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

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

<b>set </b><font color="Maroon">tVG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>;<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="E1:118"/><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="E2:118"/><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><b>that </b><br/><a NAME="E3:118"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 <b>and </b><br/><a NAME="E4:118"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">Gs</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/>

<b>consider </b><font color="Maroon">Gns</font> being   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/>
<b>reconsider </b><font color="Maroon">sa</font> = <font color="Maroon">a</font>, <font color="Maroon">sb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gns</font> <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>;<br/>
<a NAME="E7:118"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="txt" href="chord.html#E9:118"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E8:118"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="txt" href="chord.html#E19:118"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E9:118"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</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">Gns</font>,<span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Gb</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">Gns</font>,<span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span>;<br/>
<a NAME="E12:118"/><i><font color="Green">E68</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">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E13:118"/><i><font color="Green">E69</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E14:118"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</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">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>, <font color="Maroon">B</font> = <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</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="E16:118"/>
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#E8:118"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E17:118"/><b>then </b><i><font color="Green">E70</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="118_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="E1:118_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:118_1"/>
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="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:118_1"/>
<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="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:118"/><b>then </b><i><font color="Green">E72</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_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="E1:118_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:118_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="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:118_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="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:118"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">B</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#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Gb</font> = <font color="Maroon">Gb</font> as   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gs</font>, <font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D6" target="_self">CHORD:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E24:118"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E25:118"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">xg</font> = <font color="Maroon">x</font>, <font color="Maroon">yg</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/>
<a NAME="E27:118"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gs</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E28:118"/><i><font color="Green">E81</font></i>: 
not <font color="Maroon">xg</font>,<font color="Maroon">yg</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 
<b>by </b><i>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/>
<b>set </b><font color="Maroon">xy</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>;<br/>
<b>set </b><font color="Maroon">Bx</font> = <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xg</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">Ax</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">xg</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>consider </b><font color="Maroon">Gax</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">xg</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E30:118"/><i><font color="Green">E82</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gax</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">xg</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/>
<b>consider </b><font color="Maroon">Ga1</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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/>
<i><font color="Green">E83</font></i>: <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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> = 
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>
<a NAME="E33:118"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E34:118"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E35:118"/><b>then </b>
( <font color="Maroon">x</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#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</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#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x</font>, <font color="Maroon">ya</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Ga1</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">xag</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E38:118"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">xag</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E39:118"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">xg</font>,<font color="Maroon">xag</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E40:118"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E41:118"/><b>then </b>
<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">(<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#E42:118"><i><font color="Green">E112</font></i></a></i>;<br/>
<a NAME="E42:118"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">Gax</font> is <a href="glib_002.html#V1">connected</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">yag</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E44:118"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">yag</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E45:118"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">yg</font>,<font color="Maroon">yag</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E46:118"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">yag</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">xg</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E45:118"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E47:118"/>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E48:118"/><b>then </b>
not <font color="Maroon">yg</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gax</font>
 
<b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E49:118"/><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">Gax</font></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, , <a class="txt" href="chord.html#E46:118"><i><font color="Green">E115</font></i></a></i>;<br/>
<a NAME="E50:118"/><b>then </b>
<font color="Maroon">Ga1</font> is <a href="glib_002.html#V1">connected</a>
 
