<?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>let </b><font color="Maroon" title="c2">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:117"/><i><font color="Green" title="E91">A1</font></i>: 
( not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c2">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:117"/><i><font color="Green" title="E92">A2</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E3:117"/><i><font color="Green" title="E93">A3</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E4:117"/><i><font color="Green" title="E94">A4</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</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="txt" href="chord.html#E1:117"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E2:117"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E3:117"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="glib_001.html#T152" title="GLIB_001:th.152">GLIB_001:152</a></i>;<br/>
<div><i><font color="Green" title="E95">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_1"/><i><font color="Green" title="E96">A6</font></i>: 
<font color="Maroon" title="c6">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="c2">P</font>
 ;<br/><a NAME="E2:117_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">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="c2">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a>, <a class="txt" href="chord.html#E1:117_1"><i><font color="Green" title="E96">A6</font></i></a></i>;<br/><a NAME="E3:117_1"/><b>then </b>
<font color="Maroon" title="c6">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="c2">P</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:117_1"/>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:117"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:117"/><i><font color="Green" title="E96">A7</font></i>: 
( <font color="Maroon" title="c5">Q</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="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c5">Q</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="c3">x</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E3:117"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></i>;<br/>
<a NAME="E8:117"/><b>then </b><i><font color="Green" title="E97">A8</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</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#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/>
<a NAME="E9:117"/><i><font color="Green" title="E98">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="chord.html#E3:117"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1 &amp; 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies for <font color="Olive" title="b1">j</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>  st <font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 holds <br/>for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> );<br/>
<a NAME="E10:117"/><i><font color="Green" title="E99">A10</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:117_2"/><i><font color="Green" title="E100">A11</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ;<br/>

<b>assume </b><a NAME="E2:117_2"/><i><font color="Green" title="E101">A12</font></i>: 
( 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font> &amp; 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">j</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:117_2"/><i><font color="Green" title="E102">A13</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/>

<b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E4:117_2"/><i><font color="Green" title="E103">A14</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E5:117_2"/><i><font color="Green" title="E104">A15</font></i>: 
( <font color="Maroon" title="c6">k</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="c7">j</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="E6:117_2"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E2:117_2"><i><font color="Green" title="E101">A12</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:117_2"/><b>then </b><i><font color="Green" title="E105">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4
 
<b>by </b><i><a class="txt" href="chord.html#E3:117_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:117_2"/><i><font color="Green" title="E106">A17</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c2">P</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/>
<a NAME="E9:117_2"/><b>then </b><i><font color="Green" title="E107">A18</font></i>: 
<font color="Maroon" title="c7">j</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="c2">P</font>
 
<b>by </b><i><a class="txt" href="chord.html#E7:117_2"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_2_1"/>
( <font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 or <font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> )
 </span><b>by </b><i><a class="txt" href="chord.html#E7:117_2"><i><font color="Green" title="E105">A16</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="117_2_1_1"><b>suppose </b></a><a NAME="E1:117_2_1_1"/>
<font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 ;<br/><div class="add"><a NAME="E2:117_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a>, <a class="txt" href="chord.html#E8:117_2"><i><font color="Green" title="E106">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:117_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:117_2"><i><font color="Green" title="E103">A14</font></i></a>, <a class="txt" href="chord.html#E4:117"><i><font color="Green" title="E94">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_2_1_2"><b>suppose </b></a><a NAME="E1:117_2_1_2"/><i><font color="Green" title="E108">A19</font></i>: 
<font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:117_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E3:117_2_1_2"/><b>then </b><i><font color="Green" title="E109">A20</font></i>: 
