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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">H</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><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">H</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E27">A2</font></i>: 
( not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E28">A3</font></i>: 
( not <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> )
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E29">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E5:30"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c3">H</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#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c3">H</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="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">J</font> = <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E6:30"/><i><font color="Green" title="E31">A6</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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T29" title="GLIB_001:th.29">GLIB_001:29</a></i>;<br/>
<div><i><font color="Green" title="E32">A7</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" title="c5">m</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="E1:30_1"/>
<font color="Maroon" title="c5">m</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="E2:30_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E3:30_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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="E4:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c5">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:30"/><i><font color="Green" title="E33">A8</font></i>: 
for <font color="Olive" title="b1">m</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">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">m</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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> holds <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp; not <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">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" title="c5">m</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="E1:30_2"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">m</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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 ;<br/>

<a NAME="E2:30_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="txt" href="chord.html#E1:30_2"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp; not <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:30_2"><i><font color="Green" title="E34">A9</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>
<a NAME="E9:30"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</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="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</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="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></i>;<br/>
<a NAME="E10:30"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:30_3"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T29" title="GLIB_001:th.29">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" title="E36">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:30_3_1"/>
1 <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></i>;<br/><a NAME="E3:30_3_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:30_3_1"><i><font color="Green" title="E36">A12</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="E4:30_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font></span>)</span> <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#E2:30_3"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" 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="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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></span>)</span> <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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E6:30_3_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 &amp; 0 <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="c3">H</font> )
 ;<br/><b>hence </b><a NAME="E7:30_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:30_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="chord.html#E2:30_3"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:30"/><b>then </b><i><font color="Green" title="E35">A13</font></i>: 
not <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/>
<a NAME="E13:30"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">H</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="glib_001.html#D27" title="GLIB_001:def.27">GLIB_001:def 27</a></i>;<br/>
<a NAME="E14:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="chord.html#E1:30"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E15:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T86" title="GLIB_001:th.86">GLIB_001:86</a></i>;<br/>
<a NAME="E16:30"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D27" title="GLIB_001:def.27">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" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be  <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>assume </b><a NAME="E1:30_4"/><i><font color="Green" title="E37">A15</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 ;<br/><a NAME="E2:30_4"/><i><font color="Green" title="E38">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</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="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" title="c5">m</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; <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> ) or ( <font color="Maroon" title="c5">m</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; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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" title="c5">m</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; <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/><div class="add"><a NAME="E2:30_4_1_1"/><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="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">n</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <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="c2">P</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#E2:30"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="chord.html#E7:30"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">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" title="E39">A17</font></i>: 
( <font color="Maroon" title="c5">m</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; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> )
 ;<br/><div class="add"><a NAME="E2:30_4_1_2"/><b>then </b><i><font color="Green" title="E40">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c5">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:30"><i><font color="Green" title="E32">A7</font></i></a></i>;<br/><a NAME="E3:30_4_1_2"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp; not <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="txt" href="chord.html#E1:30_4_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="chord.html#E8:30"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:30_4_1_2"/><i><font color="Green" title="E41">A19</font></i>: 
( <font color="Maroon" title="c7">j</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="c3">H</font> &amp; <font color="Maroon" title="c6">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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E6:30_4_1_2"/><i><font color="Green" title="E42">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <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="c3">H</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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">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" title="c7">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">j</font> as  <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/><a NAME="E3:30_4_1_2_1"/>
<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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">jj</font> = <font color="Maroon" title="c7">j</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/><b>reconsider </b><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c8">jj</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <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/><a NAME="E10:30_4_1_2"/><i><font color="Green" title="E43">A21</font></i>: 
<font color="Maroon" title="c9">j1</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="c3">H</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:30_4_1_2"/><i><font color="Green" title="E44">A22</font></i>: 
<font color="Maroon" title="c9">j1</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><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" title="E45">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 ;<br/><a NAME="E2:30_4_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E1:30_4_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="chord.html#E2:30_4_1_2"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="chord.html#E10:30_4_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="chord.html#E6:30_4_1_2"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E3:30_4_1_2_2"/><b>then </b><i><font color="Green" title="E46">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green" title="E47">A25</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" title="E48">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E2:30_4_1_2_2_1"/>
