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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>set </b><font color="Maroon" title="c2">tVG</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:118"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <b>and </b><br/><a NAME="E2:118"/><i><font color="Green" title="E92">A2</font></i>: 
not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c5">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:118"/><i><font color="Green" title="E93">A3</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>
 <b>and </b><br/><a NAME="E4:118"/><i><font color="Green" title="E94">A4</font></i>: 
not <font color="Maroon" title="c5">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">Gs</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>;<br/>

<b>consider </b><font color="Maroon" title="c7">Gns</font> being   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">sa</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c9">sb</font> = <font color="Maroon" title="c4">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c7">Gns</font> <b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="chord.html#T77" target="_self" title="CHORD:th.77">Th77</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>;<br/>
<b>set </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>;<br/>
<a NAME="E7:118"/><i><font color="Green" title="E95">A5</font></i>: 
<font color="Maroon" title="c5">S</font> is    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="chord.html#T70" target="_self" title="CHORD:th.70">Th70</a></i>;<br/>
<a NAME="E8:118"/><i><font color="Green" title="E96">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="chord.html#T76" target="_self" title="CHORD:th.76">Th76</a></i>;<br/>
<a NAME="E9:118"/><i><font color="Green" title="E97">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E7:118"><i><font color="Green" title="E95">A5</font></i></a>, <a class="ref" href="chord.html#T75" target="_self" title="CHORD:th.75">Th75</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">Ga</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c7">Gns</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c13">Gb</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c7">Gns</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span>;<br/>
<a NAME="E12:118"/><i><font color="Green" title="E98">A8</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c12">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E13:118"/><i><font color="Green" title="E99">A9</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c13">Gb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E14:118"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c7">Gns</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>, <font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c7">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:118"/>
not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/>
<a NAME="E17:118"/><b>then </b><i><font color="Green" title="E100">A10</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:118_1"/><i><font color="Green" title="E101">A11</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 ;<br/><a NAME="E2:118_1"/>
not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:118"><i><font color="Green" title="E97">A7</font></i></a>, <a class="txt" href="chord.html#E1:118_1"><i><font color="Green" title="E101">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:118_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:118_1"><i><font color="Green" title="E101">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:118"/><b>then </b><i><font color="Green" title="E101">A12</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:118_2"/><i><font color="Green" title="E102">A13</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 ;<br/><a NAME="E2:118_2"/>
not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:118"><i><font color="Green" title="E97">A7</font></i></a>, <a class="txt" href="chord.html#E1:118_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:118_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:118_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:118"/><b>then </b><i><font color="Green" title="E102">A14</font></i>: 
<font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Ga</font> = <font color="Maroon" title="c12">Ga</font> as   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c14">A</font> <b>by </b><i><a class="txt" href="chord.html#E17:118"><i><font color="Green" title="E100">A10</font></i></a>, <a class="txt" href="chord.html#E19:118"><i><font color="Green" title="E101">A12</font></i></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Gb</font> = <font color="Maroon" title="c13">Gb</font> as   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c15">B</font> <b>by </b><i><a class="txt" href="chord.html#E17:118"><i><font color="Green" title="E100">A10</font></i></a>, <a class="txt" href="chord.html#E21:118"><i><font color="Green" title="E102">A14</font></i></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></i>;<br/>
<b>let </b><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c6">Gs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">CHORD:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E24:118"/><i><font color="Green" title="E103">A15</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">y</font>
 <b>and </b><br/><a NAME="E25:118"/><i><font color="Green" title="E104">A16</font></i>: 
not <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">xg</font> = <font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c21">yg</font> = <font color="Maroon" title="c19">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/>
<a NAME="E27:118"/><i><font color="Green" title="E105">A17</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">Gs</font>
 
<b>by </b><i><a class="txt" href="chord.html#E4:118"><i><font color="Green" title="E94">A4</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E28:118"/><i><font color="Green" title="E106">A18</font></i>: 
not <font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E4:118"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="chord.html#E25:118"><i><font color="Green" title="E104">A16</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/>
<b>set </b><font color="Maroon" title="c22">xy</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c23">Bx</font> = <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c24">Ax</font> = <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c25">Gax</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E30:118"/><i><font color="Green" title="E107">A19</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c25">Gax</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">Ga1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<i><font color="Green" title="E108">A20</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<font color="Maroon" title="c14">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
;<br/>
<a NAME="E33:118"/><b>then </b><i><font color="Green" title="E109">A21</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c26">Ga1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E34:118"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E35:118"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">xa</font> = <font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c28">ya</font> = <font color="Maroon" title="c19">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c26">Ga1</font> <b>by </b><i><a class="txt" href="chord.html#E32:118"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">xag</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E38:118"/><i><font color="Green" title="E110">A22</font></i>: 
<font color="Maroon" title="c29">xag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>and </b><br/><a NAME="E39:118"/><i><font color="Green" title="E111">A23</font></i>: 
<font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c29">xag</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E3:118"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></i>;<br/>
<a NAME="E40:118"/>
not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 
<b>by </b><i><a class="txt" href="chord.html#E19:118"><i><font color="Green" title="E101">A12</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E41:118"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">Ga</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E12:118"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="chord.html#E38:118"><i><font color="Green" title="E110">A22</font></i></a>, <a class="txt" href="chord.html#E39:118"><i><font color="Green" title="E111">A23</font></i></a></i>;<br/>
<a NAME="E42:118"/><b>then </b><i><font color="Green" title="E112">A24</font></i>: 
