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<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="chord.html#V6" target="_self">chordal</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:120"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:120"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>assume </b><a NAME="E3:120"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/>



<b>assume </b><a NAME="E4:120"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/>

<b>assume </b><a NAME="E5:120"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> or  ex <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Olive">c</font>,<font color="Olive">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1"/>
( <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1"><b>suppose </b></a><a NAME="E1:120_1_1"/>
<font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:120_1_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or <font color="Maroon">a</font>,<font color="Olive">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/><a NAME="E3:120_1_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:120_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2"><b>suppose </b></a><a NAME="E1:120_1_2"/><i><font color="Green">E66</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">A</font> = <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>;<br/><b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> </span>}</span> ;<br/><a NAME="E3:120_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/><a NAME="E4:120_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> </span>}</span> 
 ;<br/><a NAME="E5:120_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E6:120_1_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="ref" href="chord.html#T21" target="_self">Th21</a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:120_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:120_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E4:120_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a></i>;<br/><b>thus </b><a NAME="E5:120_1_2_1"/>
<font color="Maroon">x</font> is <a href="ordinal1.html#V7">natural</a>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V5">natural-membered</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="membered.html#D5">MEMBERED:def 5</a></i>;<br/><a NAME="E9:120_1_2"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E11:120_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E12:120_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E14:120_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E15:120_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">c</font>,<font color="Maroon">y</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/><a NAME="E16:120_1_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , , </i>;<br/><b>consider </b><font color="Maroon">Ga</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font></span>)</span>;<br/><a NAME="E18:120_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>set </b><font color="Maroon">tVG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>;<br/><a NAME="E19:120_1_2"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">a1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:120_1_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E1:120_1_2"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E22:120_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:120_1_2_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:120_1_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:120_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E24:120_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">Ga</font> = <font color="Maroon">Ga</font> as   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">A</font> <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, , </i>;<br/><b>set </b><font color="Maroon">Ay</font> = <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>consider </b><font color="Maroon">Gay</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><a NAME="E27:120_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gay</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E28:120_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E29:120_1_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">ca</font> = <font color="Maroon">c</font>, <font color="Maroon">ya</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gay</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon">yaa</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E32:120_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">yaa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E33:120_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">yaa</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E34:120_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E35:120_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="chord.html#E38:120_1_2"><i><font color="Green">E112</font></i></a></i>;<br/><a NAME="E36:120_1_2"/><b>then </b>
<font color="Maroon">Gay</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">Wa</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Gay</font><b> such that </b><br/><a NAME="E38:120_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">Wa</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">ca</font>,<font color="Maroon">ya</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><a NAME="E39:120_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">Wa</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ca</font> &amp; <font color="Maroon">Wa</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ya</font> )
 <b>by </b><i><a class="txt" href="chord.html#E39:120_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">Gay</font><b> such that </b><br/><a NAME="E41:120_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">Wa</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">Wa</font> <a href="glib_001.html#K4">.last()</a> 
 <b>and </b><br/><a NAME="E42:120_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">P</font> is <a href="chord.html#V1" target="_self">minlength</a>
 <b>by </b><i/>;<br/><a NAME="E43:120_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ca</font>
 <b>by </b><i><a class="txt" href="chord.html#E42:120_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="chord.html#E41:120_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E44:120_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E42:120_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a>, <a class="txt" href="chord.html#E41:120_1_2"><i><font color="Green">E114</font></i></a></i>;<br/><a NAME="E45:120_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E46:120_1_2"/><b>then </b>
not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E47:120_1_2"/><b>then </b><i><font color="Green">E119</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> 3
 <b>by </b><i><a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_3"/>
 <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">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:120_1_2_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><a NAME="E3:120_1_2_3"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="txt" href="chord.html#E47:120_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E4:120_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E5:120_1_2_3"/><b>then </b>
<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> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">Gay</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E6:120_1_2_3"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 2 <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="chord.html#E48:120_1_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="E7:120_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><a NAME="E49:120_1_2"/>
5 <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#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#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="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E50:120_1_2"/><b>then </b>
