font color="Green">E355</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:179_5"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, , <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/>
<a NAME="E21:179_5"/><b>then </b>
( <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="chord.html#V5">chordal</a> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, , , , <a class="ref" href="chord.html#T97">CHORD:97</a></i>;<br/>
<a NAME="E22:179_5"/><b>then </b><i><font color="Green">E357</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="glib_001.html#V5">Path-like</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="glib_001.html#V1">closed</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="chord.html#V5">chordal</a> )
 
<b>by </b><i><a class="ref" href="chord.html#T91">CHORD:91</a>, <a class="ref" href="glib_001.html#T121">GLIB_001:121</a>, <a class="ref" href="glib_001.html#T149">GLIB_001:149</a></i>;<br/>
<a NAME="E23:179_5"/><i><font color="Green">E358</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> &amp; 5 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> &amp; 7 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:179_5"/><i><font color="Green">E359</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:179_5"/><i><font color="Green">E360</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/>
<a NAME="E26:179_5"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6">-</a> 5</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6">-</a> 7</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="lexbfs.html#T11" target="_self">Th11</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, , <a class="ref" href="glib_001.html#T25">GLIB_001:25</a></i>;<br/>
<a NAME="E27:179_5"/><b>then </b><i><font color="Green">E361</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K3">.first()</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:179_5"/><i><font color="Green">E362</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green">Lemma127</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, <a class="txt" href="lexbfs.html#E31:179"><i><font color="Green">E116</font></i></a>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:179_5_2"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:179_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T93">GLIB_001:93</a></i>;<br/><a NAME="E3:179_5_2"/><b>then </b><i><font color="Green">E364</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_5_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1_1"><b>suppose </b></a><a NAME="E1:179_5_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:179_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:179_5_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1_2"><b>suppose </b></a><a NAME="E1:179_5_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E30:179_5"/>
<font color="Maroon">S<sub>2</sub></font>[4]
 <b>by </b><i><a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:179"/><i><font color="Green">E365</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:179_6"/><i><font color="Green">E366</font></i>: 
4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font>
 <b>and </b><br/><a NAME="E2:179_6"/><i><font color="Green">E367</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">kk</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">kk</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">P</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>, <font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v3</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v4</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:179_6"/><i><font color="Green">E368</font></i>: 
( <font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> is <a href="chord.html#V5">chordal</a> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>and </b><br/><a NAME="E6:179_6"/><i><font color="Green">E369</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">v4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  )
 <b>and </b><br/><a NAME="E7:179_6"/><i><font color="Green">E370</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v1</font> )
 <b>and </b><br/><a NAME="E8:179_6"/><i><font color="Green">E371</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 <b>and </b><br/><a NAME="E9:179_6"/><i><font color="Green">E372</font></i>: 
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v4</font> &amp; <font color="Olive">x</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Maroon">v1</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:179_6"/><i><font color="Green">E373</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/>
<a NAME="E11:179_6"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 4
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:179_6"/><b>then </b><i><font color="Green">E374</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 8 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:179_6"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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 &amp; 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">ez</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:179_6"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">ez</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="chord.html#T92">CHORD:92</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:179_6_1"/><i><font color="Green">E376</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v4</font>,<font color="Maroon">v3</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:179_6_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:179_6_1"/><b>then </b><i><font color="Green">E378</font></i>: 
1 <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">P</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="chord.html#T92">CHORD:92</a></i>;<br/><b>thus </b><a NAME="E4:179_6_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E17:179_6"/><b>then </b><i><font color="Green">E379</font></i>: 
( <font color="Maroon">v4</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">v4</font>,<font color="Maroon">v3</font> <a href="chord.html#R1">are_adjacent</a>  )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T11" target="_self">Th11</a>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">v5</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E19:179_6"/><i><font color="Green">E380</font></i>: 
( <font color="Maroon">v5</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font> )
 <b>and </b><br/><a NAME="E20:179_6"/><i><font color="Green">E381</font></i>: 
( <font color="Maroon">v5</font>,<font color="Maroon">v3</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">v5</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E21:179_6"/><i><font color="Green">E382</font></i>: 
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v5</font> &amp; <font color="Olive">x</font>,<font color="Maroon">v3</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , , </i>;<br/>
<div><i><font color="Green">E383</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:179_6_2"/><i><font color="Green">E384</font></i>: 
<font color="Maroon">v5</font>,<font color="Maroon">v1</font> <a href="chord.html#R1">are_adjacent</a> 
 ;<br/><a NAME="E2:179_6_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:179_6_2"/><b>then </b><i><font color="Green">E385</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>consider </b><font color="Maroon">v6</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:179_6_2"/><i><font color="Green">E386</font></i>: 
( <font color="Maroon">v6</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v6</font> )
 <b>and </b><br/><a NAME="E6:179_6_2"/><i><font color="Green">E387</font></i>: 
( <font color="Maroon">v6</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">v6</font>,<font color="Maroon">v1</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E7:179_6_2"/>
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v6</font> &amp; <font color="Olive">x</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Maroon">v1</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v6</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, , , , , </i>;<br/><a NAME="E8:179_6_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v6</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, , </i>;<br/><b>hence </b><a NAME="E9:179_6_2"/>
contradiction
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:179_6"/><i><font color="Green">E388</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">v5</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, , <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">Qr</font> = <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> ;<br/>
<a NAME="E25:179_6"/><i><font color="Green">E389</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E26:179_6"/><i><font color="Green">E390</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<a NAME="E27:179_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E28:179_6"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E29:179_6"/><b>then </b><i><font color="Green">E391</font></i>: 
( <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <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#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, </i>;<br/>
<a NAME="E30:179_6"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E31:179_6"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E32:179_6"/><b>then </b><i><font color="Green">E392</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> 3 &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E33:179_6"/><i><font color="Green">E393</font></i>: 
( <font color="Maroon">v4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">v5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<a NAME="E34:179_6"/><i><font color="Green">E394</font></i>: 
not <font color="Maroon">v5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E35:179_6"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>,<font color="Maroon">v5</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/>
<a NAME="E36:179_6"/><b>then </b>
( <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="chord.html#V5">chordal</a> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, , <a class="ref" href="chord.html#T97">CHORD:97</a></i>;<br/>
<a NAME="E37:179_6"/><b>then </b><i><font color="Green">E395</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="glib_001.html#V5">Path-like</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="glib_001.html#V1">closed</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  is <a href="chord.html#V5">chordal</a> )
 
