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

<b>let </b><font color="Maroon">S</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
( <font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
(  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font> &amp;  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:44_1"/>
 <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>assume </b><a NAME="E2:44"/>
( <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:44"/><b>then </b>
(  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font> &amp;  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:44"/>
 <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