<b>by </b><i><a class="txt" href="chord.html#E44:118"><i><font color="Green">E113</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">Wa</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Ga1</font><b> such that </b><br/><a NAME="E52:118"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">Wa</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">xa</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="E53:118"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">Wa</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">xa</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#E52:118"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<b>consider </b><font color="Maroon">Pa</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">Ga1</font><b> such that </b><br/><a NAME="E55:118"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">Pa</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="E56:118"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">Pa</font> is <a href="chord.html#V1" target="_self">minlength</a>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Gbx</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">B</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xg</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E58:118"/><i><font color="Green">E122</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gbx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xg</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/>
<b>consider </b><font color="Maroon">Gb1</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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/>
<i><font color="Green">E123</font></i>: <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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> = 
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>
<a NAME="E61:118"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gb1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E62:118"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E63:118"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xb</font> = <font color="Maroon">x</font>, <font color="Maroon">yb</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gb1</font> <b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green">E122</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon">xbg</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E66:118"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">xbg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E67:118"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">xg</font>,<font color="Maroon">xbg</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E68:118"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E69:118"/><b>then </b>
<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">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="txt" href="chord.html#E61:118"><i><font color="Green">E124</font></i></a>, <a class="txt" href="chord.html#E66:118"><i><font color="Green">E125</font></i></a></i>;<br/>
<a NAME="E70:118"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">Gbx</font> is <a href="glib_002.html#V1">connected</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">ybg</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E72:118"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">ybg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E73:118"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">yg</font>,<font color="Maroon">ybg</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E74:118"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">ybg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xg</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E70:118"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E75:118"/>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <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">x</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#D4">XBOOLE_0:def 4</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E76:118"/><b>then </b>
not <font color="Maroon">yg</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gbx</font>
 
<b>by </b><i><a class="txt" href="chord.html#E56:118"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E77:118"/><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">Gbx</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E56:118"><i><font color="Green">E121</font></i></a>, <a class="txt" href="chord.html#E73:118"><i><font color="Green">E129</font></i></a>, <a class="txt" href="chord.html#E72:118"><i><font color="Green">E128</font></i></a></i>;<br/>
<a NAME="E78:118"/><b>then </b>
<font color="Maroon">Gb1</font> is <a href="glib_002.html#V1">connected</a>
 
<b>by </b><i><a class="txt" href="chord.html#E67:118"><i><font color="Green">E126</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">Wb</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Gb1</font><b> such that </b><br/><a NAME="E80:118"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">Wb</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">yb</font>,<font color="Maroon">xb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/>
<a NAME="E81:118"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">Wb</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ya</font> &amp; <font color="Maroon">Wb</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E74:118"><i><font color="Green">E130</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<b>consider </b><font color="Maroon">Pb</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">Gb1</font><b> such that </b><br/><a NAME="E83:118"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">Pb</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">Wb</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">Wb</font> <a href="glib_001.html#K4">.last()</a> 
 <b>and </b><br/><a NAME="E84:118"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">Pb</font> is <a href="chord.html#V1" target="_self">minlength</a>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Pag</font> = <font color="Maroon">Pa</font>, <font color="Maroon">Pbg</font> = <font color="Maroon">Pb</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="E86:118"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">Pag</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Pag</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</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">Pag</font></span>)</span> )
 
;<br/>
<a NAME="E87:118"/>
( <font color="Maroon">Pa</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pa</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Pa</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pa</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">Pa</font></span>)</span> )
 
;<br/>
<a NAME="E88:118"/><b>then </b><i><font color="Green">E203</font></i>: 
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xg</font> &amp; <font color="Maroon">Pag</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">Pag</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yg</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E53:118"><i><font color="Green">E118</font></i></a>, <a class="txt" href="chord.html#E55:118"><i><font color="Green">E119</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E89:118"/><i><font color="Green">E204</font></i>: 
not <font color="Maroon">Pag</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="E90:118"/><b>then </b><i><font color="Green">E205</font></i>: 
not <font color="Maroon">Pag</font> is <a href="glib_001.html#V8">Cycle-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E91:118"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pag</font> <a href="glib_001.html#K13">.vertices()</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pag</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="E92:118"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">Pbg</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Pbg</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</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">Pbg</font></span>)</span> )
 
;<br/>
<a NAME="E93:118"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">Pb</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Pb</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</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">Pb</font></span>)</span> )
 
;<br/>
<a NAME="E94:118"/><b>then </b><i><font color="Green">E209</font></i>: 
( <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">yg</font> &amp; <font color="Maroon">Pbg</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">Pbg</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xg</font> )
 