<b>by </b><i><a class="ref" href="chord.html#T91">CHORD:91</a>, <a class="ref" href="glib_001.html#T121">GLIB_001:121</a>, <a class="ref" href="glib_001.html#T149">GLIB_001:149</a></i>;<br/>
<a NAME="E38:179_6"/><i><font color="Green">E417</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E39:179_6"/><i><font color="Green">E418</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/>
<a NAME="E40:179_6"/><b>then </b>
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T25">GLIB_001:25</a></i>;<br/>
<a NAME="E41:179_6"/><b>then </b><i><font color="Green">E419</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Maroon">v4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">v5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K3">.first()</a>  )
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T23">GLIB_001:23</a></i>;<br/>
<div><i><font color="Green">E420</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:179_6_3"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:179_6_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T93">GLIB_001:93</a></i>;<br/><a NAME="E3:179_6_3"/><b>then </b><i><font color="Green">E422</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v5</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_6_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v5</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1_1"><b>suppose </b></a><a NAME="E1:179_6_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:179_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:179_6_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1_2"><b>suppose </b></a><a NAME="E1:179_6_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v5</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_6_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v5</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E43:179_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D14" target="_self">Def14</a>, , , , <a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:179"/><i><font color="Green">E423</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<a NAME="E50:179"/>
( 4 <a href="xxreal_0.html#R1">&lt;=</a> 11 &amp; 11 <a href="xxreal_0.html#R1">&lt;=</a> 11 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E51:179"/><b>then </b>
4 <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> 11
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E52:179"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<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> 11]
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">P</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>, <font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v3</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v4</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E54:179"/><i><font color="Green">E424</font></i>: 
( <font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> is <a href="chord.html#V5">chordal</a> )
 <b>and </b><br/><a NAME="E55:179"/><i><font color="Green">E425</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</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> 11</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>and </b><br/><a NAME="E56:179"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">v4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  )
 <b>and </b><br/><a NAME="E57:179"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v1</font> )
 <b>and </b><br/><a NAME="E58:179"/>
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v4</font> &amp; <font color="Olive">x</font>,<font color="Maroon">v2</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">x</font>,<font color="Maroon">v1</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v4</font>
 ;<br/>
<a NAME="E59:179"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</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> 10</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/>
<a NAME="E60:179"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 21</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 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>
 
<b>by </b><i><a class="ref" href="glib_001.html#D14">GLIB_001:def 14</a></i>;<br/>
<a NAME="E61:179"/><b>then </b>
<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> 11 <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/>
<b>hence </b><a NAME="E62:179"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


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