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

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x6</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E1:126"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x6</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 ;<br/>

<b>set </b><font color="Maroon">p15</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>;<br/>
<a NAME="E2:126"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 5 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">x5</font> )
 
<b>by </b><i/>;<br/>
<b>thus </b> <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p0"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x6</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, </i>
<br/>.= 
5 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="finseq_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
6

;<br/>

<a NAME="E4:126"/>
 <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> 5
 
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:126"/>
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">x5</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , </i>;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 6 = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p0"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x3</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x5</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a></i>
<br/>.= 
<font color="Maroon">x6</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>
;<br/>


</div>