<font color="Maroon" title="c25">Gax</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
<b>by </b><i><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">yag</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E44:118"/><i><font color="Green" title="E113">A25</font></i>: 
<font color="Maroon" title="c30">yag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>and </b><br/><a NAME="E45:118"/><i><font color="Green" title="E114">A26</font></i>: 
<font color="Maroon" title="c21">yg</font>,<font color="Maroon" title="c30">yag</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E3:118"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></i>;<br/>
<a NAME="E46:118"/><i><font color="Green" title="E115">A27</font></i>: 
<font color="Maroon" title="c30">yag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E44:118"><i><font color="Green" title="E113">A25</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E47:118"/>
( not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="chord.html#E19:118"><i><font color="Green" title="E101">A12</font></i></a>, <a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E48:118"/><b>then </b>
not <font color="Maroon" title="c21">yg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c25">Gax</font>
 
<b>by </b><i><a class="txt" href="chord.html#E30:118"><i><font color="Green" title="E107">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E49:118"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c25">Gax</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E30:118"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="chord.html#E45:118"><i><font color="Green" title="E114">A26</font></i></a>, <a class="txt" href="chord.html#E46:118"><i><font color="Green" title="E115">A27</font></i></a></i>;<br/>
<a NAME="E50:118"/><b>then </b>
<font color="Maroon" title="c26">Ga1</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
<b>by </b><i><a class="txt" href="chord.html#E42:118"><i><font color="Green" title="E112">A24</font></i></a>, <a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c31">Wa</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c26">Ga1</font><b> such that </b><br/><a NAME="E52:118"/><i><font color="Green" title="E116">A28</font></i>: 
<font color="Maroon" title="c31">Wa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c27">xa</font>,<font color="Maroon" title="c28">ya</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/>
<a NAME="E53:118"/><i><font color="Green" title="E117">A29</font></i>: 
( <font color="Maroon" title="c31">Wa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">xa</font> &amp; <font color="Maroon" title="c31">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ya</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E52:118"><i><font color="Green" title="E116">A28</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c32">Pa</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c26">Ga1</font><b> such that </b><br/><a NAME="E55:118"/><i><font color="Green" title="E118">A30</font></i>: 
<font color="Maroon" title="c32">Pa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c31">Wa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c31">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>and </b><br/><a NAME="E56:118"/><i><font color="Green" title="E119">A31</font></i>: 
<font color="Maroon" title="c32">Pa</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">Gbx</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E58:118"/><i><font color="Green" title="E120">A32</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c33">Gbx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">Gb1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<i><font color="Green" title="E121">A33</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<font color="Maroon" title="c15">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
;<br/>
<a NAME="E61:118"/><b>then </b><i><font color="Green" title="E122">A34</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c34">Gb1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E62:118"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E63:118"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c35">xb</font> = <font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c36">yb</font> = <font color="Maroon" title="c19">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c34">Gb1</font> <b>by </b><i><a class="txt" href="chord.html#E60:118"><i><font color="Green" title="E121">A33</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c37">xbg</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E66:118"/><i><font color="Green" title="E123">A35</font></i>: 
<font color="Maroon" title="c37">xbg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <b>and </b><br/><a NAME="E67:118"/><i><font color="Green" title="E124">A36</font></i>: 
<font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c37">xbg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E3:118"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="chord.html#T83" target="_self" title="CHORD:th.83">Th83</a></i>;<br/>
<a NAME="E68:118"/>
not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 
<b>by </b><i><a class="txt" href="chord.html#E21:118"><i><font color="Green" title="E102">A14</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E69:118"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c17">Gb</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E13:118"><i><font color="Green" title="E99">A9</font></i></a>, <a class="txt" href="chord.html#E66:118"><i><font color="Green" title="E123">A35</font></i></a>, <a class="txt" href="chord.html#E67:118"><i><font color="Green" title="E124">A36</font></i></a></i>;<br/>
<a NAME="E70:118"/><b>then </b><i><font color="Green" title="E125">A37</font></i>: 
<font color="Maroon" title="c33">Gbx</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
<b>by </b><i><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c38">ybg</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E72:118"/><i><font color="Green" title="E126">A38</font></i>: 
<font color="Maroon" title="c38">ybg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <b>and </b><br/><a NAME="E73:118"/><i><font color="Green" title="E127">A39</font></i>: 
<font color="Maroon" title="c21">yg</font>,<font color="Maroon" title="c38">ybg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E3:118"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="chord.html#T83" target="_self" title="CHORD:th.83">Th83</a></i>;<br/>
<a NAME="E74:118"/><i><font color="Green" title="E128">A40</font></i>: 
<font color="Maroon" title="c38">ybg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E72:118"><i><font color="Green" title="E126">A38</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E75:118"/>
( not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> &amp; not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="chord.html#E21:118"><i><font color="Green" title="E102">A14</font></i></a>, <a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E27:118"><i><font color="Green" title="E105">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E76:118"/><b>then </b>
not <font color="Maroon" title="c21">yg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c33">Gbx</font>
 
<b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green" title="E120">A32</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E77:118"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c33">Gbx</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E58:118"><i><font color="Green" title="E120">A32</font></i></a>, <a class="txt" href="chord.html#E73:118"><i><font color="Green" title="E127">A39</font></i></a>, <a class="txt" href="chord.html#E74:118"><i><font color="Green" title="E128">A40</font></i></a></i>;<br/>
<a NAME="E78:118"/><b>then </b>
<font color="Maroon" title="c34">Gb1</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
<b>by </b><i><a class="txt" href="chord.html#E70:118"><i><font color="Green" title="E125">A37</font></i></a>, <a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c39">Wb</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c34">Gb1</font><b> such that </b><br/><a NAME="E80:118"/><i><font color="Green" title="E129">A41</font></i>: 
<font color="Maroon" title="c39">Wb</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c36">yb</font>,<font color="Maroon" title="c35">xb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/>
<a NAME="E81:118"/><i><font color="Green" title="E130">A42</font></i>: 
