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

<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">pe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">p</font> = <font color="Maroon">pe</font>;<br/>
<b>assume </b><a NAME="E1:22"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">v1</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">v2</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>, <font color="Maroon">v2</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">vs</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E4:22"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_1"/>
( 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">pe</font> )
 ;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:22_1"/>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E4:22_1"/>
<font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_4.html#R1">orientedly_joins</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</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>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">pe</font>
 
<b>by </b><i>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E54</font></i>: 
( 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> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E2:22_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:22_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:22_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:22_2"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:22_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</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="E7:22_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:22_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_3"/>
( 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">pe</font> )
 ;<br/><a NAME="E2:22_3"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E3:22_3"/>
<font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_4"/><i><font color="Green">E64</font></i>: 
( 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> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:22_4_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:22_4_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:22_4_1_1"/><b>then </b>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E5:22_4_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:22_4_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:22_4_1_2"/><b>then </b>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:22_4_1_2"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_5"/>
( 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">pe</font> )
 ;<br/><a NAME="E2:22_5"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>take </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><b>thus </b><a NAME="E3:22_5"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K2">*&gt;</a></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>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>thus </b><a NAME="E4:22_5"/>
<font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="graph_1.html#D9">GRAPH_1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:22"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> holds <br/> not the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:22"/>
<font color="Maroon">pe</font> is  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>
 <b>by </b><i>, , , , , , <a class="ref" href="graph_1.html#D11">GRAPH_1:def 11</a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a>, <a class="ref" href="graph_4.html#D7">GRAPH_4:def 7</a></i>;<br/>


</div>
