<?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">W1</font>, <font color="Maroon" title="c3">W2</font> be   <a href="glib_001.html#NM4" title="GLIB_001:NM.4">Trail</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">W1</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">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">W</font> = <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><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#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:18_1"/><i><font color="Green" title="E17">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 <b>and </b><br/><a NAME="E2:18_1"/><i><font color="Green" title="E18">A4</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>
 <b>and </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 ;<br/><a NAME="E4:18_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="helly.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="helly.html#E2:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:18_1"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
<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">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E3:18_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:18_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E2:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="helly.html#E3:18_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:18_1"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
<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">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <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">W1</font> or  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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">W2</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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) )
 </span><b>by </b><i><a class="txt" href="helly.html#E5:18_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
<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">W1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:18_1_1_1"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c2">W1</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">W1</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="helly.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="helly.html#E2:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/><a NAME="E4:18_1_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">W1</font>
 <b>by </b><i><a class="txt" href="helly.html#E2:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="helly.html#E2:18_1_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:18_1_1_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">W1</font>
 <b>by </b><i><a class="txt" href="helly.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:18_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">W1</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">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span> <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/><b>hence </b><a NAME="E7:18_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span> <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="helly.html#E1:18_1_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="helly.html#E3:18_1_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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">W2</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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:18_1_1_2"/><i><font color="Green" title="E22">A11</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">W2</font>
 <b>and </b><br/><a NAME="E4:18_1_1_2"/><i><font color="Green" title="E23">A12</font></i>: 
<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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">k</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="helly.html#E4:18_1_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="chain_1.html#T7" title="CHAIN_1:th.7">CHAIN_1:7</a></i>;<br/><a NAME="E6:18_1_1_2"/><i><font color="Green" title="E24">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="helly.html#E3:18_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="helly.html#E4:18_1_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E7:18_1_1_2"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c3">W2</font>
 <b>by </b><i><a class="txt" href="helly.html#E3:18_1_1_2"><i><font color="Green" title="E22">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="E8:18_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:18_1_1_2"/><b>then </b><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c3">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E7:18_1_1_2"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1"/>
( <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">W1</font> or  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">l</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">W2</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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">l</font> ) )
 </span><b>by </b><i><a class="txt" href="helly.html#E7:18_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="glib_001.html#T35" title="GLIB_001:th.35">GLIB_001:35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1"/><i><font color="Green" title="E27">A16</font></i>: 
<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">W1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E28">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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">W1</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/><a NAME="E3:18_1_1_2_1_1"/>
( 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">W1</font> )
 <b>by </b><i><a class="txt" href="helly.html#E1:18_1_1_2_1_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:18_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">W1</font> <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">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span> <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="helly.html#E2:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="helly.html#E6:18_1_1_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="helly.html#E9:18_1_1_2"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="helly.html#E2:18_1_1_2_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">l</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">W2</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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">l</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:18_1_1_2_1_2"/><i><font color="Green" title="E27">A18</font></i>: 
<font color="Maroon" title="c9">l</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">W2</font>
 <b>and </b><br/><a NAME="E4:18_1_1_2_1_2"/><i><font color="Green" title="E28">A19</font></i>: 
<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">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c9">l</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="helly.html#E4:18_1_1_2_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="chain_1.html#T7" title="CHAIN_1:th.7">CHAIN_1:7</a></i>;<br/><a NAME="E6:18_1_1_2_1_2"/><i><font color="Green" title="E29">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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="c3">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="helly.html#E3:18_1_1_2_1_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="helly.html#E4:18_1_1_2_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E7:18_1_1_2_1_2"/><i><font color="Green" title="E30">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:18_1_1_2_1_2"/>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="helly.html#E2:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="helly.html#E4:18_1_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="helly.html#E4:18_1_1_2_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:18_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E10:18_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</font></span>)</span> <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="helly.html#E6:18_1_1_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="helly.html#E7:18_1_1_2"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="helly.html#E6:18_1_1_2_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="helly.html#E7:18_1_1_2_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">W2</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#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/>


</div>