( <font color="Maroon" title="c39">Wb</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ya</font> &amp; <font color="Maroon" title="c39">Wb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">xa</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E80:118"><i><font color="Green" title="E129">A41</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c40">Pb</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c34">Gb1</font><b> such that </b><br/><a NAME="E83:118"/><i><font color="Green" title="E131">A43</font></i>: 
<font color="Maroon" title="c40">Pb</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c39">Wb</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c39">Wb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>and </b><br/><a NAME="E84:118"/><i><font color="Green" title="E132">A44</font></i>: 
<font color="Maroon" title="c40">Pb</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c41">Pag</font> = <font color="Maroon" title="c32">Pa</font>, <font color="Maroon" title="c42">Pbg</font> = <font color="Maroon" title="c40">Pb</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></i>;<br/>
<a NAME="E86:118"/><i><font color="Green" title="E133">A45</font></i>: 
( <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> )
 
;<br/>
<a NAME="E87:118"/>
( <font color="Maroon" title="c32">Pa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c32">Pa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Pa</font></span>)</span> )
 
;<br/>
<a NAME="E88:118"/><b>then </b><i><font color="Green" title="E134">A46</font></i>: 
( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">xg</font> &amp; <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yg</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E53:118"><i><font color="Green" title="E117">A29</font></i></a>, <a class="txt" href="chord.html#E55:118"><i><font color="Green" title="E118">A30</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E89:118"/><b>then </b><i><font color="Green" title="E135">A47</font></i>: 
not <font color="Maroon" title="c41">Pag</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E90:118"/><b>then </b><i><font color="Green" title="E136">A48</font></i>: 
not <font color="Maroon" title="c41">Pag</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E91:118"/><i><font color="Green" title="E137">A49</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/>
<a NAME="E92:118"/><i><font color="Green" title="E138">A50</font></i>: 
( <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> )
 
;<br/>
<a NAME="E93:118"/><i><font color="Green" title="E139">A51</font></i>: 
( <font color="Maroon" title="c40">Pb</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c40">Pb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pb</font></span>)</span> )
 
;<br/>
<a NAME="E94:118"/><b>then </b><i><font color="Green" title="E140">A52</font></i>: 
( <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yg</font> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">xg</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E81:118"><i><font color="Green" title="E130">A42</font></i></a>, <a class="txt" href="chord.html#E83:118"><i><font color="Green" title="E131">A43</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E95:118"/><b>then </b><i><font color="Green" title="E141">A53</font></i>: 
not <font color="Maroon" title="c42">Pbg</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E96:118"/><b>then </b><i><font color="Green" title="E142">A54</font></i>: 
not <font color="Maroon" title="c42">Pbg</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E97:118"/><i><font color="Green" title="E143">A55</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/>
<b>set </b><font color="Maroon" title="c43">P</font> = <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font>;<br/>
<a NAME="E98:118"/><i><font color="Green" title="E144">A56</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="ref" href="glib_001.html#T29" title="GLIB_001:th.29">GLIB_001:29</a></i>;<br/>
<a NAME="E99:118"/><i><font color="Green" title="E145">A57</font></i>: 
not <font color="Maroon" title="c41">Pag</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/>
<a NAME="E100:118"/><i><font color="Green" title="E146">A58</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E28:118"><i><font color="Green" title="E106">A18</font></i></a>, <a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="ref" href="chord.html#T46" target="_self" title="CHORD:th.46">Th46</a></i>;<br/>
<a NAME="E101:118"/><i><font color="Green" title="E147">A59</font></i>: 
not <font color="Maroon" title="c42">Pbg</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/>
<a NAME="E102:118"/><i><font color="Green" title="E148">A60</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E28:118"><i><font color="Green" title="E106">A18</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="ref" href="chord.html#T46" target="_self" title="CHORD:th.46">Th46</a></i>;<br/>
<a NAME="E103:118"/><i><font color="Green" title="E149">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">Pa</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/>
<a NAME="E2:118_5_1"/><b>then </b><i><font color="Green" title="E150">A62</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E33:118"><i><font color="Green" title="E109">A21</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:118_5_1"/>
<font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pb</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/>
<a NAME="E4:118_5_1"/><b>then </b><i><font color="Green" title="E151">A63</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E61:118"><i><font color="Green" title="E122">A34</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24" title="XBOOLE_1:th.24">XBOOLE_1:24</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<b>by </b><i><a class="txt" href="chord.html#E8:118"><i><font color="Green" title="E96">A6</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:118_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E2:118_5_1"><i><font color="Green" title="E150">A62</font></i></a>, <a class="txt" href="chord.html#E4:118_5_1"><i><font color="Green" title="E151">A63</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">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" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.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" title="c44">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:118_5_2"/>
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<a NAME="E2:118_5_2"/><b>then </b>
( <font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> or <font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:118_5_2"/>
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E91:118"><i><font color="Green" title="E137">A49</font></i></a>, <a class="txt" href="chord.html#E97:118"><i><font color="Green" title="E143">A55</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E150">A64</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.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" title="c41">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c44">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118_6"/><i><font color="Green" title="E151">A65</font></i>: 
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:118_6"/>
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E3:118_6"><i><font color="Green" title="E151">A65</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:118_6"/><b>then </b><i><font color="Green" title="E152">A66</font></i>: 
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pa</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T111" title="GLIB_001:th.111">GLIB_001:111</a></i>;<br/>
<a NAME="E6:118_6"/>
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E3:118_6"><i><font color="Green" title="E151">A65</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:118_6"/><b>then </b><i><font color="Green" title="E153">A67</font></i>: 
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">Pb</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T111" title="GLIB_001:th.111">GLIB_001:111</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c45">a1</font>, <font color="Maroon" title="c46">a2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c26">Ga1</font>, <font color="Maroon" title="c47">na</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:118_6"/><i><font color="Green" title="E154">A68</font></i>: 
<font color="Maroon" title="c47">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>and </b><br/><a NAME="E10:118_6"/><i><font color="Green" title="E155">A69</font></i>: 
<font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">na</font>
 <b>and </b><br/><a NAME="E11:118_6"/>
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E12:118_6"/><i><font color="Green" title="E156">A70</font></i>: 
