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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:119"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  holds <br/>for <font color="Olive">S</font> being   <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  st <font color="Olive">S</font> is <a href="chord.html#V4" target="_self">minimal</a> &amp; not <font color="Olive">S</font> is <a href="xboole_0.html#V1">empty</a> holds <br/>for <font color="Olive">G2</font> being  <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Olive">S</font> holds  <font color="Olive">G2</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D11" target="_self">CHORD:def 11</a><br/>

<b>assume </b><a NAME="E2:119"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; <font color="Maroon">P</font> is <a href="glib_001.html#V8">Cycle-like</a> )
 ;<br/>

<a NAME="E3:119"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E4:119"/>
<font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <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="E5:119"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:119"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <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#NR2" target="_self">&gt;=</a> 8 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:119"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9
 
<b>by </b><i><a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <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 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:119_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">P</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">n</font>
 ;<br/><a NAME="E2:119_1"/>
( <font color="Maroon">m</font> <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">P</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:119_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <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="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E4:119_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:119_2"/>
(  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">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font> or for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_2_1"><b>suppose </b></a><a NAME="E1:119_2_1"/><i><font color="Green">E67</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">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:119_2_1"/><i><font color="Green">E68</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">P</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:119_2_1"/><i><font color="Green">E69</font></i>: 
(  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">P</font> )
 <b>by </b><i/>;<br/><a NAME="E4:119_2_1"/>
<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#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:119_2_1"/><b>then </b>
(  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">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 )
 ;<br/><b>hence </b><a NAME="E6:119_2_1"/>
<font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_2_2"><b>suppose </b></a><a NAME="E1:119_2_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:119_2_2"/>
<font color="Maroon">m</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/><b>then reconsider </b><font color="Maroon">Pm</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E4:119_2_2"/>
<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>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">Pn</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E6:119_2_2"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">Pm</font>,<font color="Maroon">Pn</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><b>consider </b><font color="Maroon">S</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">Pm</font>,<font color="Maroon">Pn</font><b> such that </b><br/><a NAME="E8:119_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 <b>by </b><i><a class="txt" href="chord.html#E13:119_2_2"><i><font color="Green">E76</font></i></a></i>;<br/><b>set </b><font color="Maroon">P15</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">P5l</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>;<br/><a NAME="E9:119_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">m</font> <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>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:119_2_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E11:119_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , <a class="txt" href="chord.html#E1:119_2_2"><i><font color="Green">E70</font></i></a></i>;<br/><b>consider </b><font color="Maroon">G2</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><a NAME="E13:119_2_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">G2</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, , , , <a class="ref" href="chord.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E14:119_2_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:119_2_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">m</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#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E17:119_2_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="chord.html#T22" target="_self">Th22</a>, , </i>;<br/><a NAME="E18:119_2_2"/><i><font color="Green">E83</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 <a href="xcmplx_0.html#K6">-</a> 2 )
 <b>by </b><i>, , </i>;<br/><a NAME="E19:119_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E20:119_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</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">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, , <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E21:119_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E22:119_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E23:119_2_2"/><i><font color="Green">E112</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</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/><a NAME="E24:119_2_2"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<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>
 <b>by </b><i><a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E25:119_2_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T119">GLIB_001:119</a></i>;<br/><a NAME="E26:119_2_2"/>
<font color="Maroon">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">Pn</font>,<font color="Maroon">Pm</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">l</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E28:119_2_2"/><i><font color="Green">E114</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</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">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E29:119_2_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E23:119_2_2"><i><font color="Green">E112</font></i></a>, </i>;<br/><a NAME="E30:119_2_2"/><i><font color="Green">E117</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E25:119_2_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">l</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="txt" href="chord.html#E29:119_2_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">l2</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E33:119_2_2"/>
( <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 0 &amp; <font color="Maroon">l</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">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="txt" href="chord.html#E25:119_2_2"><i><font color="Green">E113</font></i></a></i>;<br/><a NAME="E34:119_2_2"/><b>then </b>
<font color="Maroon">l</font> <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">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E35:119_2_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l2</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">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><b>reconsider </b><font color="Maroon">aa</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">G2</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i>, <a class="txt" href="chord.html#E28:119_2_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E38:119_2_2"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 5</span>)</span> <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> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 5</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E39:119_2_2"/>
<font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</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> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E25:119_2_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E40:119_2_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">l2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</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> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 5</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E30:119_2_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="E41:119_2_2"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="txt" href="chord.html#E29:119_2_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E42:119_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:119_2_2_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font>
 ;<br/><a NAME="E2:119_2_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</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="txt" href="chord.html#E38:119_2_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="txt" href="chord.html#E35:119_2_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:119_2_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, , <a class="txt" href="chord.html#E40:119_2_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><a NAME="E44:119_2_2"/><b>then </b>
<font color="Maroon">aa</font>,<font color="Maroon">bb</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E46:119_2_2"/><i><font color="Green">E125</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> 3,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font></span>)</span>,<font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E47:119_2_2"/>
<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">k</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l2</font></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="txt" href="chord.html#E40:119_2_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><b>hence </b><a NAME="E48:119_2_2"/>
<font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="txt" href="chord.html#E38:119_2_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="chord.html#E35:119_2_2"><i><font color="Green">E118</font></i></a>, , <a class="txt" href="chord.html#E17:119_2_2"><i><font color="Green">E82</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
