<?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="E1:220"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> is <a href="lexbfs.html#V22" target="_self">with_property_T</a>
 <b>and </b><br/><a NAME="E2:220"/><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>set </b><font color="Maroon">VLG</font> =  <a href="glib_003.html#K7">the_VLabel_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="E3:220"/><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/>

<a NAME="E4:220"/><i><font color="Green">E39</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="E5:220"/><i><font color="Green">E40</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="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<a NAME="E6:220"/><i><font color="Green">E42</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="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T65">FUNCT_1:65</a></i>;<br/>
<a NAME="E7:220"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <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="ref" href="chord.html#T104">CHORD:104</a></i>;<br/>
<a NAME="E8:220"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:220"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K2">rng</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="finseq_1.html#K2">Seg</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="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 &amp; not <font color="Maroon">a<sub>1</sub></font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">a<sub>1</sub></font> is <a href="chord.html#V5">chordal</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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="glib_001.html#K3">.first()</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="glib_001.html#K4">.last()</a> </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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span> &amp;  ex <font color="Olive">i</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> 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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) ) );<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">R</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:220_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">R</font>]
 ;<br/><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#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</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> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:220_1"/><i><font color="Green">E49</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>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:220_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:220_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:220_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:220_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:220_1"/><i><font color="Green">E89</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>2</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/><b>consider </b><font color="Maroon">k</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="E5:220_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E6:220_1"/><i><font color="Green">E91</font></i>: 
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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, );<br/></i><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:220_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:220_1"/><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E11:220_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> 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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 <b>and </b><br/><a NAME="E12:220_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/><a NAME="E15:220_1"/>
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="E16:220_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K5">.vertexAt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D8">GLIB_001:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/><a NAME="E18:220_1"/><i><font color="Green">E96</font></i>: 
( <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="E20:220_1"/><i><font color="Green">E97</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="chord.html#T92">CHORD:92</a></i>;<br/><a NAME="E21:220_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_2"><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="E1:220_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:220_1_2"/>
<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="E3:220_1_2"/><b>then </b>
(  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <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> 1,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>,<font color="Maroon">G</font> iff 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="chord.html#T92">CHORD:92</a></i>;<br/><b>hence </b><a NAME="E4:220_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div><a NAME="E23:220_1"/><b>then </b><i><font color="Green">E101</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E25:220_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">d</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">d</font> )
 <b>and </b><br/><a NAME="E26:220_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a>  )
 <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#E20"><i><font color="Green">Lemma128</font></i></a>, <a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, , <a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a></i>;<br/><a NAME="E27:220_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</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="E28:220_1"/><i><font color="Green">E116</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">d</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><i><font color="Green">E117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:220_1_3"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">dn</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:220_1_3"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">dn</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">dn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:220_1_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">dn</font> <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="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:220_1_3"/>
( <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">dn</font> )
 <b>by </b><i>, <a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="chord.html#D3">CHORD:def 3</a>, <a class="ref" href="chord.html#T2">CHORD:2</a></i>;<br/><a NAME="E6:220_1_3"/><b>then </b>
<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> <font color="Maroon">dn</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:220_1_3"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">dn</font> &amp; <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 <b>by </b><i>, , , <a class="ref" href="chord.html#T4">CHORD:4</a></i>;<br/><a NAME="E8:220_1_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">dn</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_3_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">dn</font> or <font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_3_1_1"><b>suppose </b></a><a NAME="E1:220_1_3_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">dn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="lexbfs.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_3_1_2"><b>suppose </b></a><a NAME="E1:220_1_3_1_2"/>