<font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E13:118_6"/><i><font color="Green" title="E157">A71</font></i>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font>,<font color="Maroon" title="c26">Ga1</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:118_6"><i><font color="Green" title="E152">A66</font></i></a>, <a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c48">b1</font>, <font color="Maroon" title="c49">b2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c34">Gb1</font>, <font color="Maroon" title="c50">nb</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:118_6"/>
<font color="Maroon" title="c50">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>and </b><br/><a NAME="E16:118_6"/>
<font color="Maroon" title="c48">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">nb</font>
 <b>and </b><br/><a NAME="E17:118_6"/>
<font color="Maroon" title="c44">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E18:118_6"/>
<font color="Maroon" title="c49">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E19:118_6"/><i><font color="Green" title="E158">A72</font></i>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c48">b1</font>,<font color="Maroon" title="c49">b2</font>,<font color="Maroon" title="c34">Gb1</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:118_6"><i><font color="Green" title="E153">A67</font></i></a>, <a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/>
<a NAME="E20:118_6"/><i><font color="Green" title="E159">A73</font></i>: 
( <font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c48">b1</font>,<font color="Maroon" title="c49">b2</font>,<font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E13:118_6"><i><font color="Green" title="E157">A71</font></i></a>, <a class="txt" href="chord.html#E19:118_6"><i><font color="Green" title="E158">A72</font></i></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/>
<a NAME="E21:118_6"/><b>then </b><i><font color="Green" title="E160">A74</font></i>: 
( ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">b1</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">b2</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">b2</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">b1</font> ) )
 
<b>by </b><i><a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></i>;<br/>
<a NAME="E22:118_6"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c26">Ga1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E32:118"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E23:118_6"/><b>then </b><i><font color="Green" title="E161">A75</font></i>: 
( ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> or <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> or <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E24:118_6"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c34">Gb1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E60:118"><i><font color="Green" title="E121">A33</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E25:118_6"/><b>then </b><i><font color="Green" title="E162">A76</font></i>: 
( ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> or <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> or <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) )
 
<b>by </b><i><a class="txt" href="chord.html#E21:118_6"><i><font color="Green" title="E160">A74</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> ) )
 </span><b>by </b><i><a class="txt" href="chord.html#E8:118"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="chord.html#E23:118_6"><i><font color="Green" title="E161">A75</font></i></a>, <a class="txt" href="chord.html#E25:118_6"><i><font color="Green" title="E162">A76</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E163">A77</font></i>: 
( ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) )
 ;<br/><div class="add"><a NAME="E2:118_6_1_1"/>
<font color="Maroon" title="c47">na</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c47">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></i>;<br/><b>hence </b><a NAME="E3:118_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E24:118"><i><font color="Green" title="E103">A15</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E9:118_6"><i><font color="Green" title="E154">A68</font></i></a>, <a class="txt" href="chord.html#E10:118_6"><i><font color="Green" title="E155">A69</font></i></a>, <a class="txt" href="chord.html#E12:118_6"><i><font color="Green" title="E156">A70</font></i></a>, <a class="txt" href="chord.html#E1:118_6_1_1"><i><font color="Green" title="E163">A77</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">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" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) or ( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:118_6_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E28:118"><i><font color="Green" title="E106">A18</font></i></a>, <a class="txt" href="chord.html#E20:118_6"><i><font color="Green" title="E159">A73</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</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" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">Pbg</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="ref" href="chord.html#T28" target="_self" title="CHORD:th.28">Th28</a></i>;<br/>
<a NAME="E106:118"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="chord.html#E100:118"><i><font color="Green" title="E146">A58</font></i></a>, <a class="txt" href="chord.html#E102:118"><i><font color="Green" title="E148">A60</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E107:118"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E108:118"/><b>then </b><i><font color="Green" title="E151">A78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E109:118"/>
<font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 
<b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E89:118"><i><font color="Green" title="E135">A47</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E95:118"><i><font color="Green" title="E141">A53</font></i></a>, <a class="txt" href="chord.html#E99:118"><i><font color="Green" title="E145">A57</font></i></a>, <a class="txt" href="chord.html#E101:118"><i><font color="Green" title="E147">A59</font></i></a>, <a class="txt" href="chord.html#E103:118"><i><font color="Green" title="E149">A61</font></i></a>, <a class="txt" href="chord.html#E104:118"><i><font color="Green" title="E150">A64</font></i></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></i>;<br/>
<a NAME="E110:118"/><b>then </b>
<font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 
<b>by </b><i><a class="txt" href="chord.html#E108:118"><i><font color="Green" title="E151">A78</font></i></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c44">m</font>, <font color="Maroon" title="c45">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E112:118"/><i><font color="Green" title="E152">A79</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">n</font>
 <b>and </b><br/><a NAME="E113:118"/><i><font color="Green" title="E153">A80</font></i>: 
<font color="Maroon" title="c45">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>
 <b>and </b><br/><a NAME="E114:118"/><i><font color="Green" title="E154">A81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font>
 <b>and </b><br/><a NAME="E115:118"/><i><font color="Green" title="E155">A82</font></i>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E116:118"/><i><font color="Green" title="E156">A83</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="chord.html#D10" target="_self" title="CHORD:def.10">Def10</a></i>;<br/>
<a NAME="E117:118"/><i><font color="Green" title="E157">A84</font></i>: 
( <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E118:118"/><i><font color="Green" title="E158">A85</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">n</font>
 
<b>by </b><i><a class="txt" href="chord.html#E112:118"><i><font color="Green" title="E152">A79</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c46">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E120:118"/><i><font color="Green" title="E159">A86</font></i>: 
<font color="Maroon" title="c46">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E115:118"><i><font color="Green" title="E155">A82</font></i></a></i>;<br/>
<a NAME="E121:118"/><i><font color="Green" title="E160">A87</font></i>: 
<font color="Maroon" title="c46">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="chord.html#E120:118"><i><font color="Green" title="E159">A86</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/>
