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

<a NAME="E1:22_1_1"/>
<span class="p1"><a href="yellow_3.html#K1" target="_self">["</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">R</font></span><a href="yellow_3.html#K1" target="_self">"]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:22_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K1" target="_self">["</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">R</font></span><a href="yellow_3.html#K1" target="_self">"]</a></span>
 ;<br/>

<a NAME="E2:22_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_1_1_1"/><i><font color="Green">E20</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<a NAME="E5:22_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22_1_1_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:22_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:22_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:22_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:22_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:22_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E16:22_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E17:22_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:22_1_1_1"/>
<font color="Maroon">b</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E19:22_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, , , , , , <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>
<b>hence </b><a NAME="E20:22_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_1_1"/>
<span class="p1"><a href="yellow_3.html#K1" target="_self">["</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">R</font></span><a href="yellow_3.html#K1" target="_self">"]</a></span> is    <a href="relset_1.html#M2">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>


</div>