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_3_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D14" target="_self">Def14</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E30:220_1"/><i><font color="Green">E122</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">d</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">c</font>
 <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>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <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> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">d</font>,<font color="Maroon">G</font> );<br/><b>consider </b><font color="Maroon">el</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E32:220_1"/><i><font color="Green">E327</font></i>: 
<font color="Maroon">el</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">d</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/><a NAME="E33:220_1"/><i><font color="Green">E330</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>3</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E34:220_1"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>] &amp; ( 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>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="lexbfs.html#D14" target="_self">Def14</a>);<br/></i><b>then consider </b><font color="Maroon">j</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="E36:220_1"/><i><font color="Green">E331</font></i>: 
( not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> &amp; 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <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> <font color="Maroon">j</font>,<font color="Maroon">d</font>,<font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E37:220_1"/><i><font color="Green">E332</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 <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font>
 ;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><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="E40:220_1"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">d</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E41:220_1"/><i><font color="Green">E335</font></i>: 
<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#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="chord.html#T2">CHORD:2</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> ;<br/><a NAME="E43:220_1"/><i><font color="Green">E336</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E44:220_1"/><i><font color="Green">E338</font></i>: 
( <font color="Maroon">C</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">C</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E45:220_1"/>
<font color="Maroon">C</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T95">GLIB_001:95</a></i>;<br/><a NAME="E46:220_1"/><b>then </b><i><font color="Green">E339</font></i>: 
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i/>;<br/><a NAME="E47:220_1"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">d</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a>, </i>;<br/><div><i><font color="Green">E342</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_4"/><i><font color="Green">E343</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:220_1_4"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T2">CHORD:2</a></i>;<br/><a NAME="E3:220_1_4"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self">Def16</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:220_1_4"/><b>then </b>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>hence </b><a NAME="E5:220_1_4"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><a NAME="E49:220_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> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E50:220_1"/><b>then </b><i><font color="Green">E344</font></i>: 
( not <font color="Maroon">C</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">C</font> is <a href="chord.html#V5">chordal</a> )
 <b>by </b><i>, , <a class="ref" href="chord.html#T93">CHORD:93</a></i>;<br/><a NAME="E51:220_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E52:220_1"/><b>then </b><i><font color="Green">E345</font></i>: 
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><i><font color="Green">E346</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:220_1_5"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</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">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:220_1_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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
 <b>by </b><i>, </i>;<br/><a NAME="E3:220_1_5"/><b>then </b><i><font color="Green">E348</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T4">CHORD:4</a></i>;<br/><a NAME="E4:220_1_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 5
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, , , <a class="ref" href="chord.html#D3">CHORD:def 3</a></i>;<br/><a NAME="E5:220_1_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:220_1_5"/><b>then </b><i><font color="Green">E349</font></i>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:220_1_5"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">cc</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E9:220_1_5"/><i><font color="Green">E351</font></i>: 
( 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">cc</font> &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E10:220_1_5"/><b>then </b><i><font color="Green">E352</font></i>: 
<font color="Maroon">cc</font> <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="E11:220_1_5"/>
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cc</font>,<font color="Maroon">d</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a>, , <a class="ref" href="lexbfs.html#T15" target="_self">Th15</a>, </i>;<br/><b>hence </b><a NAME="E12:220_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="lexbfs.html#T15" target="_self">Th15</a>, <a class="ref" href="lexbfs.html#T16" target="_self">Th16</a>, , </i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">C</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self">Def16</a>, , , <a class="ref" href="lexbfs.html#T14" target="_self">Th14</a>, <a class="ref" href="chord.html#T97">CHORD:97</a></i>;<br/><a NAME="E55:220_1"/><i><font color="Green">E353</font></i>: 
( not <font color="Maroon">D</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">D</font> is <a href="chord.html#V5">chordal</a> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self">Def16</a>, , , <a class="ref" href="lexbfs.html#T14" target="_self">Th14</a>, <a class="ref" href="chord.html#T97">CHORD:97</a></i>;<br/><a NAME="E56:220_1"/><i><font color="Green">E354</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E57:220_1"/><b>then </b><i><font color="Green">E355</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E58:220_1"/><i><font color="Green">E357</font></i>: 