<font color="Maroon" title="c7">j</font> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">jj</font> = <font color="Maroon" title="c7">j</font> 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>  <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="c10">lP2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 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/><b>set </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font>;<br/><a NAME="E6:117_2_1_2"/><i><font color="Green" title="E110">A21</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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="txt" href="chord.html#E3:117_2_1_2"><i><font color="Green" title="E109">A20</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E7:117_2_1_2"/><i><font color="Green" title="E111">A22</font></i>: 
<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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:117_2_1_2"><i><font color="Green" title="E108">A19</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E8:117_2_1_2"/><b>then </b><i><font color="Green" title="E112">A23</font></i>: 
3 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E6:117_2_1_2"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:117_2_1_2"/><i><font color="Green" title="E113">A24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <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="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <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="c5">Q</font> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E3:117_2_1_2"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E10:117_2_1_2"/><i><font color="Green" title="E114">A25</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <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="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <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="c3">x</font> )
 <b>by </b><i><a class="txt" href="chord.html#E7:117"><i><font color="Green" title="E96">A7</font></i></a>, <a class="txt" href="chord.html#E3:117_2_1_2"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E11:117_2_1_2"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="glib_001.html#D8" title="GLIB_001:def.8">GLIB_001:def 8</a></i>;<br/><a NAME="E12:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:117_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E9:117_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="glib_001.html#T90" title="GLIB_001:th.90">GLIB_001:90</a></i>;<br/><a NAME="E13:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2"><i><font color="Green" title="E113">A24</font></i></a>, <a class="txt" href="chord.html#E9:117_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a></i>;<br/><a NAME="E14:117_2_1_2"/><b>then </b><i><font color="Green" title="E115">A26</font></i>: 
not <font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</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#E10:117_2_1_2"><i><font color="Green" title="E114">A25</font></i></a>, <a class="txt" href="chord.html#E2:117"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><div><i><font color="Green" title="E116">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">i</font> be  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_2_1_2_1"/><i><font color="Green" title="E117">A28</font></i>: 
<font color="Maroon" title="c12">i</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E2:117_2_1_2_1"/><i><font color="Green" title="E118">A29</font></i>: 
<font color="Maroon" title="c12">i</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="E3:117_2_1_2_1"/>
<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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E4:117_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E1:117_2_1_2_1"><i><font color="Green" title="E117">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:117_2_1_2_1"/><b>then </b><i><font color="Green" title="E119">A30</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</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="c12">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</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="c5">Q</font> )
 <b>by </b><i><a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="txt" href="chord.html#E3:117_2_1_2"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E2:117_2_1_2_1"><i><font color="Green" title="E118">A29</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E6:117_2_1_2_1"/><b>then </b><i><font color="Green" title="E120">A31</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 <b>by </b><i><a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:117_2_1_2_1"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></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> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:117_2_1_2_1"><i><font color="Green" title="E117">A28</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:117_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E6:117_2_1_2"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E9:117_2_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</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="c12">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a>, <a class="txt" href="chord.html#E5:117_2_1_2_1"><i><font color="Green" title="E119">A30</font></i></a>, <a class="txt" href="chord.html#E6:117_2_1_2_1"><i><font color="Green" title="E120">A31</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font>;<br/><a NAME="E16:117_2_1_2"/><i><font color="Green" title="E117">A32</font></i>: 
<font color="Maroon" title="c8">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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:117_2"><i><font color="Green" title="E103">A14</font></i></a>, <a class="txt" href="chord.html#E10:117_2_1_2"><i><font color="Green" title="E114">A25</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><div><i><font color="Green" title="E118">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_2_1_2_2"/><i><font color="Green" title="E119">A34</font></i>: 