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:30_4_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/><a NAME="E3:30_4_1_2_2_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</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="E4:30_4_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c3">H</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">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:30_4_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="chord.html#E10:30_4_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="chord.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">GLIB_001:148</a></i>;<br/><b>hence </b><a NAME="E5:30_4_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:30_4_1_2_2"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a>, <a class="txt" href="chord.html#E1:30_4_1_2_2_1"><i><font color="Green" title="E48">A26</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" title="E48">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <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> 
 <b>by </b><i><a class="txt" href="chord.html#E3:30_4_1_2_2"><i><font color="Green" title="E46">A24</font></i></a>, <a class="txt" href="chord.html#E4:30"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:30_4_1_2_2"/><i><font color="Green" title="E49">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <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="c3">H</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="chord.html#E1:30_4_1_2_2"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="chord.html#E4:30_4_1_2_2"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="chord.html#E3:30_4_1_2_2"><i><font color="Green" title="E46">A24</font></i></a>, <a class="txt" href="chord.html#E4:30"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div><i><font color="Green" title="E50">A29</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" title="E51">A30</font></i>: 
<font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span>
 ;<br/><a NAME="E2:30_4_1_2_2_2"/><i><font color="Green" title="E52">A31</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a>, <a class="txt" href="chord.html#E6:30_4_1_2_2"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/><a NAME="E3:30_4_1_2_2_2"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:30_4_1_2_2_2"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="chord.html#E6:30"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="chord.html#E5:30_4_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/><a NAME="E4:30_4_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c9">j1</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="c3">H</font>
 <b>by </b><i><a class="txt" href="chord.html#E10:30_4_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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="txt" href="chord.html#E11:30_4_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="chord.html#E2:30_4_1_2_2_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="chord.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 ;<br/><a NAME="E2:30_4_1_2_2_3"/><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="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</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#E1:30_4_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="chord.html#E2:30"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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="txt" href="chord.html#E1:30_4_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="chord.html#E7:30"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="chord.html#E5:30_4_1_2_2"><i><font color="Green" title="E48">A27</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:30_4_1_2_2"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E7:30_4_1_2_2"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="chord.html#E2:30_4"><i><font color="Green" title="E38">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><b>end;</b></div><b>hence </b><a NAME="E13:30_4_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">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" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_4_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</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><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" title="E39">A32</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> )
 ;<br/><div class="add"><a NAME="E2:30_4_1_4"/><b>then </b>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp; not <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:30_4"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="chord.html#E8:30"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E4:30_4_1_4"/><i><font color="Green" title="E40">A33</font></i>: 
( <font color="Maroon" title="c7">k</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="c3">H</font> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">k</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E5:30_4_1_4"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp; not <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="txt" href="chord.html#E1:30_4_1_4"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="chord.html#E8:30"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:30_4_1_4"/><i><font color="Green" title="E41">A34</font></i>: 
( <font color="Maroon" title="c8">j</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="c3">H</font> &amp; <font color="Maroon" title="c6">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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><a NAME="E8:30_4_1_4"/><i><font color="Green" title="E42">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E4:30_4_1_4"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E9:30_4_1_4"/><i><font color="Green" title="E43">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E7:30_4_1_4"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="chord.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">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" title="c7">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c7">k</font> as  <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/><a NAME="E3:30_4_1_4_1"/>
<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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:30_4_1_4"><i><font color="Green" title="E40">A33</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">kk</font> = <font color="Maroon" title="c7">k</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/><b>reconsider </b><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c9">kk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <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/><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" title="c8">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c8">j</font> as  <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/><a NAME="E3:30_4_1_4_2"/>
<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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:30_4_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E7:30_4_1_4"><i><font color="Green" title="E41">A34</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">jj</font> = <font color="Maroon" title="c8">j</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/><b>reconsider </b><font color="Maroon" title="c12">j1</font> = <font color="Maroon" title="c11">jj</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E16:30_4_1_4"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="txt" href="chord.html#E4:30_4_1_4"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="chord.html#E7:30_4_1_4"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="chord.html#E1:30_4"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/><a NAME="E17:30_4_1_4"/><b>then </b><i><font color="Green" title="E44">A37</font></i>: 
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j1</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="E18:30_4_1_4"/>
<font color="Maroon" title="c12">j1</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="c3">H</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:30_4_1_4"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E19:30_4_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">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="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="chord.html#E8:30_4_1_4"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="chord.html#E9:30_4_1_4"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="chord.html#E4:30_4_1_4"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="chord.html#E7:30_4_1_4"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="chord.html#E17:30_4_1_4"><i><font color="Green" title="E44">A37</font></i></a>, <a class="txt" href="chord.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">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" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 
<b>by </b><i><a class="txt" href="chord.html#E16:30"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/>
<b>hence </b><a NAME="E19:30"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">H</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#E10:30"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="chord.html#E12:30"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/>


</div>
