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

<b>let </b><font color="Maroon">N</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">N</font>, <font color="Maroon">S</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">N</font>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">R</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E2:54"/>
( <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> implies <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> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E21</font></i>: 
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4" target="_self">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E3:54_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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/>
<a NAME="E4:54_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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="E5:54_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:54_1"/><b>then </b>
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">e</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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/>
<b>hence </b><a NAME="E7:54_1"/>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E3:54"/><i><font color="Green">E27</font></i>: 
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E4:54"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4" target="_self">dom</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:54"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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/>
<a NAME="E7:54"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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="E8:54"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K10" target="_self">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:54"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E11:54"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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="E12:54"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:54"/>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