<a NAME="E122:118"/><i><font color="Green" title="E161">A88</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E123:118"/><i><font color="Green" title="E162">A89</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">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" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> &amp; <font color="Maroon" title="c45">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> ) or ( <font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">n</font> ) or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">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" title="E163">A90</font></i>: 
( <font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> &amp; <font color="Maroon" title="c45">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_1"/><b>then </b>
( <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font> &amp; <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font> )
 <b>by </b><i><a class="txt" href="chord.html#E122:118"><i><font color="Green" title="E161">A88</font></i></a>, <a class="txt" href="chord.html#E123:118"><i><font color="Green" title="E162">A89</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:118_7_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><a NAME="E4:118_7_1"/><b>then </b>
<font color="Maroon" title="c41">Pag</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E90:118"><i><font color="Green" title="E136">A48</font></i></a>, <a class="txt" href="chord.html#E112:118"><i><font color="Green" title="E152">A79</font></i></a>, <a class="txt" href="chord.html#E115:118"><i><font color="Green" title="E155">A82</font></i></a>, <a class="txt" href="chord.html#E1:118_7_1"><i><font color="Green" title="E163">A90</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></i>;<br/><a NAME="E5:118_7_1"/><b>then </b>
<font color="Maroon" title="c32">Pa</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E32:118"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></i>;<br/><b>hence </b><a NAME="E6:118_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E56:118"><i><font color="Green" title="E119">A31</font></i></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</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" title="E163">A91</font></i>: 
( <font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">n</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_2"/><b>then </b>
<font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="txt" href="chord.html#E122:118"><i><font color="Green" title="E161">A88</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:118_7_2"/><b>then </b><i><font color="Green" title="E164">A92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><a NAME="E4:118_7_2"/>
( <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> &amp; not <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font> )
 <b>by </b><i><a class="txt" href="chord.html#E113:118"><i><font color="Green" title="E153">A80</font></i></a>, <a class="txt" href="chord.html#E123:118"><i><font color="Green" title="E162">A89</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c47">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:118_7_2"/><i><font color="Green" title="E165">A93</font></i>: 
<font color="Maroon" title="c47">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>and </b><br/><a NAME="E7:118_7_2"/><i><font color="Green" title="E166">A94</font></i>: 
<font color="Maroon" title="c45">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">n1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E8:118_7_2"/><i><font color="Green" title="E167">A95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E6:118_7_2"><i><font color="Green" title="E165">A93</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E9:118_7_2"/><i><font color="Green" title="E168">A96</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c26">Ga1</font>
 <b>by </b><i><a class="txt" href="chord.html#E117:118"><i><font color="Green" title="E157">A84</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></i>;<br/><a NAME="E10:118_7_2"/>
<font color="Maroon" title="c47">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c48">n1</font> = <font color="Maroon" title="c47">n1</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c49">n11</font> = <font color="Maroon" title="c48">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E13:118_7_2"/><i><font color="Green" title="E169">A97</font></i>: 
<font color="Maroon" title="c49">n11</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:118_7_2"><i><font color="Green" title="E165">A93</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:118_7_2"/><b>then </b><i><font color="Green" title="E170">A98</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c34">Gb1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">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" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) or ( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> ) or ( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> ) or ( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="chord.html#E33:118"><i><font color="Green" title="E109">A21</font></i></a>, <a class="txt" href="chord.html#E61:118"><i><font color="Green" title="E122">A34</font></i></a>, <a class="txt" href="chord.html#E9:118_7_2"><i><font color="Green" title="E168">A96</font></i></a>, <a class="txt" href="chord.html#E14:118_7_2"><i><font color="Green" title="E170">A98</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="E171">A99</font></i>: 
( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 or <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1"><i><font color="Green" title="E171">A99</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E172">A100</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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" title="E173">A101</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">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#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c44">m</font> or 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">m</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_1_1_1_1"><i><font color="Green" title="E173">A101</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c44">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" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">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="txt" href="chord.html#E114:118"><i><font color="Green" title="E154">A81</font></i></a>, <a class="txt" href="chord.html#E122:118"><i><font color="Green" title="E161">A88</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_1"><i><font color="Green" title="E172">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E174">A102</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">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" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="txt" href="chord.html#E122:118"><i><font color="Green" title="E161">A88</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:118_7_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E175">A103</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_1_1_1_1_1_2"><i><font color="Green" title="E174">A102</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_1_1_1_2"/><i><font color="Green" title="E176">A104</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_1_1_1_2"><i><font color="Green" title="E174">A102</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:118_7_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E177">A105</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><a NAME="E7:118_7_2_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E4:118_7_2_1_1_1_1_1_1_2"><i><font color="Green" title="E176">A104</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E8:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E9:118_7_2_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:118"><i><font color="Green" title="E144">A56</font></i></a>, <a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><b>hence </b><a NAME="E10:118_7_2_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E116:118"><i><font color="Green" title="E156">A83</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_1"><i><font color="Green" title="E172">A100</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2_1_1_1_1_1_1_2"><i><font color="Green" title="E175">A103</font></i></a>, <a class="txt" href="chord.html#E6:118_7_2_1_1_1_1_1_1_2"><i><font color="Green" title="E177">A105</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</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" title="c41">Pag</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E90:118"><i><font color="Green" title="E136">A48</font></i></a>, <a class="txt" href="chord.html#E121:118"><i><font color="Green" title="E160">A87</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_1"><i><font color="Green" title="E172">A100</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c32">Pa</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E32:118"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></i>;<br/><b>hence </b><a NAME="E5:118_7_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E56:118"><i><font color="Green" title="E119">A31</font></i></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</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" title="E172">A106</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">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" title="E173">A107</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 ;<br/><b>set </b><font color="Maroon" title="c50">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">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" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> or <font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_1_1_2_1"><i><font color="Green" title="E173">A107</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E174">A108</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">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" title="E175">A109</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E176">A110</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:118_7_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E177">A111</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font>,<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E117:118"><i><font color="Green" title="E157">A84</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_2_1_1_1"><i><font color="Green" title="E174">A108</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E7:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:118_7_2_1_1_1_2_1_1_1"><i><font color="Green" title="E176">A110</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E178">A112</font></i>: 
<font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><a NAME="E9:118_7_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c44">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E2:118_7_2_1_1_1_2_1_1_1"><i><font color="Green" title="E175">A109</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E11:118_7_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:118"><i><font color="Green" title="E144">A56</font></i></a>, <a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a></i>;<br/><b>hence </b><a NAME="E12:118_7_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E116:118"><i><font color="Green" title="E156">A83</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_2"><i><font color="Green" title="E172">A106</font></i></a>, <a class="txt" href="chord.html#E5:118_7_2_1_1_1_2_1_1_1"><i><font color="Green" title="E177">A111</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2_1_1_1_2_1_1_1"><i><font color="Green" title="E178">A112</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_1_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c44">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">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="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</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" title="c41">Pag</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E90:118"><i><font color="Green" title="E136">A48</font></i></a>, <a class="txt" href="chord.html#E115:118"><i><font color="Green" title="E155">A82</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_1_1_2"><i><font color="Green" title="E172">A106</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></i>;<br/><a NAME="E4:118_7_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c32">Pa</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E32:118"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></i>;<br/><b>hence </b><a NAME="E5:118_7_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E56:118"><i><font color="Green" title="E119">A31</font></i></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</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" title="E171">A113</font></i>: 
( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c50">ac</font> = <font color="Maroon" title="c32">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c16">Ga</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:118_7_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c51">aa</font> = <font color="Maroon" title="c3">a</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c16">Ga</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c52">bc</font> = <font color="Maroon" title="c40">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c17">Gb</font> <b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_2"><i><font color="Green" title="E171">A113</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:118_7_2_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c53">bb</font> = <font color="Maroon" title="c4">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c17">Gb</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon" title="c54">WA</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">Ga</font><b> such that </b><br/><a NAME="E9:118_7_2_1_2"/><i><font color="Green" title="E172">A114</font></i>: 
<font color="Maroon" title="c54">WA</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c51">aa</font>,<font color="Maroon" title="c50">ac</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c55">WB</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c17">Gb</font><b> such that </b><br/><a NAME="E11:118_7_2_1_2"/><i><font color="Green" title="E173">A115</font></i>: 
<font color="Maroon" title="c55">WB</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c52">bc</font>,<font color="Maroon" title="c53">bb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c56">WA</font> = <font color="Maroon" title="c54">WA</font>, <font color="Maroon" title="c57">WB</font> = <font color="Maroon" title="c55">WB</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">Gns</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>set </b><font color="Maroon" title="c58">WAB</font> = <font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font>;<br/><a NAME="E13:118_7_2_1_2"/>
<font color="Maroon" title="c46">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c52">bc</font>,<font color="Maroon" title="c7">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:118"><i><font color="Green" title="E101">A12</font></i></a>, <a class="txt" href="chord.html#E21:118"><i><font color="Green" title="E102">A14</font></i></a>, <a class="txt" href="chord.html#E120:118"><i><font color="Green" title="E159">A86</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2_1_2"><i><font color="Green" title="E171">A113</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></i>;<br/><a NAME="E14:118_7_2_1_2"/><b>then </b><i><font color="Green" title="E174">A116</font></i>: 
<font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c52">bc</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c59">WAs</font> = <font color="Maroon" title="c56">WA</font>, <font color="Maroon" title="c60">WBs</font> = <font color="Maroon" title="c57">WB</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">Gns</font> ;<br/><b>set </b><font color="Maroon" title="c61">WaB</font> = <font color="Maroon" title="c59">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c62">Wab</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c59">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font></span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c60">WBs</font>;<br/><a NAME="E16:118_7_2_1_2"/>
<font color="Maroon" title="c59">WAs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c51">aa</font>,<font color="Maroon" title="c50">ac</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:118_7_2_1_2"><i><font color="Green" title="E172">A114</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><a NAME="E17:118_7_2_1_2"/><b>then </b><i><font color="Green" title="E175">A117</font></i>: 
<font color="Maroon" title="c59">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c51">aa</font>,<font color="Maroon" title="c52">bc</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:118_7_2_1_2"><i><font color="Green" title="E174">A116</font></i></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></i>;<br/><a NAME="E18:118_7_2_1_2"/>
<font color="Maroon" title="c60">WBs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c52">bc</font>,<font color="Maroon" title="c53">bb</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:118_7_2_1_2"><i><font color="Green" title="E173">A115</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">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" title="c59">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Gns</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c50">ac</font>,<font color="Maroon" title="c46">e</font>,<font color="Maroon" title="c52">bc</font></span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c60">WBs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="chord.html#E17:118_7_2_1_2"><i><font color="Green" title="E175">A117</font></i></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">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#E1:118"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:118"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</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"/>
( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> )
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3"/><b>then </b>
( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> or <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>then </b><i><font color="Green" title="E171">A118</font></i>: <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> = 
<font color="Maroon" title="c18">x</font>
<b>by </b><i><a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E117:118"><i><font color="Green" title="E157">A84</font></i></a>, <a class="txt" href="chord.html#E1:118_7_2"><i><font color="Green" title="E163">A91</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>
<b>by </b><i><a class="txt" href="chord.html#E81:118"><i><font color="Green" title="E130">A42</font></i></a>, <a class="txt" href="chord.html#E83:118"><i><font color="Green" title="E131">A43</font></i></a>, <a class="txt" href="chord.html#E93:118"><i><font color="Green" title="E139">A51</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">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" title="E172">A119</font></i>: 
<font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 ;<br/><b>set </b><font color="Maroon" title="c50">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">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" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font> or <font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_3_2"><i><font color="Green" title="E172">A119</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E173">A120</font></i>: 
<font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green" title="E174">A121</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green" title="E175">A122</font></i>: 
<font color="Maroon" title="c49">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font>,<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>,<font color="Maroon" title="c34">Gb1</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:118_7_2_1_3_2_1_1"><i><font color="Green" title="E173">A120</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E6:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>,<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font>,<font color="Maroon" title="c34">Gb1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><a NAME="E7:118_7_2_1_3_2_1_1"/><b>then </b><i><font color="Green" title="E176">A123</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span>,<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><a NAME="E8:118_7_2_1_3_2_1_1"/><i><font color="Green" title="E177">A124</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n11</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E4:118_7_2_1_3_2_1_1"><i><font color="Green" title="E175">A122</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E9:118_7_2_1_3_2_1_1"/><i><font color="Green" title="E178">A125</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n11</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E10:118_7_2_1_3_2_1_1"/>
<font color="Maroon" title="c49">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E2:118_7_2_1_3_2_1_1"><i><font color="Green" title="E174">A121</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:118_7_2_1_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:118_7_2_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:118"><i><font color="Green" title="E144">A56</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2_1_3_2_1_1"><i><font color="Green" title="E177">A124</font></i></a>, <a class="txt" href="chord.html#E9:118_7_2_1_3_2_1_1"><i><font color="Green" title="E178">A125</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><b>hence </b><a NAME="E13:118_7_2_1_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E116:118"><i><font color="Green" title="E156">A83</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2_1_3"><i><font color="Green" title="E171">A118</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2_1_3_2_1_1"><i><font color="Green" title="E176">A123</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_2_1_2"><b>suppose </b></a><a NAME="E1:118_7_2_1_3_2_1_2"/>
<font color="Maroon" title="c49">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c49">n11</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E3:118_7_2_1_3_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E114:118"><i><font color="Green" title="E154">A81</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E13:118_7_2"><i><font color="Green" title="E169">A97</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2_1_3"><i><font color="Green" title="E171">A118</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="c42">Pbg</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E96:118"><i><font color="Green" title="E142">A54</font></i></a>, <a class="txt" href="chord.html#E121:118"><i><font color="Green" title="E160">A87</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2_1_3"><i><font color="Green" title="E171">A118</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></i>;<br/><a NAME="E6:118_7_2_1_3"/><b>then </b>
<font color="Maroon" title="c40">Pb</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E60:118"><i><font color="Green" title="E121">A33</font></i></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</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#E84:118"><i><font color="Green" title="E132">A44</font></i></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</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" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:118_7_2_1_4"/><b>then </b>
( ( <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> or <font color="Maroon" title="c41">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) &amp; ( <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> or <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:118_7_2_1_4"/><b>then </b>
<font color="Maroon" title="c20">xg</font>,<font color="Maroon" title="c21">yg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E114:118"><i><font color="Green" title="E154">A81</font></i></a>, <a class="txt" href="chord.html#E115:118"><i><font color="Green" title="E155">A82</font></i></a>, <a class="txt" href="chord.html#E3:118_7_2"><i><font color="Green" title="E164">A92</font></i></a>, <a class="txt" href="chord.html#E7:118_7_2"><i><font color="Green" title="E166">A94</font></i></a>, <a class="txt" href="chord.html#E8:118_7_2"><i><font color="Green" title="E167">A95</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:118_7_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:118"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="chord.html#E25:118"><i><font color="Green" title="E104">A16</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</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" title="E163">A126</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">m</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c47">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:118_7_3"/><i><font color="Green" title="E164">A127</font></i>: 
<font color="Maroon" title="c44">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E4:118_7_3"/><i><font color="Green" title="E165">A128</font></i>: 
<font color="Maroon" title="c47">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:118_7_3"/>
<font color="Maroon" title="c45">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:118"><i><font color="Green" title="E158">A85</font></i></a>, <a class="txt" href="chord.html#E1:118_7_3"><i><font color="Green" title="E163">A126</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:118_7_3"/><b>then </b>
( <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> &amp; not <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">Pag</font> )
 <b>by </b><i><a class="txt" href="chord.html#E113:118"><i><font color="Green" title="E153">A80</font></i></a>, <a class="txt" href="chord.html#E123:118"><i><font color="Green" title="E162">A89</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c48">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:118_7_3"/><i><font color="Green" title="E166">A129</font></i>: 
<font color="Maroon" title="c48">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>and </b><br/><a NAME="E9:118_7_3"/><i><font color="Green" title="E167">A130</font></i>: 
