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

<a NAME="E1:22_1_1"/>
<span class="p1"><a href="yellow_3.html#K1" title="YELLOW_3:func.1">["</a><span class="default"><font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">R</font></span><a href="yellow_3.html#K1" title="YELLOW_3:func.1">"]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K1" title="YELLOW_3:func.1">["</a><span class="default"><font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">R</font></span><a href="yellow_3.html#K1" title="YELLOW_3:func.1">"]</a></span>
 ;<br/>

<a NAME="E2:22_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T10" target="_self" title="YELLOW_3:th.10">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:22_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<a NAME="E5:22_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:22_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="yellow_3.html#T10" target="_self" title="YELLOW_3:th.10">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">s</font>, <font color="Maroon" title="c11">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:22_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:22_1_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:22_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="yellow_3.html#T10" target="_self" title="YELLOW_3:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c15">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:22_1_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:22_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="yellow_3.html#T10" target="_self" title="YELLOW_3:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a2</font>, <font color="Maroon" title="c17">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:22_1_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a2</font>,<font color="Maroon" title="c17">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:22_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="yellow_3.html#T10" target="_self" title="YELLOW_3:th.10">Th10</a></i>;<br/>
<a NAME="E14:22_1_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a2</font>,<font color="Maroon" title="c17">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E9:22_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="yellow_3.html#E11:22_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="yellow_3.html#E13:22_1_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E15:22_1_1_1"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E4:22_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_3.html#E11:22_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E16:22_1_1_1"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E7:22_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="yellow_3.html#E11:22_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="yellow_3.html#E14:22_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E17:22_1_1_1"/><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E4:22_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_3.html#E13:22_1_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="yellow_3.html#E14:22_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:22_1_1_1"/>
<font color="Maroon" title="c13">b</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E7:22_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="yellow_3.html#E13:22_1_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="yellow_3.html#E14:22_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E19:22_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E4:22_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_3.html#E7:22_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="yellow_3.html#E14:22_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="yellow_3.html#E15:22_1_1_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="yellow_3.html#E16:22_1_1_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="yellow_3.html#E17:22_1_1_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="mcart_1.html#T11" title="MCART_1:th.11">MCART_1:11</a></i>;<br/>
<b>hence </b><a NAME="E20:22_1_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow_3.html#E9:22_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">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" title="YELLOW_3:func.1">["</a><span class="default"><font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">R</font></span><a href="yellow_3.html#K1" title="YELLOW_3:func.1">"]</a></span> is    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>


</div>