0 <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><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <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> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>set </b><font color="Maroon">d</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/><a NAME="E52:120_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E53:120_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gay</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E54:120_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
( <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">A</font> or <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_4"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:120_1_2_4"/><b>then </b>
<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="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="chord.html#E48:120_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:120_1_2_4"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 &amp; 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 5 )
 ;<br/><b>hence </b><a NAME="E4:120_1_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E57:120_1_2"/>
<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="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">d</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>,<font color="Maroon">Gay</font>
 <b>by </b><i><a class="txt" href="chord.html#E48:120_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E58:120_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<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="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">d</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><a NAME="E59:120_1_2"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">y</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E60:120_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E61:120_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E62:120_1_2"/><i><font color="Green">E130</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><b>set </b><font color="Maroon">gds</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>;<br/><b>set </b><font color="Maroon">gcs</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_5"/><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:120_1_2_5"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E60:120_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:120_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="xboole_0.html#R2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E61:120_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="chord.html#E58:120_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:120_1_2_5"/><b>then </b><i><font color="Green">E199</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_1.html#T24">TREES_1:24</a></i>;<br/><a NAME="E5:120_1_2_5"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="chord.html#E52:120_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E62:120_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E65:120_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E66:120_1_2"/><i><font color="Green">E201</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E67:120_1_2"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E61:120_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E68:120_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E62:120_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="chord.html#E61:120_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E70:120_1_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E62:120_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="chord.html#E55:120_1_2"><i><font color="Green">E126</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon">Gs</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>reconsider </b><font color="Maroon">xs</font> = <font color="Maroon">x</font>, <font color="Maroon">ys</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gs</font> <b>by </b><i>, <a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E73:120_1_2"/>
<font color="Maroon">Gs</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, , , <a class="ref" href="chord.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E74:120_1_2"/><b>then </b>
<font color="Maroon">xs</font>,<font color="Maroon">ys</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">ej</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E76:120_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">ej</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">xs</font>,<font color="Maroon">ys</font>,<font color="Maroon">Gs</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E77:120_1_2"/>
<font color="Maroon">ej</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><a NAME="E78:120_1_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</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/><a NAME="E79:120_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E61:120_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E80:120_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E81:120_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="chord.html#E52:120_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, , <a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E82:120_1_2"/><b>then </b><i><font color="Green">E208</font></i>: 
not <font color="Maroon">d</font>,<font color="Maroon">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T29" target="_self">Th29</a>, <a class="ref" href="chord.html#T13" target="_self">Th13</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>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">G</font> );<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>() <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>;<br/><a NAME="E83:120_1_2"/><i><font color="Green">E209</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>()
 ;<br/><a NAME="E84:120_1_2"/><i><font color="Green">E210</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E85:120_1_2"/><b>then </b><i><font color="Green">E211</font></i>: 
1 <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/><a NAME="E86:120_1_2"/><i><font color="Green">E212</font></i>: 
not <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 is <a href="abian.html#V1">even</a>
 ;<br/><a NAME="E87:120_1_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">x</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green">E117</font></i></a></i>;<br/><a NAME="E88:120_1_2"/><b>then </b><i><font color="Green">E213</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E90:120_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E91:120_1_2"/><i><font color="Green">E215</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">i</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="ref" href="chord.html#T31" target="_self">Th31</a>, );<br/></i><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="chord.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E93:120_1_2"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T32" target="_self">Th32</a>, , <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E94:120_1_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="chord.html#T30" target="_self">Th30</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T32" target="_self">Th32</a></i>;<br/><div><i><font color="Green">E218</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_6"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>
 ;<br/><a NAME="E2:120_1_2_6"/><i><font color="Green">E220</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T32" target="_self">Th32</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:120_1_2_6"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T34" target="_self">Th34</a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E4:120_1_2_6"/>