<b>by </b><i><a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="chord.html#T29" target="_self">Th29</a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E95:118"/><i><font color="Green">E210</font></i>: 
not <font color="Maroon">Pbg</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="E96:118"/><b>then </b><i><font color="Green">E211</font></i>: 
not <font color="Maroon">Pbg</font> is <a href="glib_001.html#V8">Cycle-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E97:118"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pbg</font> <a href="glib_001.html#K13">.vertices()</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pbg</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/>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font>;<br/>
<a NAME="E98:118"/><i><font color="Green">E213</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="glib_001.html#T29">GLIB_001:29</a></i>;<br/>
<a NAME="E99:118"/><i><font color="Green">E214</font></i>: 
not <font color="Maroon">Pag</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="E100:118"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">Pag</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, , , , <a class="ref" href="chord.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E101:118"/><i><font color="Green">E216</font></i>: 
not <font color="Maroon">Pbg</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="E102:118"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">Pbg</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, , , , <a class="ref" href="chord.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E103:118"/><i><font color="Green">E218</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</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"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:118_5"/>
<span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</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"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:118_5_1"/>
<font color="Maroon">Pag</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pa</font> <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="E2:118_5_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">Pag</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:118_5_1"/>
<font color="Maroon">Pbg</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</font> <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="E4:118_5_1"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">Pbg</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E60:118"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<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#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:118_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</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"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:118_5"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></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">Pag</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">Pbg</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:118_5_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E2:118_5_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:118_5_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T31" target="_self">Th31</a>, <a class="ref" href="chord.html#T33" target="_self">Th33</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>
<a NAME="E104:118"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">Pag</font> <a href="glib_001.html#K14">.edges()</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Pbg</font> <a href="glib_001.html#K14">.edges()</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:118_6"/>
<span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</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> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><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="E3:118_6"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</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">Pbg</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="E4:118_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pag</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:118_6"/><b>then </b><i><font color="Green">E313</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pa</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/>
<a NAME="E6:118_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pbg</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:118_6"/><b>then </b><i><font color="Green">E314</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Ga1</font>, <font color="Maroon">a2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Ga1</font>, <font color="Maroon">na</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="E9:118_6"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">na</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">Pag</font>
 <b>and </b><br/><a NAME="E10:118_6"/><i><font color="Green">E316</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">na</font>
 <b>and </b><br/><a NAME="E11:118_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">na</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E12:118_6"/><i><font color="Green">E317</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">na</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E13:118_6"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">Ga1</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gb1</font>, <font color="Maroon">b2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gb1</font>, <font color="Maroon">nb</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="E15:118_6"/>
<font color="Maroon">nb</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">Pbg</font>
 <b>and </b><br/><a NAME="E16:118_6"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nb</font>
 <b>and </b><br/><a NAME="E17:118_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nb</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E18:118_6"/>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nb</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E19:118_6"/><i><font color="Green">E319</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">Gb1</font>
 <b>by </b><i><a class="ref" href="chord.html#T38" target="_self">Th38</a>, <a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/>
<a NAME="E20:118_6"/><i><font color="Green">E320</font></i>: 
( <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="chord.html#T40" target="_self">Th40</a>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/>
<a NAME="E21:118_6"/><b>then </b><i><font color="Green">E321</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> ) )
 
<b>by </b><i><a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/>
<a NAME="E22:118_6"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga1</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#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E23:118_6"/><b>then </b><i><font color="Green">E322</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) &amp; ( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E24:118_6"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gb1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green">E122</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E25:118_6"/><b>then </b><i><font color="Green">E323</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) &amp; ( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_6_1"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 </span><b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="chord.html#T41" target="_self">Th41</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_6_1_1"><b>suppose </b></a><a NAME="E1:118_6_1_1"/><i><font color="Green">E324</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 ;<br/><div class="add"><a NAME="E2:118_6_1_1"/>
<font color="Maroon">na</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">na</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T41">XREAL_1:41</a></i>;<br/><b>hence </b><a NAME="E3:118_6_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, , , , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_6_1_2"><b>suppose </b></a><a NAME="E1:118_6_1_2"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:118_6_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E105:118"/>
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Pbg</font> <a href="glib_001.html#K15">.length()</a> </span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E106:118"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="chord.html#D2" target="_self">Def2</a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E107:118"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E108:118"/><b>then </b><i><font color="Green">E325</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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="E109:118"/>
<font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font> is <a href="glib_001.html#V8">Cycle-like</a>
 
