<?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="E1:42"/>
( <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>  implies <font color="Maroon">S</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>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_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>  )
 ;<br/>

<b>then </b><font color="Maroon">S</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> = 
<font color="Maroon">S</font> <a href="relat_1.html#K5">*</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<b>hence </b><a NAME="E3:42_1"/>
<font color="Maroon">S</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> 
 ;<br/>


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

<b>thus </b><a NAME="E2:42"/>
( <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>  implies <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">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_2"/>
( <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>  )
 ;<br/>

<b>then </b><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">S</font> = 
<a href="xboole_0.html#K1">{}</a>  <a href="relat_1.html#K5">*</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<b>hence </b><a NAME="E3:42_2"/>
<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">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


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


</div>