( <font color="Maroon">D</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">D</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E59:220_1"/><i><font color="Green">E358</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 3 <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/><div><i><font color="Green">E359</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:220_1_6"/><i><font color="Green">E360</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:220_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T2">CHORD:2</a></i>;<br/><a NAME="E3:220_1_6"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D17" target="_self">Def17</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:220_1_6"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div><a NAME="E61:220_1"/><i><font color="Green">E361</font></i>: 
 ex <font color="Olive">i</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_7_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1_1"><b>suppose </b></a><a NAME="E1:220_1_7_1_1"/><i><font color="Green">E362</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">j</font>
;<br/><div><i><font color="Green">E363</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_7_1_1_1"/><i><font color="Green">E364</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E2:220_1_7_1_1_1"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_7_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3">CHORD:3</a></i>;<br/><a NAME="E4:220_1_7_1_1_1"/><b>then </b><i><font color="Green">E365</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E5:220_1_7_1_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E6:220_1_7_1_1_1"/><b>then </b><i><font color="Green">E366</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E7:220_1_7_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, , , <a class="ref" href="chord.html#T4">CHORD:4</a></i>;<br/><a NAME="E8:220_1_7_1_1_1"/><b>then </b><i><font color="Green">E367</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T15" target="_self">Th15</a>, , <a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E9:220_1_7_1_1_1"/>
( 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> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><a NAME="E10:220_1_7_1_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D18" target="_self">Def18</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E11:220_1_7_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#D22" target="_self">Def22</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_7_1_1_2"/><i><font color="Green">E368</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:220_1_7_1_1_2"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_7_1_1_2"/><b>then </b>
( <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:220_1_7_1_1_2"/><b>then </b><i><font color="Green">E369</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E5:220_1_7_1_1_2"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E6:220_1_7_1_1_2"/><b>then </b><i><font color="Green">E370</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E7:220_1_7_1_1_2"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, <a class="ref" href="lexbfs.html#D18" target="_self">Def18</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E8:220_1_7_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, , <a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:220_1_7_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a>, <a class="ref" href="lexbfs.html#D19" target="_self">Def19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1_2"><b>suppose </b></a><a NAME="E1:220_1_7_1_2"/><i><font color="Green">E371</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">i</font>
;<br/><a NAME="E2:220_1_7_1_2"/><i><font color="Green">E372</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><div><i><font color="Green">E373</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_7_1_2_1"/><i><font color="Green">E374</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E2:220_1_7_1_2_1"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_7_1_2_1"/><b>then </b><i><font color="Green">E375</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3">CHORD:3</a></i>;<br/><a NAME="E4:220_1_7_1_2_1"/><b>then </b><i><font color="Green">E376</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E5:220_1_7_1_2_1"/><b>then </b><i><font color="Green">E378</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E6:220_1_7_1_2_1"/><i><font color="Green">E379</font></i>: 
<font color="Maroon">e</font> <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="lexbfs.html#D21" target="_self">Def21</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_7_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> or <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 </span><b>by </b><i>, <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="220_1_7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:220_1_7_1_2_1_1_1"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_7_1_2_1_1_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> or <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 </span><b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self">Def24</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="220_1_7_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:220_1_7_1_2_1_1_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_7_1_2_1_1_1_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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="lexbfs.html#D22" target="_self">Def22</a>, <a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_7_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:220_1_7_1_2_1_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_7_1_2_1_1_1_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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="lexbfs.html#D23" target="_self">Def23</a>, , <a class="ref" href="lexbfs.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:220_1_7_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D25" target="_self">Def25</a>, <a class="ref" href="lexbfs.html#T15" target="_self">Th15</a>, , <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="220_1_7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:220_1_7_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:220_1_7_1_2_1_1_2"/><b>then </b><i><font color="Green">E381</font></i>: 
<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, , <a class="ref" href="chord.html#T3">CHORD:3</a></i>;<br/><a NAME="E3:220_1_7_1_2_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:220_1_7_1_2_1_1_2"/><b>then </b><i><font color="Green">E382</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D25" target="_self">Def25</a>, <a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E5:220_1_7_1_2_1_1_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <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="lexbfs.html#D25" target="_self">Def25</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:220_1_7_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D23" target="_self">Def23</a>, <a class="ref" href="lexbfs.html#D26" target="_self">Def26</a>, <a class="ref" href="lexbfs.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_7_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_7_1_2_2"/><i><font color="Green">E383</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:220_1_7_1_2_2"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_7_1_2_2"/><b>then </b>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="lexbfs.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E4:220_1_7_1_2_2"/><b>then </b>
