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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font>;<br/>



<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>case </b></a><a NAME="E1:10_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:10_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:10_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E5:10_1_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_1"/><i><font color="Green">E37</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> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:10_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:10_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:10_1_1_1_1"/>
<font color="Maroon">f</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#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E6:10_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="graph_1.html#R2">joins</a> <font color="Olive">x'</font>,<font color="Olive">y'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_2"/><i><font color="Green">E39</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">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<a NAME="E2:10_1_1_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_1_2"/><b>then </b>
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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:10_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</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">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E5:10_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:10_1_1_1_2"/><b>then </b><i><font color="Green">E42</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">f</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:10_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:10_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:10_1_1_1_2"/><i><font color="Green">E59</font></i>: 
1 <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="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:10_1_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:10_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:10_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</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> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>, <font color="Maroon">b</font> = <font color="Maroon">f</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> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E14:10_1_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E15:10_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<a NAME="E16:10_1_1_1_2"/><b>then </b>
( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E17:10_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#D9">GRAPH_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E18:10_1_1_1_2"/>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Olive">x'</font>,<font color="Olive">y'</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E7:10_1_1_1"/><b>then </b>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> ) &amp;  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="Olive">p</font> holds <br/><font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="graph_1.html#R2">joins</a> <font color="Olive">x'</font>,<font color="Olive">y'</font> ) ) ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:10_1_1_1"/>
 <a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font> is    <a href="graph_1.html#M2">Chain</a> of  <a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D11">GRAPH_1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>case </b></a><a NAME="E1:10_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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="E3:10_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:10_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E5:10_1_1_2"/>
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E6:10_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:10_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E8:10_1_1_2"/><b>then </b>
 <a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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/><b>hence </b><a NAME="E9:10_1_1_2"/>
 <a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font> is  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of  <a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="graph_1.html#T14">GRAPH_1:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:10"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> holds <br/>the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <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#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E65</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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<a NAME="E2:10_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:10_2"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E4:10_2"/><b>then </b>
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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:10_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</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">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E6:10_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:10_2"/><b>then </b><i><font color="Green">E70</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">f</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:10_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:10_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:10_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:10_2"/><i><font color="Green">E73</font></i>: 
1 <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="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:10_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:10_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:10_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</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> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>, <font color="Maroon">b</font> = <font color="Maroon">f</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> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E16:10_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E17:10_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<a NAME="E18:10_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E19:10_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:10_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</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> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:10_2"/><i><font color="Green">E78</font></i>: 
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="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E22:10_2"/>
<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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E23:10_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="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:10_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:10_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<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</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E26:10_2"/>
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
 <a href="jgraph_1.html#K3" target="_self">PairF</a> <font color="Maroon">f</font> is  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of  <a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/>


</div>
