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

<b>reconsider </b><font color="Maroon">D</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">Ch1</font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1;<br/>
<a NAME="E2:61_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">P1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:61_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">P1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<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>for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,0</span>)</span>;<br/>
<a NAME="E6:61_1_1"/><i><font color="Green">E51</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>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</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>  <a href="numbers.html#K5">NAT</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="61_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="E1:61_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>
 ;<br/>

<a NAME="E2:61_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:61_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:61_1_1_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E5:61_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:61_1_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_fin.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> =  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">f</font>,0 as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
;<br/>

<b>thus </b><a NAME="E8:61_1_1_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>, <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">XFS1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:61_1_1"/><i><font color="Green">E58</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>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">XFS1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E4:61_1_1"><i><font color="Green">E50</font></i></a>);<br/></i>
<a NAME="E9:61_1_1"/><i><font color="Green">E59</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">XFS1</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">XFS1</font> = <font color="Maroon">XFS1</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">XFS1</font> = <font color="Maroon">XFS1</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS1</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E12:61_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E13:61_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E14:61_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">Ch</font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>;<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>for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,<font color="Maroon">x</font></span>)</span>;<br/>
<a NAME="E15:61_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> holds <br/> ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x1</font>,<font color="Olive">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 ;<br/>

<a NAME="E2:61_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:61_1_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E5:61_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:61_1_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_fin.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> =  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
;<br/>

<b>thus </b><a NAME="E8:61_1_1_2"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x1</font>, <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>] )
 <b>by </b><i><a class="txt" href="card_fin.html#E8:61_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">XFS</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:61_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x1</font>,<font color="Maroon">XFS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E18:61_1_1"/><i><font color="Green">E71</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">XFS</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">XFS</font> = <font color="Maroon">XFS</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">XFS</font> = <font color="Maroon">XFS</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E8:61_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">XFS</font>
;<br/>

<a NAME="E21:61_1_1"/>
(  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E22:61_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E23:61_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E24:61_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E9:61_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">f1</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><br/> for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>  st <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( <font color="Olive">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/>( ( <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 ) ) ) );<br/>
<a NAME="E25:61_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> holds <br/> ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x1</font>,<font color="Olive">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>
 ;<br/>

<a NAME="E2:61_1_1_3"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E13:61_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E4:61_1_1_3"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) );<br/>
<a NAME="E5:61_1_1_3"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">fd</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">fd</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">d</font>,<font color="Olive">fd</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_1_1_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:61_1_1_3_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:61_1_1_3_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:61_1_1_3_1"/><b>then </b>
( ( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 ) &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1_1_3_1"/>
 ex <font color="Olive">fd</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">fd</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">d</font>,<font color="Olive">fd</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E7:61_1_1_3"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">d</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E15:61_1_1"><i><font color="Green">E63</font></i></a>);<br/></i>
<a NAME="E8:61_1_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_1_1_3_2"/><i><font color="Green">E86</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:61_1_1_3_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_1_3_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:61_1_1_3_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:61_1_1_3_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:61_1_1_3_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61_1_1_3_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1_1_3_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


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

<b>thus </b><a NAME="E3:61_1_1_3_2"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_1_3_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:61_1_1_3_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:61_1_1_3_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:61_1_1_3_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:61_1_1_3_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1_1_3_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E9:61_1_1_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E10:61_1_1_3"/><b>then </b>
<font color="Maroon">P</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a></i>;<br/>
<a NAME="E11:61_1_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E14:61_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:61_1_1_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<a NAME="E14:61_1_1_3"/>
(  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E15:61_1_1_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, </i>;<br/>
<b>hence </b><a NAME="E16:61_1_1_3"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a></i>;<br/>

<b>let </b><font color="Maroon">f1'</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>;<br/>



<b>assume </b><a NAME="E17:61_1_1_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>thus </b><a NAME="E18:61_1_1_3"/>
<font color="Maroon">f'</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1'</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E14:61_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="card_fin.html#T4" target="_self">Th4</a></i>;<br/>