( <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E5:220_1_7_1_2_2"/><b>then </b><i><font color="Green">E384</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E6:220_1_7_1_2_2"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D18" target="_self">Def18</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:220_1_7_1_2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:220_1_7_1_2_2"/><b>then </b>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E9:220_1_7_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:220_1_7_1_2"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="lexbfs.html#D19" target="_self">Def19</a>, , <a class="ref" href="lexbfs.html#D18" target="_self">Def18</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="lexbfs.html#D20" target="_self">Def20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E62:220_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">c</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">d</font>
 <b>by </b><i>, , , , <a class="ref" href="lexbfs.html#T15" target="_self">Th15</a>, , <a class="ref" href="lexbfs.html#T16" target="_self">Th16</a>, , <a class="ref" href="lexbfs.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E63:220_1"/><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">c</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">d</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">D</font> <a href="glib_001.html#K6">.reverse()</a>  as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> ;<br/><a NAME="E65:220_1"/><i><font color="Green">E386</font></i>: 
( <font color="Maroon">R</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <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="lexbfs.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_001.html#T23">GLIB_001:23</a></i>;<br/><a NAME="E66:220_1"/><b>then </b><i><font color="Green">E387</font></i>: 
( not <font color="Maroon">R</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">R</font> 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#D24">GLIB_001:def 24</a></i>;<br/><a NAME="E67:220_1"/><i><font color="Green">E388</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/><a NAME="E68:220_1"/><b>then </b>
3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E69:220_1"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E70:220_1"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T26">GLIB_001:26</a></i>;<br/><a NAME="E71:220_1"/><b>then </b><i><font color="Green">E389</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T17" target="_self">Th17</a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_8_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_8_1_1"><b>suppose </b></a><a NAME="E1:220_1_8_1_1"/><i><font color="Green">E390</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_8_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_1_8_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 </span><b>by </b><i>, <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="220_1_8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:220_1_8_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_8_1_1_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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="220_1_8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:220_1_8_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:220_1_8_1_1_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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="lexbfs.html#D22" target="_self">Def22</a>, <a class="ref" href="lexbfs.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:220_1_8_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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>, <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="220_1_8_1_2"><b>suppose </b></a><a NAME="E1:220_1_8_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:220_1_8_1_2"/><b>then </b>
( 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> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E3:220_1_8_1_2"/><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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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">P</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E4:220_1_8_1_2"/><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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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">b</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E5:220_1_8_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E73:220_1"/><b>then </b><i><font color="Green">E391</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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D21" target="_self">Def21</a>, <a class="ref" href="lexbfs.html#D15" target="_self">Def15</a>, <a class="ref" href="lexbfs.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E74:220_1"/><i><font color="Green">E392</font></i>: 
for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> holds <br/>( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:220_1_9"/><i><font color="Green">E393</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:220_1_9"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="chord.html#T2">CHORD:2</a></i>;<br/>
<a NAME="E3:220_1_9"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self">Def24</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:220_1_9"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T26">GLIB_001:26</a></i>;<br/>

<a NAME="E5:220_1_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D25" target="_self">Def25</a>, <a class="ref" href="glib_001.html#T26">GLIB_001:26</a></i>;<br/>
<b>hence </b><a NAME="E6:220_1_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E75:220_1"/>
 ex <font color="Olive">i</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> 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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_10"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:220_1_10"/><i><font color="Green">E395</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 <b>and </b><br/><a NAME="E3:220_1_10"/><i><font color="Green">E417</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:220_1_10"/><i><font color="Green">E418</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> 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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D17" target="_self">Def17</a></i>;<br/>
<b>set </b><font color="Maroon">ir</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<a NAME="E5:220_1_10"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self">Def24</a>, <a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/>
<a NAME="E6:220_1_10"/><b>then </b><i><font color="Green">E419</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E7:220_1_10"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self">Def24</a>, <a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/>
<a NAME="E8:220_1_10"/><b>then </b><i><font color="Green">E420</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:220_1_10"/><b>then </b><i><font color="Green">E421</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E50"><i><font color="Green">Lemma187</font></i></a>, <a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ir</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">ir</font>
;<br/>

