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

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>given </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:101"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:101"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:101"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E4:101"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">P</font> is <a href="glib_001.html#V8">Cycle-like</a> implies ( (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) ) )
 ;<br/>

<b>take </b>
<font color="Maroon">m</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D10" target="_self">CHORD:def 10</a><br/>

<b>take </b>
<font color="Maroon">n</font>
;<br/>

<a NAME="E5:101"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:101"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>thus </b><a NAME="E7:101"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green">E83</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:101"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a></i>;<br/>

<a NAME="E9:101"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:101_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:101_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:101"/><b>then </b><i><font color="Green">E68</font></i>: 
not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:101_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:101_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, , <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_2"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><a NAME="E4:101_2"/><b>then </b>
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E5:101_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a>, , , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E13:101"/>
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E33:101"><i><font color="Green">E85</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E14:101"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>and </b><br/><a NAME="E15:101"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:101"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>and </b><br/><a NAME="E18:101"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E19:101"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:101"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D15">GLIB_001:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1;<br/>
<a NAME="E21:101"/>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E22:101"/><b>then </b>
2 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E23:101"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E24:101"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E26:101"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font>
 
;<br/>
<a NAME="E27:101"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E28:101"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T79">GLIB_001:79</a></i>;<br/>
<a NAME="E29:101"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E30:101"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E31:101"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/>
<a NAME="E32:101"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E33:101"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3"/>
( ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> ) )
 </span><b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, , , , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1"><b>suppose </b></a><a NAME="E1:101_3_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_1"><b>suppose </b></a><a NAME="E1:101_3_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2"><b>suppose </b></a><a NAME="E1:101_3_1_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_1_1_2_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:101_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E4:101_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E5:101_3_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E31:101"><i><font color="Green">E83</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:101_3_1_1_2_1_3"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:101_3_1_1_2_1_3"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E4:101_3_1_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_3"><b>suppose </b></a><a NAME="E1:101_3_1_1_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2"><b>suppose </b></a><a NAME="E1:101_3_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:101_3_2_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1_1_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:101_3_2_1_1_1_2"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><a NAME="E3:101_3_2_1_1_1_2"/><b>then </b>
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E4:101_3_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_3"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:101_3_2_1_1_1_3"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_2_1_1_1_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:101_3_2_1_1_1_3"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><a NAME="E5:101_3_2_1_1_1_3"/><b>then </b>
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E6:101_3_2_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E32:101"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a>, , , , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1_2_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_3"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:101_3_2_1_2_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_2_1_2_1_3"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> )
 ;<br/><a NAME="E4:101_3_2_1_2_1_3"/><b>then </b>
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E5:101_3_2_1_2_1_3"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
