<?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>, <font color="Maroon">H</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E33</font></i>: 
( not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a> &amp; not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">H</font> is <a href="glib_001.html#V3">trivial</a> &amp; not <font color="Maroon">H</font> is <a href="glib_001.html#V1">closed</a> )
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E5:30"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">H</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">H</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  )
 ;<br/>

<b>set </b><font color="Maroon">J</font> = <font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font>;<br/>
<a NAME="E6:30"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T29">GLIB_001:29</a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:30_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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="heyting3.html#T1">HEYTING3:1</a>, <a class="txt" href="chord.html#E2:30"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E3:30_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <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">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T33">GLIB_001:33</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:30"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">m</font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Olive">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> holds <br/>( <font color="Olive">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> )
 ;<br/>

<a NAME="E2:30_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
( <font color="Maroon">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">m</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="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T31">GLIB_001:31</a></i>;<br/>
<a NAME="E10:30"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:30_3"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T29">GLIB_001:29</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_3_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:30_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E3:30_3_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:30_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span> <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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:30_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:30_3_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>hence </b><a NAME="E7:30_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:30_3"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T1" target="_self">Th1</a>, , <a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:30"/><b>then </b><i><font color="Green">E74</font></i>: 
not <font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/>
<a NAME="E13:30"/><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_001.html#K14">.edges()</a>  &amp; <font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">H</font> <a href="glib_001.html#K12">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="glib_001.html#D27">GLIB_001:def 27</a></i>;<br/>
<a NAME="E14:30"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E15:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="glib_001.html#K12">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T86">GLIB_001:86</a></i>;<br/>
<a NAME="E16:30"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font> is <a href="glib_001.html#V4">Trail-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D27">GLIB_001:def 27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:30_4"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> )
 ;<br/><a NAME="E2:30_4"/><i><font color="Green">E81</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_4_1"/>
( ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_1"><b>suppose </b></a><a NAME="E1:30_4_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><a NAME="E2:30_4_1_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/><b>hence </b><a NAME="E3:30_4_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> implies ( <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2"><b>suppose </b></a><a NAME="E1:30_4_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><a NAME="E2:30_4_1_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <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">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E3:30_4_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="txt" href="chord.html#E2:30"><i><font color="Green">E33</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">j</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="E5:30_4_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp; <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="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E6:30_4_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_2_1"/>
not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:30_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">jj</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E10:30_4_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:30_4_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">j1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_2_2"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:30_4_1_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E3:30_4_1_2_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_2_2_1"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <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="glib_001.html#K4">.last()</a> 
 ;<br/><a NAME="E2:30_4_1_2_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:30_4_1_2_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:30_4_1_2_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:30_4_1_2_2_1"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/><b>hence </b><a NAME="E6:30_4_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a>, <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:30_4_1_2_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <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="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T12" target="_self">Th12</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:30_4_1_2_2"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="chord.html#T12" target="_self">Th12</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_2_2_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&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#K7">.append</a> <font color="Maroon">H</font></span>)</span>
 ;<br/><a NAME="E2:30_4_1_2_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</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">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a>, </i>;<br/><a NAME="E3:30_4_1_2_2_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T2" target="_self">Th2</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:30_4_1_2_2_2"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:30_4_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="txt" href="chord.html#E3:30"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_2_2_3"/><i><font color="Green">E124</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:30_4_1_2_2_3"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, , , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/><b>hence </b><a NAME="E3:30_4_1_2_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, , <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:30_4_1_2_2"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:30_4_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> implies ( <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_3"><b>suppose </b></a><a NAME="E1:30_4_1_3"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_4_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> implies ( <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_4"><b>suppose </b></a><a NAME="E1:30_4_1_4"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><a NAME="E2:30_4_1_4"/>
( <font color="Maroon">m</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#K7">.append</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">m</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="chord.html#T4" target="_self">Th4</a>, , <a class="txt" href="chord.html#E2:30"><i><font color="Green">E33</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:30_4_1_4"/><i><font color="Green">E126</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">H</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E5:30_4_1_4"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="txt" href="chord.html#E2:30"><i><font color="Green">E33</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">j</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="E7:30_4_1_4"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp; <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="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35">GLIB_001:35</a></i>;<br/><a NAME="E8:30_4_1_4"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><a NAME="E9:30_4_1_4"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_001.html#T34">GLIB_001:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_4_1"/>
not <font color="Maroon">k</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:30_4_1_4_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">kk</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4_1_4_2"/>
not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:30_4_1_4_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_4_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">jj</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E16:30_4_1_4"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="chord.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E17:30_4_1_4"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:30_4_1_4"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E19:30_4_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> implies ( <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> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E18:30"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font> is <a href="glib_001.html#V5">Path-like</a>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/>
<b>hence </b><a NAME="E19:30"/>
<font color="Maroon">P</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">H</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>


</div>