<div><i><font color="Green">E422</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_10_1"/><i><font color="Green">E423</font></i>: 
( <font color="Maroon">ir</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 ;<br/><b>set </b><font color="Maroon">jr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/><b>set </b><font color="Maroon">kr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/><a NAME="E2:220_1_10_1"/><i><font color="Green">E424</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_10_1"/><b>then </b><i><font color="Green">E425</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><b>reconsider </b><font color="Maroon">jr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="lexbfs.html#D27" target="_self">Def27</a>, <a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E5:220_1_10_1"/><i><font color="Green">E474</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D23" target="_self">Def23</a>, </i>;<br/><b>reconsider </b><font color="Maroon">kr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E7:220_1_10_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, </i>;<br/><a NAME="E8:220_1_10_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:220_1_10_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kr</font> &amp; <font color="Maroon">kr</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">jr</font> &amp; <font color="Maroon">jr</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="chord.html#T2">CHORD:2</a>, , </i>;<br/><b>hence </b><a NAME="E10:220_1_10_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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D26" target="_self">Def26</a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ja</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_1_10_2"/><i><font color="Green">E476</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ja</font> &amp; <font color="Maroon">ja</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ir</font> )
 ;<br/><b>set </b><font color="Maroon">jr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ja</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/><b>set </b><font color="Maroon">kr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/><a NAME="E2:220_1_10_2"/><i><font color="Green">E477</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_1_10_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, </i>;<br/><a NAME="E4:220_1_10_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:220_1_10_2"/><b>then </b><i><font color="Green">E478</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ja</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">kr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="lexbfs.html#D27" target="_self">Def27</a>, <a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E7:220_1_10_2"/><i><font color="Green">E479</font></i>: 
<font color="Maroon">ja</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D27" target="_self">Def27</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:220_1_10_2"/><b>then </b><i><font color="Green">E480</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ja</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <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> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ja</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><b>reconsider </b><font color="Maroon">jr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ja</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="lexbfs.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E10:220_1_10_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ja</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/><a NAME="E11:220_1_10_2"/><b>then </b>
<font color="Maroon">jr</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:220_1_10_2"/><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> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kr</font></span>)</span> <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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">jr</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D25" target="_self">Def25</a></i>;<br/><b>hence </b><a NAME="E13:220_1_10_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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ja</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D27" target="_self">Def27</a>, <a class="ref" href="lexbfs.html#D23" target="_self">Def23</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:220_1_10"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ir</font> &amp; <font color="Maroon">ir</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">ir</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> 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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">j</font>, <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ir</font> 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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) )
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E50"><i><font color="Green">Lemma187</font></i></a>, <a class="ref" href="glib_001.html#T22">GLIB_001:22</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E76:220_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D19" target="_self">Def19</a>, <a class="ref" href="lexbfs.html#D20" target="_self">Def20</a>, , <a class="ref" href="lexbfs.html#D18" target="_self">Def18</a>, <a class="ref" href="lexbfs.html#D22" target="_self">Def22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:220"/><i><font color="Green">E481</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</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;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:220_2"/><i><font color="Green">E482</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a>  )
 ;<br/><b>let </b><font color="Maroon">va</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">vb</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">vc</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:220_2"/><i><font color="Green">E486</font></i>: 
( <font color="Maroon">va</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">vb</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">vc</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="E3:220_2"/><i><font color="Green">E487</font></i>: 
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">va</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">vb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">vc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vb</font> &amp; <font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vc</font> )
 ;<br/><b>assume </b><a NAME="E4:220_2"/><i><font color="Green">E488</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a> 
 ;<br/><a NAME="E5:220_2"/><i><font color="Green">E489</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/><a NAME="E6:220_2"/>
