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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y1</font>, <font color="Olive">y2</font>, <font color="Olive">y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> ) ) );<br/>
<a NAME="E1:22_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E3:22_1_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:22_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><b>take </b><font color="Maroon">y</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>thus </b><a NAME="E6:22_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T66">FINSEQ_1:66</a></i>;<br/><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>take </b><font color="Maroon">y2</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>take </b><font color="Maroon">y3</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><a NAME="E7:22_1_1_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E8:22_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:22_1_1_1_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>thus </b><a NAME="E10:22_1_1_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> ) )
 <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E3:22_1_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 3 </span>}</span>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:22_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><b>set </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>set </b><font color="Maroon">y2</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2;<br/><b>set </b><font color="Maroon">y3</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3;<br/><a NAME="E6:22_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E7:22_1_1_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span>;<br/><a NAME="E9:22_1_1_1_1_2"/>
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/><a NAME="E10:22_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> &amp; 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> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">z1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">z2</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:22_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/><b>hence </b><a NAME="E13:22_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3
;<br/><b>thus </b><a NAME="E14:22_1_1_1_1_2"/>
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E15:22_1_1_1_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> ) )
 <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Ar</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ar</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:22_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ar</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> 
 <b>and </b><br/><a NAME="E5:22_1_1"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Ar</font> = <font color="Maroon">Ar</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y1</font>, <font color="Olive">y2</font>, <font color="Olive">y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) ) );<br/>
<a NAME="E7:22_1_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_2_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:22_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E3:22_1_1_2_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:22_1_1_2_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><b>set </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>set </b><font color="Maroon">y2</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>set </b><font color="Maroon">y3</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>take </b><font color="Maroon">y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E6:22_1_1_2_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E7:22_1_1_2_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/><a NAME="E10:22_1_1_2_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_4.html#K2">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/><a NAME="E11:22_1_1_2_1_1"/><b>then </b>
( 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> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:22_1_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
;<br/><b>thus </b><a NAME="E13:22_1_1_2_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E14:22_1_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_2_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:22_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E3:22_1_1_2_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 3 </span>}</span>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:22_1_1_2_1_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><b>set </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1;<br/><b>set </b><font color="Maroon">y2</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2;<br/><b>set </b><font color="Maroon">y3</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3;<br/><a NAME="E6:22_1_1_2_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E7:22_1_1_2_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E9:22_1_1_2_1_2"/>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/><a NAME="E10:22_1_1_2_1_2"/><b>then </b>
( 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> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E11:22_1_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2
;<br/><b>take </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3
;<br/><b>thus </b><a NAME="E12:22_1_1_2_1_2"/>
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E13:22_1_1_2_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">R</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:22_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E10:22_1_1"/><i><font color="Green">E100</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E11:22_1_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b><font color="Maroon">S</font> =  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<font color="Maroon">Ar</font>,<font color="Maroon">R</font> #);<br/>

<b>thus </b><a NAME="E13:22_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>thus </b><a NAME="E14:22_1_1"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_3"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/><a NAME="E4:22_1_1_3"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T118">FINSEQ_2:118</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:22_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:22_1_1_3"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:22_1_1_3"/><b>then </b>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp;  ex <font color="Olive">y1</font>, <font color="Olive">y2</font>, <font color="Olive">y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> ) ) ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:22_1_1_3"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:22_1_1_3"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E11:22_1_1_3"/><i><font color="Green">E104</font></i>: 
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E12:22_1_1_3"/>
( <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E13:22_1_1_3"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E16:22_1_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E19:22_1_1"/>
( <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="finseq_4.html#K3">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T124">FINSEQ_2:124</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:22_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:22_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:22_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E24:22_1_1"/><i><font color="Green">E107</font></i>: 
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E25:22_1_1"/><i><font color="Green">E108</font></i>: 
( <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E26:22_1_1"/><i><font color="Green">E109</font></i>: 
( <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y2</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y3</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D10">FINSEQ_1:def 10</a></i>;<br/>
<a NAME="E27:22_1_1"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y2</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y3</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E28:22_1_1"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y2</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E29:22_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<b>hence </b><a NAME="E30:22_1_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>

<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E32:22_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E33:22_1_1"/><i><font color="Green">E110</font></i>: 
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y1</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E34:22_1_1"/>
( <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E35:22_1_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div>
