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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>







<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K8" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="afinsq_1.html#K8" target="_self">%&gt;</a></span> implies (  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K8" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="afinsq_1.html#K8" target="_self">%&gt;</a></span>
 ;<br/>

<b>hence </b> <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <span class="p2"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" 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">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
2 <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">z</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#T4" target="_self">Th4</a></i>
<br/>.= 
2 <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/>.= 
3

;<br/><br/>

<a NAME="E3:67_1"/><i><font color="Green">E33</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="E4:67_1"/><b>then </b><i><font color="Green">E43</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">x</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<b>thus </b><font color="Maroon">p</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">x</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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="afinsq_1.html#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</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>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<a NAME="E6:67_1"/><i><font color="Green">E44</font></i>: 
1 <a href="hidden.html#R2">in</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E7:67_1"/>
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:67_1"/><b>then </b><i><font color="Green">E46</font></i>: 
1 <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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E3:67_1"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="afinsq_1.html#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>
;<br/>

<a NAME="E10:67_1"/><i><font color="Green">E47</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">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:67_1"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" 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">z</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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" 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#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">z</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>, </i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i/>
;<br/>


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

<b>assume </b><a NAME="E2:67"/><i><font color="Green">E48</font></i>: 
(  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 ;<br/>

<b>then </b><i><font color="Green">E49</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font> = 
2 <a href="nat_1.html#K1">+</a> 1
<b>by </b><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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" 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#T4" target="_self">Th4</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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" 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">z</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#T1" target="_self">Th1</a></i>
;<br/>
<a NAME="E4:67"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</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">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:67_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">k</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#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span>
 ;<br/>

<a NAME="E2:67_3"/>
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:67_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> 2
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:67_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:67_3"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:67_3"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:67"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> holds <br/><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="afinsq_1.html#K1" target="_self">len</a> <span class="p3"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</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">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:67_4"/>
<font color="Maroon">k</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">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 ;<br/>

<a NAME="E2:67_4"/><b>then </b>
<font color="Maroon">k</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="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E3:67_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</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><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="afinsq_1.html#K1" target="_self">len</a> <span class="p3"><a href="afinsq_1.html#K7" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="afinsq_1.html#K7" target="_self">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">z</font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T5" target="_self">Th5</a>, , </i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:67"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K8" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="afinsq_1.html#K8" target="_self">%&gt;</a></span>
 <b>by </b><i>, , </i>;<br/>


</div>