( <font color="Maroon">a</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; <font color="Maroon">b</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; <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> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:220_2"/><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">a</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">b</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">c</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="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:220_2"/><b>then </b><i><font color="Green">E490</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="hidden.html#R1">=</a> <font color="Maroon">va</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="hidden.html#R1">=</a> <font color="Maroon">vb</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">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vc</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="220_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:220_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">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> 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">c</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">b</font> )
 </span><b>by </b><i>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</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="220_2_1_1"><b>suppose </b></a><a NAME="E1:220_2_1_1"/><i><font color="Green">E491</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">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>
 ;<br/><div class="add"><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="E3:220_2_1_1"/><i><font color="Green">E492</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="E4:220_2_1_1"/>
( <font color="Maroon">e1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">b</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> )
 <b>and </b><br/><a NAME="E5:220_2_1_1"/>
<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">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E6:220_2_1_1"/><i><font color="Green">E493</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, , , <a class="ref" href="chord.html#T47">CHORD:47</a></i>;<br/><a NAME="E7:220_2_1_1"/><i><font color="Green">E494</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, </i>;<br/><div><i><font color="Green">E495</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_2_1_1_1"/><i><font color="Green">E496</font></i>: 
( 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:220_2_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 5
 <b>by </b><i><a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_2_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 6
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:220_2_1_1_1"/><b>then </b><i><font color="Green">E497</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 5 )
 <b>by </b><i><a class="ref" href="chord.html#T8">CHORD:8</a></i>;<br/><a NAME="E5:220_2_1_1_1"/>
( 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 6 )
 <b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:220_2_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="chord.html#T8">CHORD:8</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E498</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_2_1_1_2"/><i><font color="Green">E499</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 ;<br/><a NAME="E2:220_2_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_2_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:220_2_1_1_2"/><b>then </b><i><font color="Green">E500</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="chord.html#T7">CHORD:7</a></i>;<br/><a NAME="E5:220_2_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:220_2_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="chord.html#T7">CHORD:7</a></i>;<br/></div><b>end;</b></div><a NAME="E10:220_2_1_1"/>
not <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 is <a href="abian.html#V1">even</a>
 ;<br/><a NAME="E11:220_2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</font>]
 <b>by </b><i><a class="ref" href="lexbfs.html#T11" target="_self">Th11</a>, <a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , <a class="ref" href="chord.html#T90">CHORD:90</a>, </i>;<br/><b>hence </b><a NAME="E12:220_2_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_2_1_2"><b>suppose </b></a><a NAME="E1:220_2_1_2"/><i><font color="Green">E501</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#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">b</font>
 ;<br/><div class="add"><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="E3:220_2_1_2"/><i><font color="Green">E502</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="E4:220_2_1_2"/>
( <font color="Maroon">e1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">c</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> )
 <b>and </b><br/><a NAME="E5:220_2_1_2"/>
<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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E6:220_2_1_2"/><i><font color="Green">E503</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">a</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, , , <a class="ref" href="chord.html#T47">CHORD:47</a></i>;<br/><a NAME="E7:220_2_1_2"/><i><font color="Green">E504</font></i>: 
( <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">c</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, </i>;<br/><div><i><font color="Green">E505</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_2_1_2_1"/><i><font color="Green">E506</font></i>: 
( 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:220_2_1_2_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 5
 <b>by </b><i><a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_2_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 6
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:220_2_1_2_1"/><b>then </b><i><font color="Green">E507</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 5 )
 <b>by </b><i><a class="ref" href="chord.html#T8">CHORD:8</a></i>;<br/><a NAME="E5:220_2_1_2_1"/>
( 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 6 )
 <b>by </b><i><a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:220_2_1_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="chord.html#T8">CHORD:8</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E508</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="220_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:220_2_1_2_2"/><i><font color="Green">E509</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 ;<br/><a NAME="E2:220_2_1_2_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:220_2_1_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:220_2_1_2_2"/><b>then </b><i><font color="Green">E510</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="chord.html#T7">CHORD:7</a></i>;<br/><a NAME="E5:220_2_1_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 <b>by </b><i><a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:220_2_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, <a class="ref" href="chord.html#T7">CHORD:7</a></i>;<br/></div><b>end;</b></div><a NAME="E10:220_2_1_2"/>
not <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 is <a href="abian.html#V1">even</a>
 ;<br/><a NAME="E11:220_2_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</font>]
 <b>by </b><i><a class="ref" href="lexbfs.html#T11" target="_self">Th11</a>, <a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, <a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , <a class="ref" href="chord.html#T90">CHORD:90</a>, </i>;<br/><b>hence </b><a NAME="E12:220_2_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:220"/>
<font color="Maroon">V</font> is <a href="chord.html#V7">perfect</a>
 <b>by </b><i><a class="ref" href="chord.html#T109">CHORD:109</a></i>;<br/>


</div>