<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> 1,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T35" target="_self">Th35</a>, <a class="ref" href="chord.html#T32" target="_self">Th32</a>, </i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">Q1</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>;<br/><a NAME="E96:120_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> is <a href="chord.html#V1" target="_self">minlength</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E97:120_1_2"/><i><font color="Green">E313</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E99:120_1_2"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><a NAME="E100:120_1_2"/><i><font color="Green">E315</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="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E101:120_1_2"/><i><font color="Green">E316</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E102:120_1_2"/><i><font color="Green">E317</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T30" target="_self">Th30</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E103:120_1_2"/><i><font color="Green">E318</font></i>: 
not <font color="Maroon">Q</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E104:120_1_2"/><b>then </b>
( <font color="Maroon">Q</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">Q</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a>, <a class="ref" href="glib_001.html#D26">GLIB_001:def 26</a></i>;<br/><a NAME="E105:120_1_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E106:120_1_2"/><b>then </b><i><font color="Green">E319</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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="E107:120_1_2"/><i><font color="Green">E320</font></i>: 
not <font color="Maroon">Q</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a>, </i>;<br/><a NAME="E108:120_1_2"/><i><font color="Green">E321</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><b>set </b><font color="Maroon">cc</font> = <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> ;<br/><b>set </b><font color="Maroon">dd</font> = <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> ;<br/><a NAME="E109:120_1_2"/><i><font color="Green">E322</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="chord.html#T28" target="_self">Th28</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E110:120_1_2"/>
<font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89">GLIB_001:89</a>, <a class="ref" href="chord.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E111:120_1_2"/><b>then </b><i><font color="Green">E323</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, </i>;<br/><a NAME="E112:120_1_2"/>
<font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89">GLIB_001:89</a>, <a class="ref" href="chord.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E113:120_1_2"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, </i>;<br/><a NAME="E114:120_1_2"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">y</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E54:120_1_2"><i><font color="Green">E125</font></i></a>, , <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E115:120_1_2"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T32" target="_self">Th32</a>, , <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><div><i><font color="Green">E327</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:120_1_2_7"/><i><font color="Green">E328</font></i>: 
( <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> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> )
 ;<br/><b>assume </b><a NAME="E2:120_1_2_7"/>
 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">i</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<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="E4:120_1_2_7"/><i><font color="Green">E329</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">i</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E5:120_1_2_7"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:120_1_2_7"/><b>then </b>
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#R1">&lt;=</a> <font color="Maroon">i</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="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</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">ki1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</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>  ;<br/><a NAME="E10:120_1_2_7"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">i</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">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E11:120_1_2_7"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E12:120_1_2_7"/><b>then </b><i><font color="Green">E364</font></i>: 
( <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <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> <font color="Maroon">ki1</font> &amp; <font color="Maroon">ki1</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="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E13:120_1_2_7"/><i><font color="Green">E365</font></i>: 
<font color="Maroon">ki1</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="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_7_1"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">ki1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:120_1_2_7_1"/>
<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> <font color="Maroon">k</font></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="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> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T32">XREAL_1:32</a></i>;<br/><b>hence </b><a NAME="E3:120_1_2_7_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, , </i>;<br/></div><b>end;</b></div><a NAME="E15:120_1_2_7"/><b>then </b>
<font color="Maroon">ki1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E16:120_1_2_7"/><b>then </b><i><font color="Green">E367</font></i>: 
<font color="Maroon">ki1</font> <a href="nat_1.html#K1">+</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/>;<br/><a NAME="E17:120_1_2_7"/>
<font color="Maroon">ki1</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E18:120_1_2_7"/><b>then </b><i><font color="Green">E368</font></i>: 
<font color="Maroon">ki1</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E19:120_1_2_7"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ki1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <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> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E20:120_1_2_7"/><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> <font color="Maroon">ki1</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>,<font color="Maroon">Gay</font>
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/><b>hence </b><a NAME="E21:120_1_2_7"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T45" target="_self">Th45</a>, , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E369</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:120_1_2_8"/><i><font color="Green">E370</font></i>: 
( <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> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><b>assume </b><a NAME="E2:120_1_2_8"/>
 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">i</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<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="E4:120_1_2_8"/><i><font color="Green">E371</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">i</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E5:120_1_2_8"/><i><font color="Green">E372</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_8"/><b>then </b>
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#R1">&lt;=</a> <font color="Maroon">i</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="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</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">ki1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</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>  ;<br/><a NAME="E9:120_1_2_8"/>
