<?xml version="1.0"?>
<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">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<a NAME="E1:136"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <font color="Maroon">X</font> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:136_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <font color="Maroon">X</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:136_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <font color="Maroon">X</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="E3:136_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:136_1_1"/><i><font color="Green">E36</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E5:136_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T49" target="_self">Th49</a>, <a class="ref" href="relat_1.html#T55" target="_self">Th55</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:136_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T55" target="_self">Th55</a>, <a class="ref" href="relat_1.html#T44" target="_self">Th44</a></i>;<br/>


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

<b>assume </b><a NAME="E2:136_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
 ;<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="E4:136_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:136_1"/><i><font color="Green">E37</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E6:136_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T49" target="_self">Th49</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:136_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T49" target="_self">Th49</a>, <a class="ref" href="relat_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:136"/>
<font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relat_1.html#K9" target="_self">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