<font color="Maroon" title="c13">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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a>, <a class="txt" href="chord.html#E1:117_2_1_2_2"><i><font color="Green" title="E119">A34</font></i></a></i>;<br/><a NAME="E3:117_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c13">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="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div><a NAME="E18:117_2_1_2"/><i><font color="Green" title="E119">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</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#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green" title="E112">A23</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green" title="E115">A26</font></i></a>, <a class="ref" href="chord.html#T33" target="_self" title="CHORD:th.33">Th33</a></i>;<br/><a NAME="E19:117_2_1_2"/><i><font color="Green" title="E120">A36</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <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="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <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="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2"><i><font color="Green" title="E113">A24</font></i></a>, <a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></i>;<br/><a NAME="E20:117_2_1_2"/><i><font color="Green" title="E121">A37</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/><a NAME="E21:117_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <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> 5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E7:117_2_1_2"><i><font color="Green" title="E111">A22</font></i></a>, <a class="txt" href="chord.html#E6:117_2_1_2"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E22:117_2_1_2"/><b>then </b><i><font color="Green" title="E122">A38</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 <b>by </b><i><a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ;<br/><a NAME="E2:117_2_1_2_3"/><b>then </b>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E3:117_2_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E22:117_2_1_2"><i><font color="Green" title="E122">A38</font></i></a>, <a class="ref" href="glib_001.html#T113" title="GLIB_001:th.113">GLIB_001:113</a></i>;<br/></div><b>end;</b></div><a NAME="E24:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E18:117_2_1_2"><i><font color="Green" title="E119">A35</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="c13">m</font>, <font color="Maroon" title="c14">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="E26:117_2_1_2"/><i><font color="Green" title="E123">A39</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> )
 <b>and </b><br/><a NAME="E27:117_2_1_2"/><i><font color="Green" title="E124">A40</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E28:117_2_1_2"/><i><font color="Green" title="E125">A41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> ) &amp; (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c14">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E30:117_2_1_2"/><i><font color="Green" title="E126">A42</font></i>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E27:117_2_1_2"><i><font color="Green" title="E124">A40</font></i></a></i>;<br/><a NAME="E31:117_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a></i>;<br/><a NAME="E32:117_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c16">m1</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E35:117_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a></i>;<br/><a NAME="E36:117_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c18">n1</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E39:117_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E40:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green" title="E123">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E41:117_2_1_2"/><b>then </b><i><font color="Green" title="E127">A43</font></i>: 
<font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E42:117_2_1_2"/><i><font color="Green" title="E128">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green" title="E123">A39</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E43:117_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green" title="E123">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="chord.html#E20:117_2_1_2"><i><font color="Green" title="E121">A37</font></i></a></i>;<br/><a NAME="E44:117_2_1_2"/><b>then </b><i><font color="Green" title="E129">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E45:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">m1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></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#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E46:117_2_1_2"/><b>then </b><i><font color="Green" title="E130">A46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</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="c17">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green" title="E116">A27</font></i></a></i>;<br/><a NAME="E47:117_2_1_2"/><b>then </b><i><font color="Green" title="E131">A47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_4"/><i><font color="Green" title="E132">A48</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:117_2_1_2"><i><font color="Green" title="E120">A36</font></i></a>, <a class="txt" href="chord.html#E30:117_2_1_2"><i><font color="Green" title="E126">A42</font></i></a>, <a class="txt" href="chord.html#E46:117_2_1_2"><i><font color="Green" title="E130">A46</font></i></a>, <a class="txt" href="chord.html#E17:117_2_1_2"><i><font color="Green" title="E118">A33</font></i></a>, <a class="txt" href="chord.html#E44:117_2_1_2"><i><font color="Green" title="E129">A45</font></i></a></i>;<br/><a NAME="E3:117_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a>, <a class="txt" href="chord.html#E9:117_2"><i><font color="Green" title="E107">A18</font></i></a></i>;<br/><a NAME="E4:117_2_1_2_4"/><b>then </b><i><font color="Green" title="E133">A49</font></i>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_4_1"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ;<br/><a NAME="E2:117_2_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:117_2_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:117_2_1_2_4"><i><font color="Green" title="E132">A48</font></i></a>, <a class="txt" href="chord.html#E28:117_2_1_2"><i><font color="Green" title="E125">A41</font></i></a>, <a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green" title="E112">A23</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green" title="E115">A26</font></i></a>, <a class="ref" href="chord.html#T33" target="_self" title="CHORD:th.33">Th33</a>, <a class="ref" href="chord.html#T7" target="_self" title="CHORD:th.7">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E6:117_2_1_2_4"/><b>then </b><i><font color="Green" title="E134">A50</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:117_2_1_2_4"/><b>then </b><i><font color="Green" title="E135">A51</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font>