<font color="Maroon">i</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">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:120_1_2_8"/><b>then </b><i><font color="Green">E373</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:120_1_2_8"/><b>then </b><i><font color="Green">E374</font></i>: 
( <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <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> <font color="Maroon">ki1</font> &amp; <font color="Maroon">ki1</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="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E12:120_1_2_8"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/><a NAME="E13:120_1_2_8"/>
<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> 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> <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> 0</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E14:120_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ki1</font> &amp; <font color="Maroon">ki1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E15:120_1_2_8"/><b>then </b><i><font color="Green">E377</font></i>: 
( <font color="Maroon">ki1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">ki1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:120_1_2_8"/><i><font color="Green">E378</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ki1</font>,<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="120_1_2_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_8_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:120_1_2_8_1"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:120_1_2_8_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:120_1_2_8_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_8_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T33" target="_self">Th33</a>, <a class="ref" href="chord.html#T45" target="_self">Th45</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:120_1_2_8"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E379</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:120_1_2_9"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:120_1_2_9"/>
<font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T35" target="_self">Th35</a>, <a class="ref" href="glib_001.html#T95">GLIB_001:95</a></i>;<br/><a NAME="E3:120_1_2_9"/><b>then </b><i><font color="Green">E381</font></i>: 
<font color="Maroon">v</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/>;<br/><b>consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:120_1_2_9"/><i><font color="Green">E382</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">Q</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">v</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E6:120_1_2_9"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E7:120_1_2_9"/><b>then </b>
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#R1">&lt;=</a> <font color="Maroon">n</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="ref" href="xreal_1.html#T9">XREAL_1:9</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">kn1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</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>  ;<br/><a NAME="E10:120_1_2_9"/>
<font color="Maroon">n</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">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E11:120_1_2_9"/><b>then </b><i><font color="Green">E383</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E12:120_1_2_9"/><b>then </b><i><font color="Green">E384</font></i>: 
( <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <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> <font color="Maroon">kn1</font> &amp; <font color="Maroon">kn1</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="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E13:120_1_2_9"/><i><font color="Green">E385</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E14:120_1_2_9"/>
<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> 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> <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> 0</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:120_1_2_9"/><b>then </b><i><font color="Green">E386</font></i>: 
<font color="Maroon">kn1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T45" target="_self">Th45</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_9_1"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:120_1_2_9_1"/><i><font color="Green">E388</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:120_1_2_9_1"/><i><font color="Green">E390</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_9_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:120_1_2_9_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:120_1_2_9_1_1"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:120_1_2_9_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:120_1_2_9_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , <a class="ref" href="chord.html#D2" target="_self">Def2</a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:120_1_2_9"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E18:120_1_2_9"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">R</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E120:120_1_2"/><i><font color="Green">E391</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 7 &amp; <font color="Maroon">R</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 3 )
 <b>and </b><br/><a NAME="E121:120_1_2"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span>,<font color="Maroon">y</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span></span><a href="enumset1.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E122:120_1_2"/><i><font color="Green">E393</font></i>: 
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 7 <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  )
 <b>by </b><i>, <a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="chord.html#T40" target="_self">Th40</a>, , <a class="ref" href="chord.html#D2" target="_self">Def2</a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green">E118</font></i></a>, , , , <a class="ref" href="chord.html#T41" target="_self">Th41</a>, <a class="ref" href="chord.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E123:120_1_2"/><i><font color="Green">E394</font></i>: 
<font color="Maroon">R</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E124:120_1_2"/><i><font color="Green">E395</font></i>: 
<font color="Maroon">R</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/><a NAME="E125:120_1_2"/><i><font color="Green">E396</font></i>: 
not <font color="Maroon">R</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/><a NAME="E126:120_1_2"/><i><font color="Green">E400</font></i>: 
not <font color="Maroon">R</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><a NAME="E127:120_1_2"/><i><font color="Green">E402</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K14">.edges()</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:120_1_2_10"/>
not <font color="Maroon">Q</font> <a href="glib_001.html#K14">.edges()</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/>

<a NAME="E2:120_1_2_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</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="E4:120_1_2_10"/><i><font color="Green">E404</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:120_1_2_10"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:120_1_2_10"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:120_1_2_10"/><i><font color="Green">E410</font></i>: 
( 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> 7 &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/>
<a NAME="E9:120_1_2_10"/><i><font color="Green">E411</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 6 )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_10_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 6 )
