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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<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> ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ex <font color="Olive">q</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> );<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>
<a NAME="E1:51"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</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">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:51_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">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>given </b><font color="Maroon">p1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><a NAME="E2:51_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q1</font> )
 ;<br/>

<b>given </b><font color="Maroon">p2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><a NAME="E3:51_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">q2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font> )
 ;<br/>

<a NAME="E4:51_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E1"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="card_4.html#E1"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/><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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</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 <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="51_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="E1:51_2"/><i><font color="Green">E52</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">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:51_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:51_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="ref" href="card_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:51"/><i><font color="Green">E54</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( 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">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="txt" href="card_4.html#E1"><i><font color="Green">Lemma41</font></i></a>);<br/></i>
<b>thus </b><a NAME="E5:51"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E1:51_3"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:51_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:51_3_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>
<b>assume </b><a NAME="E4:51_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E6:51_3_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">q2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>
<b>assume </b><a NAME="E7:51_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E8:51_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T2" target="_self">Th2</a>, <a class="txt" href="card_4.html#E1:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:51_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E12:51_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">p2</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="card_4.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="card_4.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>

;<br/>
<a NAME="E14:51_3_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E15:51_3_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E16:51_3_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E17:51_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="card_4.html#T2" target="_self">Th2</a>, <a class="txt" href="card_4.html#E1:51"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


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

<b>thus </b><a NAME="E2:51_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>

<b>thus </b><a NAME="E3:51_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
  <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="51_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:51_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_3_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:51_3_2"/>
 ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ex <font color="Olive">q</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:51_3_2"/><b>then </b>
<font color="Maroon">x</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/>
<b>hence </b><a NAME="E6:51_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 ;<br/>


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

<b>let </b><font color="Maroon">x</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="E4:51_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:51_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T126">FINSEQ_2:126</a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">q1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E9:51_3"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:51_3"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E1:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:51_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</a>, <a class="txt" href="card_4.html#E1:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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

<b>hence </b><a NAME="E6:51"/>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