<font color="Maroon" title="c45">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c48">n1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E10:118_7_3"/><i><font color="Green" title="E168">A131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c48">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E8:118_7_3"><i><font color="Green" title="E166">A129</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E11:118_7_3"/><i><font color="Green" title="E169">A132</font></i>: 
<font color="Maroon" title="c47">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c48">n1</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:118"><i><font color="Green" title="E158">A85</font></i></a>, <a class="txt" href="chord.html#E3:118_7_3"><i><font color="Green" title="E164">A127</font></i></a>, <a class="txt" href="chord.html#E9:118_7_3"><i><font color="Green" title="E167">A130</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:118_7_3"/><b>then </b>
<font color="Maroon" title="c47">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:118_7_3"><i><font color="Green" title="E166">A129</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:118_7_3"/><b>then </b><i><font color="Green" title="E170">A133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E4:118_7_3"><i><font color="Green" title="E165">A128</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E14:118_7_3"/>
( <font color="Maroon" title="c48">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> &amp; <font color="Maroon" title="c47">m1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E3:118_7_3"><i><font color="Green" title="E164">A127</font></i></a>, <a class="txt" href="chord.html#E9:118_7_3"><i><font color="Green" title="E167">A130</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c49">n1</font> = <font color="Maroon" title="c48">n1</font>, <font color="Maroon" title="c50">m1</font> = <font color="Maroon" title="c47">m1</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c51">m11</font> = <font color="Maroon" title="c50">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1, <font color="Maroon" title="c52">n11</font> = <font color="Maroon" title="c49">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E17:118_7_3"/><i><font color="Green" title="E171">A134</font></i>: 
<font color="Maroon" title="c52">n11</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:118_7_3"><i><font color="Green" title="E166">A129</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:118_7_3"/>
<font color="Maroon" title="c51">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">n11</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:118_7_3"><i><font color="Green" title="E169">A132</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E19:118_7_3"/><b>then </b><i><font color="Green" title="E172">A135</font></i>: 
<font color="Maroon" title="c51">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E17:118_7_3"><i><font color="Green" title="E171">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E173">A136</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" title="E174">A137</font></i>: 
<font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c52">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" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n11</font> or <font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c52">n11</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E1:118_7_3_1"><i><font color="Green" title="E174">A137</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E175">A138</font></i>: 
<font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n11</font>
 ;<br/><div class="add"><a NAME="E2:118_7_3_1_1_1"/><b>then </b><i><font color="Green" title="E176">A139</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c51">m11</font>,<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c52">n11</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:118_7_3"><i><font color="Green" title="E172">A135</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E3:118_7_3_1_1_1"/><i><font color="Green" title="E177">A140</font></i>: 
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c51">m11</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E86:118"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="chord.html#E88:118"><i><font color="Green" title="E134">A46</font></i></a>, <a class="txt" href="chord.html#E92:118"><i><font color="Green" title="E138">A50</font></i></a>, <a class="txt" href="chord.html#E94:118"><i><font color="Green" title="E140">A52</font></i></a>, <a class="txt" href="chord.html#E19:118_7_3"><i><font color="Green" title="E172">A135</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E4:118_7_3_1_1_1"/><i><font color="Green" title="E178">A141</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c51">m11</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E5:118_7_3_1_1_1"/>
<font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:118_7_3"><i><font color="Green" title="E166">A129</font></i></a>, <a class="txt" href="chord.html#E1:118_7_3_1_1_1"><i><font color="Green" title="E175">A138</font></i></a></i>;<br/><a NAME="E6:118_7_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c51">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E7:118_7_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c51">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">Pag</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:118_7_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c42">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c42">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:118"><i><font color="Green" title="E144">A56</font></i></a>, <a class="txt" href="chord.html#E3:118_7_3_1_1_1"><i><font color="Green" title="E177">A140</font></i></a>, <a class="txt" href="chord.html#E4:118_7_3_1_1_1"><i><font color="Green" title="E178">A141</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><b>hence </b><a NAME="E9:118_7_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E116:118"><i><font color="Green" title="E156">A83</font></i></a>, <a class="txt" href="chord.html#E3:118_7_3"><i><font color="Green" title="E164">A127</font></i></a>, <a class="txt" href="chord.html#E9:118_7_3"><i><font color="Green" title="E167">A130</font></i></a>, <a class="txt" href="chord.html#E10:118_7_3"><i><font color="Green" title="E168">A131</font></i></a>, <a class="txt" href="chord.html#E13:118_7_3"><i><font color="Green" title="E170">A133</font></i></a>, <a class="txt" href="chord.html#E2:118_7_3_1_1_1"><i><font color="Green" title="E176">A139</font></i></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" title="c51">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c52">n11</font>
 ;<br/><div class="add"><a NAME="E2:118_7_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c51">m11</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c52">n11</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E3:118_7_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E11:118_7_3"><i><font color="Green" title="E169">A132</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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"/>
<font color="Maroon" title="c52">n11</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Pbg</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:118_7_3"><i><font color="Green" title="E166">A129</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:118_7_3"/><b>then </b>
<font color="Maroon" title="c42">Pbg</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E96:118"><i><font color="Green" title="E142">A54</font></i></a>, <a class="txt" href="chord.html#E115:118"><i><font color="Green" title="E155">A82</font></i></a>, <a class="txt" href="chord.html#E3:118_7_3"><i><font color="Green" title="E164">A127</font></i></a>, <a class="txt" href="chord.html#E9:118_7_3"><i><font color="Green" title="E167">A130</font></i></a>, <a class="txt" href="chord.html#E10:118_7_3"><i><font color="Green" title="E168">A131</font></i></a>, <a class="txt" href="chord.html#E13:118_7_3"><i><font color="Green" title="E170">A133</font></i></a>, <a class="txt" href="chord.html#E20:118_7_3"><i><font color="Green" title="E173">A136</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></i>;<br/><a NAME="E23:118_7_3"/><b>then </b>
<font color="Maroon" title="c40">Pb</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E60:118"><i><font color="Green" title="E121">A33</font></i></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></i>;<br/><b>hence </b><a NAME="E24:118_7_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E84:118"><i><font color="Green" title="E132">A44</font></i></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