<b>by </b><i>, , <a class="ref" href="chord.html#T34" target="_self">Th34</a>, , , <a class="ref" href="chord.html#T35" target="_self">Th35</a>, , , , , </i>;<br/>
<a NAME="E110:118"/><b>then </b>
<font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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="E112:118"/><i><font color="Green">E326</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="E113:118"/><i><font color="Green">E327</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">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span>
 <b>and </b><br/><a NAME="E114:118"/><i><font color="Green">E328</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E115:118"/><i><font color="Green">E329</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">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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="E116:118"/><i><font color="Green">E330</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K14">.edges()</a>  holds <br/> not <font color="Olive">f</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E27:118"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E117:118"/><i><font color="Green">E363</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E118:118"/><i><font color="Green">E364</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E120:118"/><i><font color="Green">E365</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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/>
<a NAME="E121:118"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <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="glib_000.html#T17">GLIB_000:17</a></i>;<br/>
<a NAME="E122:118"/><i><font color="Green">E367</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="E123:118"/><i><font color="Green">E368</font></i>: 
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="hsNdiv(this)" href="javascript:()" title="118_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7"/>
( ( <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">Pag</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> ) or ( <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">Pag</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_1"><b>suppose </b></a><a NAME="E1:118_7_1"/><i><font color="Green">E369</font></i>: 
( <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">Pag</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font> )
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="chord.html#T41" target="_self">Th41</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:118_7_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <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/><a NAME="E4:118_7_1"/><b>then </b>
<font color="Maroon">Pag</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, , <a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="chord.html#T30" target="_self">Th30</a>, <a class="txt" href="chord.html#E30:118"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E5:118_7_1"/><b>then </b>
<font color="Maroon">Pa</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E33:118"><i><font color="Green">E84</font></i></a></i>;<br/><b>hence </b><a NAME="E6:118_7_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="txt" href="chord.html#E38:118"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2"><b>suppose </b></a><a NAME="E1:118_7_2"/><i><font color="Green">E370</font></i>: 
( <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">Pag</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font>
 <b>by </b><i><a class="ref" href="chord.html#T41" target="_self">Th41</a>, <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="E3:118_7_2"/><b>then </b><i><font color="Green">E371</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</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="E4:118_7_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> &amp; not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font> )
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, , <a class="ref" href="chord.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">n1</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="E6:118_7_2"/><i><font color="Green">E372</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">Pbg</font>
 <b>and </b><br/><a NAME="E7:118_7_2"/><i><font color="Green">E373</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E8:118_7_2"/><i><font color="Green">E374</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</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>
 <b>by </b><i>, , , , , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E9:118_7_2"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga1</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E10:118_7_2"/>
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">n11</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E13:118_7_2"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">n11</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:118_7_2"/><b>then </b><i><font color="Green">E378</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gb1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_2_1"/>
( ( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) or ( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> ) )
 </span><b>by </b><i>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, <a class="txt" href="chord.html#E60:118"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1"><b>suppose </b></a><a NAME="E1:118_7_2_1_1"/><i><font color="Green">E379</font></i>: 
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_2_1_1_1"/>
( <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</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">Pag</font></span>)</span> )
 </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="118_7_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_1"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_7_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_7_2_1_1_1_1_1"/><i><font color="Green">E381</font></i>: 
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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_2_1_1_1_1_1_1"/>
( 1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> or 1 <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</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="118_7_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_1_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, </i>;<br/><b>hence </b><a NAME="E3:118_7_2_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , <a class="ref" href="chord.html#T41" target="_self">Th41</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_1_1_1_2"/><i><font color="Green">E382</font></i>: 
1 <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>
