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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>





<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> implies (  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span>
 ;<br/>

<b>hence </b> <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span>

<br/>.= 
<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"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></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"><font color="Maroon">c<sub>2</sub></font></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/>.= 
1 <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"><font color="Maroon">c<sub>2</sub></font></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#T37" target="_self">Th37</a></i>
<br/>.= 
1 <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/>.= 
2

;<br/><br/>

<a NAME="E3:66_1"/><i><font color="Green">E31</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:66_1"/><b>then </b><i><font color="Green">E32</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"><font color="Maroon">c<sub>1</sub></font></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>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:66_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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#E4:66_1"><i><font color="Green">E32</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>
<b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>
;<br/>

<a NAME="E6:66_1"/><i><font color="Green">E33</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"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E3:66_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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">1 <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:66_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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> <span class="p3"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" target="_self">Th37</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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#E6:66_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:66"/><i><font color="Green">E30</font></i>: 
(  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then </b><i><font color="Green">E31</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
1 <a href="nat_1.html#K1">+</a> 1
<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> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></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/>.= 
<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"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></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"><font color="Maroon">c<sub>2</sub></font></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#T37" target="_self">Th37</a></i>
;<br/>
<a NAME="E4:66"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> holds <br/><font color="Maroon">c<sub>3</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"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></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="66_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:66_3"/>
<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> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 ;<br/>

<a NAME="E2:66_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E3:66_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:66_3"/>
<font color="Maroon">c<sub>3</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"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></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#E2:66"><i><font color="Green">E30</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:66"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> holds <br/><font color="Maroon">c<sub>3</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> <span class="p3"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></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="66_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:66_4"/>
<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> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 ;<br/>

<a NAME="E2:66_4"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</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> <span class="p3"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" target="_self">Th37</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="afinsq_1.html#E2:66_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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#E2:66"><i><font color="Green">E30</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></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#E2:66_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
<b>by </b><i><a class="txt" href="afinsq_1.html#E3:66"><i><font color="Green">E31</font></i></a>, <a class="txt" href="afinsq_1.html#E4:66"><i><font color="Green">E32</font></i></a>, <a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span>

;<br/><br/>


</div>
