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

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

<b>thus </b><a NAME="E1:41"/>
( <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 <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></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="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E17</font></i>: 
<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/>

<i><font color="Green">E18</font></i>: <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span></span>)</span> = 
<font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:41_1"/>
 <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="sysrel.html#E2:41_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <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>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T49">RELAT_1:49</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="sysrel.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:41"/>
( <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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</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="41_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E19</font></i>: 
<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/>

<i><font color="Green">E21</font></i>: <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span></span>)</span> = 
<font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:41_2"/>
 <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:41_2"/><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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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="relat_1.html#K5">*</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="sysrel.html#E2:41_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#T48">RELAT_1:48</a></i>;<br/>
<b>hence </b><a NAME="E5:41_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> <span class="p4">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</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> 
 <b>by </b><i><a class="ref" href="sysrel.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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


</div>