 <b>by </b><i><a class="ref" href="chord.html#T41" target="_self">Th41</a>, <a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">Pag</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">Pag</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">Pag</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="jordan12.html#T3">JORDAN12:3</a>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E383</font></i>: 
<font color="Maroon">Pag</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">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E5:118_7_2_1_1_1_1_1_1_2"/><i><font color="Green">E384</font></i>: 
1 <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">Pag</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
1 <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">Pag</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:118_7_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E385</font></i>: 
<font color="Maroon">Pag</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><a NAME="E8:118_7_2_1_1_1_1_1_1_2"/><i><font color="Green">E386</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E9:118_7_2_1_1_1_1_1_1_2"/>
1 <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">Pag</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E10:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E11:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <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="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, , <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><b>hence </b><a NAME="E12:118_7_2_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:118_7_2_1_1_1_1"/><b>then </b>
<font color="Maroon">Pag</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, , <a class="ref" href="chord.html#T42" target="_self">Th42</a>, , <a class="ref" href="chord.html#T30" target="_self">Th30</a>, , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="txt" href="chord.html#E30:118"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E4:118_7_2_1_1_1_1"/><b>then </b>
<font color="Maroon">Pa</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E33:118"><i><font color="Green">E84</font></i></a></i>;<br/><b>hence </b><a NAME="E5:118_7_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="txt" href="chord.html#E38:118"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_2"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pag</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">Pag</font></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_7_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_7_2_1_1_1_2_1"/><i><font color="Green">E388</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>
 ;<br/><b>set </b><font color="Maroon">L</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_2_1_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> or <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</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="118_7_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_2_1_1_1"/><i><font color="Green">E390</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E391</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E3:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E392</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_2_1_1_1"/><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">Pag</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E393</font></i>: 
<font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">Pag</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">Pag</font></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E6:118_7_2_1_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E7:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font>
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/><a NAME="E9:118_7_2_1_1_1_2_1_1_1"/><i><font color="Green">E395</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:118_7_2_1_1_1_2_1_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">Pag</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><b>hence </b><a NAME="E13:118_7_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:118_7_2_1_1_1_2_1_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><a NAME="E3:118_7_2_1_1_1_2"/><b>then </b>
<font color="Maroon">Pag</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, , <a class="ref" href="chord.html#T30" target="_self">Th30</a>, , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="txt" href="chord.html#E30:118"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E4:118_7_2_1_1_1_2"/><b>then </b>
<font color="Maroon">Pa</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E33:118"><i><font color="Green">E84</font></i></a></i>;<br/><b>hence </b><a NAME="E5:118_7_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="txt" href="chord.html#E38:118"><i><font color="Green">E85</font></i></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="118_7_2_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_2"/><i><font color="Green">E396</font></i>: 
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ac</font> = <font color="Maroon">Pa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Ga</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:118_7_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Ga</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>reconsider </b><font color="Maroon">bc</font> = <font color="Maroon">Pb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gb</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:118_7_2_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gb</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">WA</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Ga</font><b> such that </b><br/><a NAME="E9:118_7_2_1_2"/><i><font color="Green">E400</font></i>: 
<font color="Maroon">WA</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">aa</font>,<font color="Maroon">ac</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>consider </b><font color="Maroon">WB</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Gb</font><b> such that </b><br/><a NAME="E11:118_7_2_1_2"/><i><font color="Green">E402</font></i>: 
<font color="Maroon">WB</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">bc</font>,<font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">WA</font> = <font color="Maroon">WA</font>, <font color="Maroon">WB</font> = <font color="Maroon">WB</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Gns</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><b>set </b><font color="Maroon">WAB</font> = <font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font>;<br/><a NAME="E13:118_7_2_1_2"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">ac</font>,<font color="Maroon">bc</font>,<font color="Maroon">Gns</font>
