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

<b>let </b><font color="Maroon">G</font> be  <a href="chord.html#V6" target="_self">chordal</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:117"/><i><font color="Green">E30</font></i>: 
( not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:117"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>and </b><br/><a NAME="E3:117"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E4:117"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</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="chord.html#T19" target="_self">Th19</a>, , , <a class="ref" href="glib_001.html#T152">GLIB_001:152</a></i>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_1"/><i><font color="Green">E66</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">P</font>
 ;<br/><a NAME="E2:117_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:117_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:117_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</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="E7:117"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">Q</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">Q</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<a NAME="E8:117"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R2">in</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#T89">GLIB_001:89</a></i>;<br/>
<a NAME="E9:117"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( 4 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font> &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies for <font color="Olive">j</font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font> );<br/>
<a NAME="E10:117"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st ( for <font color="Olive">a</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:117_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">a</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>]
 ;<br/>

<b>assume </b><a NAME="E2:117_2"/><i><font color="Green">E72</font></i>: 
( 4 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> &amp; 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon">j</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:117_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:117_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E5:117_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:117_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:117_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6">-</a> 4 <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 4
 
<b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:117_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <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#T45">XREAL_1:45</a></i>;<br/>
<a NAME="E9:117_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<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>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 or <font color="Maroon">j</font> <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">P</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> )
 </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="117_2_1_1"><b>suppose </b></a><a NAME="E1:117_2_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 ;<br/><div class="add"><a NAME="E2:117_2_1_1"/>
<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="chord.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E3:117_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_2_1_2"><b>suppose </b></a><a NAME="E1:117_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">j</font> <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">P</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/><div class="add"><a NAME="E2:117_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:117_2_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">j</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">jj</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><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">lP2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 as  <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>set </b><font color="Maroon">B</font> = <font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font>;<br/><a NAME="E6:117_2_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <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="E7:117_2_1_2"/><i><font color="Green">E86</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/><a NAME="E8:117_2_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E9:117_2_1_2"/><i><font color="Green">E113</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E10:117_2_1_2"/><i><font color="Green">E114</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E11:117_2_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#K5">.vertexAt</a> <font color="Maroon">j</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="glib_001.html#D8">GLIB_001:def 8</a></i>;<br/><a NAME="E12:117_2_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T90">GLIB_001:90</a></i>;<br/><a NAME="E13:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2"><i><font color="Green">E113</font></i></a>, , </i>;<br/><a NAME="E14:117_2_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
not <font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E10:117_2_1_2"><i><font color="Green">E114</font></i></a>, , <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><div><i><font color="Green">E117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_2_1_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E2:117_2_1_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:117_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E4:117_2_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:117_2_1_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , <a class="txt" href="chord.html#E1:117_2_1_2_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E6:117_2_1_2_1"/><b>then </b><i><font color="Green">E122</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:117_2_1_2_1"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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> <span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:117_2_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:117_2_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="txt" href="chord.html#E2:117_2_1_2_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font>;<br/><a NAME="E16:117_2_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="txt" href="chord.html#E10:117_2_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_2"><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="E1:117_2_1_2_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:117_2_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div><a NAME="E18:117_2_1_2"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E19:117_2_1_2"/><i><font color="Green">E127</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E20:117_2_1_2"/><i><font color="Green">E128</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E21:117_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E22:117_2_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 7
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3
 ;<br/><a NAME="E2:117_2_1_2_3"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:117_2_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/></div><b>end;</b></div><a NAME="E24:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:117_2_1_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E27:117_2_1_2"/><i><font color="Green">E198</font></i>: 
 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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E28:117_2_1_2"/><i><font color="Green">E199</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V8">Cycle-like</a> implies ( (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2"><i><font color="Green">E81</font></i></a></i>;<br/><b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E30:117_2_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E17:117_2_1_2"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E31:117_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E32:117_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</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>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m1</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E35:117_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E36:117_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</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>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E39:117_2_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E40:117_2_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E41:117_2_1_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E42:117_2_1_2"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E43:117_2_1_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</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> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/><a NAME="E44:117_2_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <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="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E45:117_2_1_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E22:117_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E46:117_2_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E47:117_2_1_2"/><b>then </b><i><font color="Green">E205</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</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"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_4"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_4"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span>,<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="txt" href="chord.html#E18:117_2_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green">E130</font></i></a>, , <a class="txt" href="chord.html#E22:117_2_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E3:117_2_1_2_4"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:117_2_1_2_4"/><b>then </b><i><font color="Green">E207</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_4_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 ;<br/><a NAME="E2:117_2_1_2_4_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:117_2_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T29" target="_self">Th29</a>, , <a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><a NAME="E6:117_2_1_2_4"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:117_2_1_2_4"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:117_2_1_2_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E9:117_2_1_2_4"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E10:117_2_1_2_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><a NAME="E49:117_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <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="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E50:117_2_1_2"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">n</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> <span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_5"/><i><font color="Green">E212</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E10:117_2_1_2"><i><font color="Green">E114</font></i></a>, </i>;<br/><a NAME="E3:117_2_1_2_5"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , , , <a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:117_2_1_2_5"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:117_2_1_2_5"/><b>then </b><i><font color="Green">E213</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:117_2_1_2_5"/>
( <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i><a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E7:117_2_1_2_5"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <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">P</font></span>)</span> <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/><b>then consider </b><font color="Maroon">kk</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:117_2_1_2_5"/><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_1"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:117_2_1_2_5_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:117_2_1_2_5_1"/><b>then </b>
( 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> 0 or 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green">E123</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E216</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_2"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:117_2_1_2_5_2"/><b>then </b><i><font color="Green">E217</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:117_2_1_2_5_2"/>
( <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E218</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_3"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:117_2_1_2_5_3"/><b>then </b><i><font color="Green">E219</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:117_2_1_2_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E13:117_2_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E22:117_2_1_2"><i><font color="Green">E129</font></i></a>, </i>;<br/><a NAME="E14:117_2_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T28" target="_self">Th28</a>, </i>;<br/><b>hence </b><a NAME="E15:117_2_1_2_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, , <a class="ref" href="chord.html#T30" target="_self">Th30</a>, <a class="ref" href="chord.html#T31" target="_self">Th31</a>, , <a class="txt" href="chord.html#E18:117_2_1_2"><i><font color="Green">E126</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E52:117_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T29" target="_self">Th29</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E53:117_2_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E54:117_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E55:117_2_1_2"/><b>then </b><i><font color="Green">E220</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <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"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E56:117_2_1_2"/><i><font color="Green">E221</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">jj</font>,<font color="Maroon">lP2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E22:117_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="chord.html#T29" target="_self">Th29</a>, </i>;<br/><a NAME="E57:117_2_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:117_2_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E58:117_2_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="chord.html#E20:117_2_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E59:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 ;<br/><b>hence </b><a NAME="E60:117_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , , <a class="txt" href="chord.html#E18:117_2_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:117"/><i><font color="Green">E313</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 ( for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3"><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="E1:117_3"/><i><font color="Green">E314</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>]
 ;<br/>