 <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="E8:117_2_1_2_4"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:117_2_1_2_4"><i><font color="Green" title="E134">A50</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E9:117_2_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:117_2_1_2_4"><i><font color="Green" title="E133">A49</font></i></a>, <a class="txt" href="chord.html#E7:117_2_1_2_4"><i><font color="Green" title="E135">A51</font></i></a>, <a class="txt" href="chord.html#E1:117"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E47:117_2_1_2"><i><font color="Green" title="E131">A47</font></i></a>, <a class="ref" href="chord.html#T92" target="_self" title="CHORD:th.92">Th92</a></i>;<br/></div><b>end;</b></div><a NAME="E49:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">n</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <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#E20:117_2_1_2"><i><font color="Green" title="E121">A37</font></i></a>, <a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green" title="E123">A39</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E50:117_2_1_2"/><b>then </b><i><font color="Green" title="E132">A52</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5"/><i><font color="Green" title="E133">A53</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 ;<br/><a NAME="E2:117_2_1_2_5"/><b>then </b><i><font color="Green" title="E134">A54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E10:117_2_1_2"><i><font color="Green" title="E114">A25</font></i></a>, <a class="txt" href="chord.html#E17:117_2_1_2"><i><font color="Green" title="E118">A33</font></i></a></i>;<br/><a NAME="E3:117_2_1_2_5"/>
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> )
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a>, <a class="txt" href="chord.html#E1:117_2_1_2_5"><i><font color="Green" title="E133">A53</font></i></a>, <a class="txt" href="chord.html#E20:117_2_1_2"><i><font color="Green" title="E121">A37</font></i></a>, <a class="txt" href="chord.html#E28:117_2_1_2"><i><font color="Green" title="E125">A41</font></i></a>, <a class="txt" href="chord.html#E16:117_2_1_2"><i><font color="Green" title="E117">A32</font></i></a>, <a class="txt" href="chord.html#E8:117_2_1_2"><i><font color="Green" title="E112">A23</font></i></a>, <a class="txt" href="chord.html#E14:117_2_1_2"><i><font color="Green" title="E115">A26</font></i></a>, <a class="ref" href="chord.html#T33" target="_self" title="CHORD:th.33">Th33</a></i>;<br/><a NAME="E4:117_2_1_2_5"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:117_2_1_2_5"/><b>then </b><i><font color="Green" title="E135">A55</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:117_2_1_2_5"/>
( <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 <b>by </b><i><a class="txt" href="chord.html#E46:117_2_1_2"><i><font color="Green" title="E130">A46</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E7:117_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">kk</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:117_2_1_2_5"/><i><font color="Green" title="E136">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/><div><i><font color="Green" title="E137">A57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_1"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:117_2_1_2_5_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="E3:117_2_1_2_5_1"/><b>then </b>
( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="chord.html#T11" target="_self" title="CHORD:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:117_2_1_2_5"><i><font color="Green" title="E133">A53</font></i></a>, <a class="txt" href="chord.html#E26:117_2_1_2"><i><font color="Green" title="E123">A39</font></i></a>, <a class="txt" href="chord.html#E6:117_2_1_2"><i><font color="Green" title="E110">A21</font></i></a>, <a class="txt" href="chord.html#E9:117_2_1_2_5"><i><font color="Green" title="E136">A56</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E138">A58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:117_2_1_2_5_2"/><b>then </b><i><font color="Green" title="E139">A59</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <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="E3:117_2_1_2_5_2"/>
( <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> )
 <b>by </b><i><a class="txt" href="chord.html#E5:117_2_1_2_5"><i><font color="Green" title="E135">A55</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2_5"><i><font color="Green" title="E136">A56</font></i></a>, <a class="txt" href="chord.html#E2:117_2_1_2_5_2"><i><font color="Green" title="E139">A59</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E139">A60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_2_1_2_5_3"/>
<font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ;<br/><a NAME="E2:117_2_1_2_5_3"/><b>then </b><i><font color="Green" title="E140">A61</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E3:117_2_1_2_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E5:117_2_1_2_5"><i><font color="Green" title="E135">A55</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E4:117_2_1_2_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E9:117_2_1_2_5"><i><font color="Green" title="E136">A56</font></i></a>, <a class="txt" href="chord.html#E3:117_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E2:117_2_1_2_5_3"><i><font color="Green" title="E140">A61</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E13:117_2_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E44:117_2_1_2"><i><font color="Green" title="E129">A45</font></i></a>, <a class="txt" href="chord.html#E17:117_2_1_2"><i><font color="Green" title="E118">A33</font></i></a></i>;<br/><a NAME="E14:117_2_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E47:117_2_1_2"><i><font color="Green" title="E131">A47</font></i></a>, <a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E15:117_2_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E12:117_2_1_2_5"><i><font color="Green" title="E139">A60</font></i></a>, <a class="txt" href="chord.html#E1:117_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="txt" href="chord.html#E9:117_2_1_2_5"><i><font color="Green" title="E136">A56</font></i></a>, <a class="txt" href="chord.html#E10:117_2_1_2_5"><i><font color="Green" title="E137">A57</font></i></a>, <a class="txt" href="chord.html#E11:117_2_1_2_5"><i><font color="Green" title="E138">A58</font></i></a>, <a class="txt" href="chord.html#E2:117_2_1_2_5"><i><font color="Green" title="E134">A54</font></i></a>, <a class="txt" href="chord.html#E30:117_2_1_2"><i><font color="Green" title="E126">A42</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E52:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E50:117_2_1_2"><i><font color="Green" title="E132">A52</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E53:117_2_1_2"/><b>then </b>