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, , , , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/><a NAME="E14:118_7_2_1_2"/><b>then </b><i><font color="Green">E404</font></i>: 
<font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">ac</font>,<font color="Maroon">bc</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><b>reconsider </b><font color="Maroon">WAs</font> = <font color="Maroon">WA</font>, <font color="Maroon">WBs</font> = <font color="Maroon">WB</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Gns</font> ;<br/><b>set </b><font color="Maroon">WaB</font> = <font color="Maroon">WAs</font> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font></span>)</span>;<br/><b>set </b><font color="Maroon">Wab</font> = <span class="p1">(<span class="default"><font color="Maroon">WAs</font> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font></span>)</span></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">WBs</font>;<br/><a NAME="E16:118_7_2_1_2"/>
<font color="Maroon">WAs</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">aa</font>,<font color="Maroon">ac</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E17:118_7_2_1_2"/><b>then </b><i><font color="Green">E409</font></i>: 
<font color="Maroon">WAs</font> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">aa</font>,<font color="Maroon">bc</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E18:118_7_2_1_2"/>
<font color="Maroon">WBs</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">bc</font>,<font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E19:118_7_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">WAs</font> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">ac</font>,<font color="Maroon">e</font>,<font color="Maroon">bc</font></span>)</span></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">WBs</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><b>hence </b><a NAME="E20:118_7_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="chord.html#E8:118"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_3"><b>suppose </b></a><a NAME="E1:118_7_2_1_3"/><i><font color="Green">E410</font></i>: 
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3"/>
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>then </b><i><font color="Green">E411</font></i>: <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, , <a class="ref" href="chord.html#T39" target="_self">Th39</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>
<br/>.= 
<font color="Maroon">Pbg</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">Pbg</font></span>)</span>
<b>by </b><i>, <a class="ref" href="chord.html#T28" target="_self">Th28</a>, <a class="ref" href="chord.html#T29" target="_self">Th29</a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_7_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_7_2_1_3_2"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 ;<br/><b>set </b><font color="Maroon">L</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_2_1_3_2_1"/>
( <font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font> or <font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</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="118_7_2_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:118_7_2_1_3_2_1_1"/><i><font color="Green">E413</font></i>: 
<font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green">E414</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon">n11</font> <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">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green">E415</font></i>: 
<font color="Maroon">n11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font>,<font color="Maroon">Pbg</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">Pbg</font></span>)</span>,<font color="Maroon">Gb1</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E6:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Pbg</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">Pbg</font></span>)</span>,<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font>,<font color="Maroon">Gb1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E7:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Pbg</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">Pbg</font></span>)</span>,<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</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="E8:118_7_2_1_3_2_1_1"/><i><font color="Green">E417</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n11</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="chord.html#T47" target="_self">Th47</a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E9:118_7_2_1_3_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n11</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E10:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green">E418</font></i>: 
1 <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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n11</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E11:118_7_2_1_3_2_1_1"/>
<font color="Maroon">n11</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">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:118_7_2_1_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><b>hence </b><a NAME="E14:118_7_2_1_3_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_3_2_1_2"/>
<font color="Maroon">n11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3_2_1_2"/><b>then </b>
<font color="Maroon">n11</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:118_7_2_1_3_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , , <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:118_7_2_1_3"/><b>then </b>
<font color="Maroon">Pbg</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T32" target="_self">Th32</a>, , , <a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="txt" href="chord.html#E30:118"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E6:118_7_2_1_3"/><b>then </b>
<font color="Maroon">Pb</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green">E122</font></i></a>, <a class="txt" href="chord.html#E33:118"><i><font color="Green">E84</font></i></a></i>;<br/><b>hence </b><a NAME="E7:118_7_2_1_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E38:118"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_2_1_4"><b>suppose </b></a><a NAME="E1:118_7_2_1_4"/>
( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">xg</font>,<font color="Maroon">yg</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_4"/><b>then </b>
( ( <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">Pag</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:118_7_2_1_4"/><b>then </b>