<a NAME="E2:117_3"/>
for <font color="Olive">a</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:117_3_1"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:117_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:117_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>]
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:117_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:117"/><i><font color="Green">E316</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="ref" href="chord.html#T21" target="_self">Th21</a>);<br/></i>
<div><i><font color="Green">E317</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_4"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">n</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">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 ;<br/><a NAME="E2:117_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <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">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T33">XREAL_1:33</a></i>;<br/><a NAME="E3:117_4"/><b>then </b>
<font color="Maroon">n</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">P</font></span>)</span> <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="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:117_4"/><i><font color="Green">E319</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><a NAME="E6:117_4"/><i><font color="Green">E320</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green">E321</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_4_1"/><i><font color="Green">E322</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 ;<br/><a NAME="E2:117_4_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 4 <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">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 4
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:117_4_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_4_2"/><i><font color="Green">E323</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:117_4_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:117_4_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:117_4_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:117_4"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="chord.html#T22" target="_self">Th22</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_5"/>
<font color="Maroon">Q</font> is <a href="chord.html#V5" target="_self">chordal</a>
 ;<br/><b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:117_5"/><i><font color="Green">E324</font></i>: 
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E4:117_5"/><i><font color="Green">E325</font></i>: 
 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">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E5:117_5"/>
( <font color="Maroon">Q</font> is <a href="glib_001.html#V8">Cycle-like</a> implies ( (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> ) ) )
 <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="E7:117_5"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E8:117_5"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:117_5"/><b>then </b><i><font color="Green">E327</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:117_5"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/><a NAME="E11:117_5"/><b>then </b><i><font color="Green">E328</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_5_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</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="117_5_1_1"><b>suppose </b></a><a NAME="E1:117_5_1_1"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:117_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:117_5_1_1"/><b>then </b>
<font color="Maroon">m</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">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:117_5_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, , <a class="ref" href="chord.html#T25" target="_self">Th25</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_5_1_2"><b>suppose </b></a><a NAME="E1:117_5_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:117_5_1_2"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">n</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">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E3:117_5_1_2"/><b>then </b>
( <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">Q</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> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:117_5_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:117"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="chord.html#V5" target="_self">chordal</a> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>


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