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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>

<b>assume </b><a NAME="E1:68"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:68"/><b>then </b>
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:68"/><i><font color="Green">E31</font></i>: 
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
;<br/>

<a NAME="E6:68"/><i><font color="Green">E32</font></i>: 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<i><font color="Green">E33</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
;<br/>
<a NAME="E2:68_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E4:68"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:68_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<b>hence </b><a NAME="E4:68_1"/>
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E1:68_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:68"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<i><font color="Green">E33</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> = 
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" target="_self">Th37</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E4:68"><i><font color="Green">E31</font></i></a>, <a class="txt" href="afinsq_1.html#E6:68"><i><font color="Green">E32</font></i></a>, <a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
;<br/>
<a NAME="E2:68_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:68_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:68_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E4:68"><i><font color="Green">E31</font></i></a>, <a class="txt" href="afinsq_1.html#E1:68_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:68_2_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E6:68"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E6:68"><i><font color="Green">E32</font></i></a>, <a class="txt" href="afinsq_1.html#E1:68_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:68_2_2_2"/>
0 <a href="hidden.html#R2">in</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:68_2_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:68_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="afinsq_1.html#E6:68"><i><font color="Green">E32</font></i></a>, <a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="afinsq_1.html#E3:68_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E4:68"><i><font color="Green">E31</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:68_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:68_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E4:68_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="afinsq_1.html#E5:68_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E1:68_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>


</div>