<font color="Maroon">xg</font>,<font color="Maroon">yg</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="chord.html#D3" target="_self">Def3</a>, , <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:118_7_2_1_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T6" target="_self">Th6</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="118_7_3"><b>suppose </b></a><a NAME="E1:118_7_3"/><i><font color="Green">E419</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:118_7_3"/><i><font color="Green">E420</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">Pag</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:118_7_3"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:118_7_3"/>
<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">Pag</font>
 <b>by </b><i><a class="ref" href="chord.html#T42" target="_self">Th42</a>, <a class="ref" href="chord.html#T40" target="_self">Th40</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:118_7_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> &amp; not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Pag</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">n1</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="E8:118_7_3"/><i><font color="Green">E422</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">Pbg</font>
 <b>and </b><br/><a NAME="E9:118_7_3"/><i><font color="Green">E423</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E10:118_7_3"/><i><font color="Green">E424</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</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>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, , , , , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E11:118_7_3"/><i><font color="Green">E425</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="chord.html#T40" target="_self">Th40</a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:118_7_3"/><b>then </b>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</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="E13:118_7_3"/><b>then </b><i><font color="Green">E426</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E14:118_7_3"/>
( <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> &amp; <font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="chord.html#T44" target="_self">Th44</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font>, <font color="Maroon">m1</font> = <font color="Maroon">m1</font> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">m11</font> = <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1, <font color="Maroon">n11</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E17:118_7_3"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">n11</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:118_7_3"/>
<font color="Maroon">m11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n11</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E19:118_7_3"/><b>then </b><i><font color="Green">E429</font></i>: 
<font color="Maroon">m11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><i><font color="Green">E430</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_7_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_7_3_1"/><i><font color="Green">E431</font></i>: 
<font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n11</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_7_3_1_1"/>
( <font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n11</font> or <font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n11</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="118_7_3_1_1_1"><b>suppose </b></a><a NAME="E1:118_7_3_1_1_1"/><i><font color="Green">E432</font></i>: 
<font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n11</font>
 ;<br/><div class="add"><a NAME="E2:118_7_3_1_1_1"/><b>then </b><i><font color="Green">E433</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m11</font>,<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n11</font>,<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="E3:118_7_3_1_1_1"/><i><font color="Green">E434</font></i>: 
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m11</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E4:118_7_3_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m11</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:118_7_3_1_1_1"/><b>then </b><i><font color="Green">E435</font></i>: 
1 <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">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m11</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:118_7_3_1_1_1"/>
<font color="Maroon">m11</font> <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">Pbg</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E7:118_7_3_1_1_1"/><b>then </b>
<font color="Maroon">m11</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">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E8:118_7_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Pag</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:118_7_3_1_1_1"/><b>then </b>
<font color="Maroon">Pbg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Pag</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">Pbg</font></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><b>hence </b><a NAME="E10:118_7_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T47" target="_self">Th47</a>, , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_7_3_1_1_2"><b>suppose </b></a><a NAME="E1:118_7_3_1_1_2"/>
<font color="Maroon">m11</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n11</font>
 ;<br/><div class="add"><a NAME="E2:118_7_3_1_1_2"/><b>then </b>
<font color="Maroon">m11</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n11</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:118_7_3_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E21:118_7_3"/><i><font color="Green">E436</font></i>: 
<font color="Maroon">n11</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Pbg</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self">Def3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:118_7_3"/>
<font color="Maroon">Pbg</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i><a class="ref" href="chord.html#T46" target="_self">Th46</a>, , <a class="ref" href="chord.html#T44" target="_self">Th44</a>, , , <a class="ref" href="chord.html#T45" target="_self">Th45</a>, , <a class="ref" href="chord.html#T32" target="_self">Th32</a>, <a class="txt" href="chord.html#E30:118"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E23:118_7_3"/><b>then </b>
<font color="Maroon">Pb</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green">E122</font></i></a>, <a class="txt" href="chord.html#E33:118"><i><font color="Green">E84</font></i></a></i>;<br/><b>hence </b><a NAME="E24:118_7_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E38:118"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
