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

<b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font>;<br/>
<b>set </b><font color="Maroon">lenW</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 ;<br/><a NAME="E2:16_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:16_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> is <a href="abian.html#V1">even</a>
 ;<br/><b>then reconsider </b><font color="Maroon">lenW</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:16_1_1_1_1"/>
<font color="Maroon">lenW</font> <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">W1</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">W2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E5:16_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:16_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><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="E1:16_1_1_1_2"/><i><font color="Green">E44</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">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">v1</font> = <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">v2</font> = <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/><b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_2_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:16_1_1_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E5:16_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E48</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> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <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">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <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">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:16_1_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E8:16_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_1_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1_1_2_1_1_2_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="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="E3:16_1_1_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:16_1_1_1_2_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">even</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="E6:16_1_1_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E7:16_1_1_1_2_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E8:16_1_1_1_2_1_1_2"/><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">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:16_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E11:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</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">W2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/><a NAME="E12:16_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</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">W2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E13:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</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">W2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E14:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="nat_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">W2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <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">W2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E17:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E89</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">W2</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E19:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</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="graph_2.html#T15">GRAPH_2:15</a></i>;<br/><a NAME="E20:16_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:16_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E22:16_1_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:16_1_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E24:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_2_1_1_2_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&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> <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</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">W1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E4:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i/>;<br/><a NAME="E5:16_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E6:16_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E7:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:16_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T15">GRAPH_2:15</a></i>;<br/><b>hence </b><a NAME="E10:16_1_1_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E2"><i><font color="Green">Lemma30</font></i></a>, , <a class="txt" href="glib_001.html#E3"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2_1_1_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">two</font> = 2 <a href="nat_1.html#K2">*</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">two</font> as  <a href="abian.html#V1">even</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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <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">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><a NAME="E7:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<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">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 ;<br/><a NAME="E8:16_1_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:16_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E11:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E2"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="graph_2.html#T15">GRAPH_2:15</a></i>;<br/><b>set </b><font color="Maroon">kaa1</font> = <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>reconsider </b><font color="Maroon">kaa1</font> = <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</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>  ;<br/><a NAME="E14:16_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">kaa1</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E15:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kaa1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T15">GRAPH_2:15</a></i>;<br/><a NAME="E16:16_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">kaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kaa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T59">NAT_1:59</a></i>;<br/><a NAME="E17:16_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">kaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:16_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E106</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E19:16_1_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">kaa1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>then </b><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kaa1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</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">W1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">kaa1</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#T2" target="_self">Th2</a>, <a class="ref" href="graph_2.html#T15">GRAPH_2:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E21:16_1_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="txt" href="glib_001.html#E3"><i><font color="Green">Lemma32</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:16_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:16_1_1_1"/>
<font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:16_1_1"/>
( ( not <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a>  implies <font color="Maroon">W1</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> holds  verum ) )
 ;<br/>


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