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

<b>assume </b><a NAME="E19:61_1_1_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>thus </b><a NAME="E20:61_1_1_3"/>
( ( <font color="Maroon">f1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">f1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">f1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">f1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, , <a class="txt" href="card_fin.html#E14:61_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="card_fin.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Perm</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span><b> such that </b><br/><a NAME="E27:61_1_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x1</font>,<font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Olive">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E12:61_1_1"><i><font color="Green">E60</font></i></a>);<br/></i>
<a NAME="E28:61_1_1"/><i><font color="Green">E183</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E29:61_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E8:61_1_1"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_4"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Perm</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Perm</font> &amp; <font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z2</font> )
 ;<br/><a NAME="E2:61_1_1_4"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, , </i>;<br/><b>then consider </b><font color="Maroon">P1z1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E4:61_1_1_4"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1z1</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">P1z1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E5:61_1_1_4"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, , </i>;<br/><b>then consider </b><font color="Maroon">P1z2</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E7:61_1_1_4"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1z2</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">P1z2</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E8:61_1_1_4"/>
( <font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Perm</font> &amp; <font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Perm</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:61_1_1_4"/><b>then </b>
( <font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> &amp; (  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E10:61_1_1_4"/><b>then </b>
<font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:61_1_1_4"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">PPermz1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E13:61_1_1_4"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">PPermz1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Perm</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">z1</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">PPermz1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E14:61_1_1_4"/>
<font color="Maroon">P1z1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">P1z2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_1_1_4_1"/><i><font color="Green">E194</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1z1</font> holds <br/><font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_1_1_4_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1z1</font>
 ;<br/>

<a NAME="E2:61_1_1_4_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1z1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_4_1_1_1"/>
( <font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E17:61_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_4_1_1_1_1"/>
<font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_1_1_4_1_1_1_1"/><b>then </b>
( <font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , , <a class="txt" href="card_fin.html#E13:61_1_1"><i><font color="Green">E61</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:61_1_1_4_1_1_1_1"/>
<font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_4_1_1_1_2"/>
<font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_1_1_4_1_1_1_2"/><b>then </b>
( <font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">PPermz1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , , <a class="txt" href="card_fin.html#E13:61_1_1"><i><font color="Green">E61</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:61_1_1_4_1_1_1_2"/>
<font color="Maroon">P1z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:61_1_1_4_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:61_1_1_4_1"/>
<font color="Maroon">P1z1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">P1z2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E15:61_1_1_4"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, , <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:61_1_1"/><b>then </b>
( <font color="Maroon">Perm</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E32:61_1_1"/><b>then </b>
( <font color="Maroon">Perm</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">Perm</font> is <a href="funct_2.html#V2">onto</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E14:61_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">Perm</font> = <font color="Maroon">Perm</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E34:61_1_1"/>
<font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Perm</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:61_1_1_5"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> holds <br/><font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">XFS</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Perm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_1_1_5_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font>
 ;<br/>

<a NAME="E2:61_1_1_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Perm</font>
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E15:61_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<a NAME="E3:61_1_1_5_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">Perm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Perm</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:61_1_1_5_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Perm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:61_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E6:61_1_1_5_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Perm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:61_1_1_5_1"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, , <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">p1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E9:61_1_1_5_1"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:61_1_1_5_1"/>
<font color="Maroon">p1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E13:61_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E11:61_1_1_5_1"/><b>then </b>
(  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font>,0 <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font>,0</span>)</span> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">Perm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Perm</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="txt" href="card_fin.html#E8:61_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E6:61_1_1"><i><font color="Green">E51</font></i></a>, , , , </i>;<br/>
<b>hence </b><a NAME="E12:61_1_1_5_1"/>
<font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">XFS</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Perm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1_1_5"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Perm</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Perm</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<a NAME="E3:61_1_1_5"/><b>then </b>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">XFS</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Perm</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fin.html#E15:61_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_1_5"/>
<font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Perm</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:61_1_1"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS1</font> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS1</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T1" target="_self">Th1</a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E36:61_1_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>


</div>