 </span><b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_1"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:120_1_2_10_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E3:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">R</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>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_001.html#T106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_2"/><i><font color="Green">E413</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:120_1_2_10_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E3:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_001.html#T106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_3"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_3"/><i><font color="Green">E414</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 6
 ;<br/><div class="add"><a NAME="E2:120_1_2_10_1_3"/>
<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 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E3:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 5,<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_001.html#T106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:120_1_2_11"/><i><font color="Green">E415</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 ;<br/><a NAME="E2:120_1_2_11"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:120_1_2_11"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E4:120_1_2_11"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span>,<font color="Maroon">y</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:120_1_2_11"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  )
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_11"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="chord.html#T28" target="_self">Th28</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E129:120_1_2"/><b>then </b><i><font color="Green">E417</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:120_1_2_12"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_12_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a>  )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_12_1_1"/><i><font color="Green">E419</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> 
 ;<br/><div class="add"><a NAME="E2:120_1_2_12_1_1"/><b>then </b><i><font color="Green">E420</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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="E3:120_1_2_12_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E4:120_1_2_12_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_12_1_2"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> 
 ;<br/><div class="add"><a NAME="E2:120_1_2_12_1_2"/><b>then </b><i><font color="Green">E422</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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="E3:120_1_2_12_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E4:120_1_2_12_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_12_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E131:120_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E132:120_1_2"/><b>then </b><i><font color="Green">E423</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>set </b><font color="Maroon">C</font> = <font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font>;<br/><a NAME="E133:120_1_2"/><i><font color="Green">E424</font></i>: 
<font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T38" target="_self">Th38</a>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, , , , , </i>;<br/><a NAME="E134:120_1_2"/><i><font color="Green">E425</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">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></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="ref" href="glib_001.html#T29">GLIB_001:29</a></i>;<br/><a NAME="E135:120_1_2"/><i><font color="Green">E426</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 6</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E136:120_1_2"/><i><font color="Green">E428</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E137:120_1_2"/><i><font color="Green">E429</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E138:120_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i>, , </i>;<br/><a NAME="E139:120_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E140:120_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E141:120_1_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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="E143:120_1_2"/><i><font color="Green">E430</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>
 <b>and </b><br/><a NAME="E144:120_1_2"/><i><font color="Green">E431</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#K7">.append</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E145:120_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E146:120_1_2"/><i><font color="Green">E432</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"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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="E147:120_1_2"/><i><font color="Green">E433</font></i>: 
( <font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="chord.html#E22:120_1_2"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E148:120_1_2"/><i><font color="Green">E434</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E149:120_1_2"/><i><font color="Green">E435</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">Q</font> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_13"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1"/>
( <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> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_1"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_1"/>
( <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> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:120_1_2_13_1_2"/><b>then </b><i><font color="Green">E436</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1_2_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_2_1_1"/>
( <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> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:120_1_2_13_1_2_1_2"/><b>then </b><i><font color="Green">E459</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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1_2_1_2_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T52" target="_self">Th52</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="120_1_2_13_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_2_1_2_1_1"/>
( <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> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:120_1_2_13_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:120_1_2_13_1_2_1_2_1_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">Q</font> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E151:120_1_2"/><i><font color="Green">E460</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#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14"/>
( <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> or <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="nat_1.html#K1">+</a> 2 or <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="nat_1.html#K1">+</a> 4 or <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="nat_1.html#K1">+</a> 6 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_1"/><i><font color="Green">E461</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">Q</font>
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:120_1_2_14_1"/><b>then </b><i><font color="Green">E462</font></i>: 
<font color="Maroon">n</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="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_1"/>
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <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">m</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 )