<font color="Maroon" title="c19">n1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span></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#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E54:117_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</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="c19">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E15:117_2_1_2"><i><font color="Green" title="E116">A27</font></i></a></i>;<br/><a NAME="E55:117_2_1_2"/><b>then </b><i><font color="Green" title="E133">A62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</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="c2">P</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="E56:117_2_1_2"/><i><font color="Green" title="E134">A63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c9">jj</font>,<font color="Maroon" title="c10">lP2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E44:117_2_1_2"><i><font color="Green" title="E129">A45</font></i></a>, <a class="txt" href="chord.html#E50:117_2_1_2"><i><font color="Green" title="E132">A52</font></i></a>, <a class="txt" href="chord.html#E17:117_2_1_2"><i><font color="Green" title="E118">A33</font></i></a></i>;<br/><a NAME="E57:117_2_1_2"/><i><font color="Green" title="E135">A64</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="txt" href="chord.html#E41:117_2_1_2"><i><font color="Green" title="E127">A43</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E58:117_2_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="txt" href="chord.html#E42:117_2_1_2"><i><font color="Green" title="E128">A44</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E59:117_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 ;<br/><b>hence </b><a NAME="E60:117_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:117"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E55:117_2_1_2"><i><font color="Green" title="E133">A62</font></i></a>, <a class="txt" href="chord.html#E56:117_2_1_2"><i><font color="Green" title="E134">A63</font></i></a>, <a class="txt" href="chord.html#E30:117_2_1_2"><i><font color="Green" title="E126">A42</font></i></a>, <a class="txt" href="chord.html#E46:117_2_1_2"><i><font color="Green" title="E130">A46</font></i></a>, <a class="txt" href="chord.html#E57:117_2_1_2"><i><font color="Green" title="E135">A64</font></i></a>, <a class="ref" href="chord.html#T92" target="_self" title="CHORD:th.92">Th92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:117"/><i><font color="Green" title="E100">A65</font></i>: 
for <font color="Olive" title="b1">k</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>   st ( for <font color="Olive" title="b2">a</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>   st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <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>assume </b><a NAME="E1:117_3"/><i><font color="Green" title="E101">A66</font></i>: 
for <font color="Olive" title="b1">a</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>   st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ;<br/>

<a NAME="E2:117_3"/>
for <font color="Olive" title="b1">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:117_3_1"/><i><font color="Green" title="E102">A67</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ;<br/>

<a NAME="E2:117_3_1"/>
<font color="Maroon" title="c7">a</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/>
<b>hence </b><a NAME="E3:117_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">a</font>]
 <b>by </b><i><a class="txt" href="chord.html#E1:117_3"><i><font color="Green" title="E101">A66</font></i></a>, <a class="txt" href="chord.html#E1:117_3_1"><i><font color="Green" title="E102">A67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:117_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 <b>by </b><i><a class="txt" href="chord.html#E10:117"><i><font color="Green" title="E99">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:117"/><i><font color="Green" title="E101">A68</font></i>: 
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="chord.html#E11:117"><i><font color="Green" title="E100">A65</font></i></a>);<br/></i>
<div><i><font color="Green" title="E102">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_4"/><i><font color="Green" title="E103">A70</font></i>: 
<font color="Maroon" title="c6">n</font> <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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 ;<br/><a NAME="E2:117_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T33" title="XREAL_1:th.33">XREAL_1:33</a></i>;<br/><a NAME="E3:117_4"/><b>then </b>
<font color="Maroon" title="c6">n</font> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <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#E1:117_4"><i><font color="Green" title="E103">A70</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:117_4"/><i><font color="Green" title="E104">A71</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/><a NAME="E6:117_4"/><i><font color="Green" title="E105">A72</font></i>: 
<font color="Maroon" title="c7">k</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/><div><i><font color="Green" title="E106">A73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_4_1"/><i><font color="Green" title="E107">A74</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 ;<br/><a NAME="E2:117_4_1"/>
