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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>







<b>thus </b><a NAME="E1:64"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">R</font> implies <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">R</font>
 ;<br/>

<b>let </b><font color="Maroon">y</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="E2:64_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:64_1"/><i><font color="Green">E23</font></i>: 
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E5:64_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E6:64_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</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>, </i>;<br/>
<a NAME="E7:64_1"/>
<font color="Maroon">R</font> <a href="relset_1.html#K10">.:</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:64_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</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/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:64_1_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</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="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<a NAME="E4:64_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:64_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a>, <a class="ref" href="relset_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E6:64_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:64_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:64"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 ;<br/>

<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="E3:64"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:64"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="relset_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<a NAME="E6:64"/>
<font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E7:64"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K6" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relset_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:64"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