 <b>by </b><i><a class="ref" href="chord.html#T48" target="_self">Th48</a>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:120_1_2_14_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:120_1_2_14_1"/><b>then </b><i><font color="Green">E463</font></i>: 
<font color="Maroon">m</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="ref" href="chord.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:120_1_2_14_1"/>
( <font color="Maroon">Q</font> <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#K7">.append</a> <font color="Maroon">R</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><b>hence </b><a NAME="E8:120_1_2_14_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="chord.html#T48" target="_self">Th48</a>, <a class="ref" href="chord.html#T34" target="_self">Th34</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_2"/><i><font color="Green">E464</font></i>: 
<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="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="chord.html#T48" target="_self">Th48</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:120_1_2_14_2"/><b>then </b><i><font color="Green">E465</font></i>: 
<font color="Maroon">m</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="ref" href="chord.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_2"/><b>then </b>
<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">y</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self">Th52</a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , <a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_14_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="chord.html#T48" target="_self">Th48</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3"><b>suppose </b></a><a NAME="E1:120_1_2_14_3"/><i><font color="Green">E466</font></i>: 
<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="nat_1.html#K1">+</a> 4
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_3"/><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#K2">+</a> 2 <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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="ref" href="chord.html#T48" target="_self">Th48</a>, </i>;<br/><a NAME="E3:120_1_2_14_3"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 4 <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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 ;<br/><a NAME="E4:120_1_2_14_3"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:120_1_2_14_3"/><b>then </b><i><font color="Green">E467</font></i>: 
<font color="Maroon">m</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="ref" href="chord.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_14_3"/><b>then </b><i><font color="Green">E468</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">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self">Th52</a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , <a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14_3_1"/>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T51" target="_self">Th51</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="120_1_2_14_3_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_3_1_1"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_14_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T38" target="_self">Th38</a>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, , , , , , <a class="ref" href="chord.html#T50" target="_self">Th50</a>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="chord.html#T47" target="_self">Th47</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_3_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_14_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T42" target="_self">Th42</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4"><b>suppose </b></a><a NAME="E1:120_1_2_14_4"/><i><font color="Green">E469</font></i>: 
<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="nat_1.html#K1">+</a> 6
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_4"/><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#K2">+</a> 2 <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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 6
 <b>by </b><i><a class="ref" href="chord.html#T48" target="_self">Th48</a>, </i>;<br/><a NAME="E3:120_1_2_14_4"/><b>then </b>
<font color="Maroon">m</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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 4
 ;<br/><a NAME="E4:120_1_2_14_4"/><b>then </b><i><font color="Green">E470</font></i>: 
<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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14_4_1"/>
( <font color="Maroon">m</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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">m</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="nat_1.html#K1">+</a> 2 )
 </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="120_1_2_14_4_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_4_1_1"/>
<font color="Maroon">m</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">Q</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_4_1_1"/><b>then </b><i><font color="Green">E471</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><a NAME="E3:120_1_2_14_4_1_1"/><b>then </b>
<font color="Maroon">m</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="ref" href="chord.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_4_1_1"/><b>then </b><i><font color="Green">E472</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">R</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> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><a NAME="E5:120_1_2_14_4_1_1"/><i><font color="Green">E473</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> 1,<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self">Th52</a>, , , , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_14_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_14_4_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="chord.html#T38" target="_self">Th38</a>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, , , , , , <a class="ref" href="chord.html#T50" target="_self">Th50</a>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="chord.html#T47" target="_self">Th47</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><a NAME="E4:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="chord.html#T38" target="_self">Th38</a>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, , , , , , <a class="ref" href="chord.html#T50" target="_self">Th50</a>, <a class="ref" href="chord.html#T52" target="_self">Th52</a>, <a class="ref" href="chord.html#T47" target="_self">Th47</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_14_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/></div><b>end;</b></div><a NAME="E7:120_1_2_14_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><b>hence </b><a NAME="E8:120_1_2_14_4_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T34" target="_self">Th34</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_4_1_2"/><i><font color="Green">E474</font></i>: 
<font color="Maroon">m</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="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:120_1_2_14_4_1_2"/><i><font color="Green">E475</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> 1,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self">Th52</a>, , , , , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E3:120_1_2_14_4_1_2"/>
( 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 <b>by </b><i><a class="ref" href="chord.html#T38" target="_self">Th38</a>, <a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E4:120_1_2_14_4_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:120_1_2_14_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Q</font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T30" target="_self">Th30</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T32" target="_self">Th32</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_14_4_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