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <b>by </b><i><a class="txt" href="chord.html#E1:117_4"><i><font color="Green" title="E103">A70</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:117_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:117_4"><i><font color="Green" title="E104">A71</font></i></a>, <a class="txt" href="chord.html#E1:117_4_1"><i><font color="Green" title="E107">A74</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_4_2"/><i><font color="Green" title="E107">A75</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:117_4_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T2" target="_self" title="CHORD:th.2">Th2</a></i>;<br/><a NAME="E3:117_4_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_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="c2">P</font></span>)</span> <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#E1:117_4_2"><i><font color="Green" title="E107">A75</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:117_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:117_4"><i><font color="Green" title="E104">A71</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:117_4"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:117_4"><i><font color="Green" title="E104">A71</font></i></a>, <a class="txt" href="chord.html#E7:117_4"><i><font color="Green" title="E106">A73</font></i></a>, <a class="txt" href="chord.html#E12:117"><i><font color="Green" title="E101">A68</font></i></a>, <a class="txt" href="chord.html#E6:117_4"><i><font color="Green" title="E105">A72</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_5"/>
<font color="Maroon" title="c5">Q</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">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="E3:117_5"/><i><font color="Green" title="E103">A76</font></i>: 
( <font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c7">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="c5">Q</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>and </b><br/><a NAME="E4:117_5"/><i><font color="Green" title="E104">A77</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> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E5:117_5"/>
( <font color="Maroon" title="c5">Q</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c7">n</font> <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="c5">Q</font> ) &amp; (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c7">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="c5">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c7">n</font> <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="c5">Q</font> ) ) )
 <b>by </b><i><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:117_5"/><i><font color="Green" title="E105">A78</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:117_5"><i><font color="Green" title="E104">A77</font></i></a></i>;<br/><a NAME="E8:117_5"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E9:117_5"/><b>then </b><i><font color="Green" title="E106">A79</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:117_5"><i><font color="Green" title="E103">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:117_5"/>
<font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E3:117_5"><i><font color="Green" title="E103">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a></i>;<br/><a NAME="E11:117_5"/><b>then </b><i><font color="Green" title="E107">A80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_5_1"/>
( <font color="Maroon" title="c7">n</font> <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="c5">Q</font> or <font color="Maroon" title="c7">n</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="c5">Q</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E3:117_5"><i><font color="Green" title="E103">A76</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="117_5_1_1"><b>suppose </b></a><a NAME="E1:117_5_1_1"/><i><font color="Green" title="E108">A81</font></i>: 
<font color="Maroon" title="c7">n</font> <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="c5">Q</font>
 ;<br/><div class="add"><a NAME="E2:117_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="txt" href="chord.html#E3:117_5"><i><font color="Green" title="E103">A76</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E3:117_5_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E4:117_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E13:117"><i><font color="Green" title="E102">A69</font></i></a>, <a class="txt" href="chord.html#E7:117_5"><i><font color="Green" title="E105">A78</font></i></a>, <a class="txt" href="chord.html#E1:117_5_1_1"><i><font color="Green" title="E108">A81</font></i></a>, <a class="txt" href="chord.html#E7:117"><i><font color="Green" title="E96">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_5_1_2"><b>suppose </b></a><a NAME="E1:117_5_1_2"/>
<font color="Maroon" title="c7">n</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="c5">Q</font>
 ;<br/><div class="add"><a NAME="E2:117_5_1_2"/><b>then </b><i><font color="Green" title="E108">A82</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E9:117"><i><font color="Green" title="E98">A9</font></i></a>, <a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><a NAME="E3:117_5_1_2"/><b>then </b>
( <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:117"><i><font color="Green" title="E95">A5</font></i></a>, <a class="txt" href="chord.html#E11:117_5"><i><font color="Green" title="E107">A80</font></i></a></i>;<br/><b>hence </b><a NAME="E4:117_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:117_5"><i><font color="Green" title="E103">A76</font></i></a>, <a class="txt" href="chord.html#E7:117_5"><i><font color="Green" title="E105">A78</font></i></a>, <a class="txt" href="chord.html#E1:117"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="chord.html#E9:117_5"><i><font color="Green" title="E106">A79</font></i></a>, <a class="txt" href="chord.html#E2:117_5_1_2"><i><font color="Green" title="E108">A82</font></i></a>, <a class="ref" href="chord.html#T92" target="_self" title="CHORD:th.92">Th92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:117"/>
( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 <b>by </b><i><a class="txt" href="chord.html#E8:117"><i><font color="Green" title="E97">A8</font></i></a>, <a class="txt" href="chord.html#E2:117"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="chord.html#E7:117"><i><font color="Green" title="E96">A7</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>


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