<?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">y</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/>





<b>thus </b><a NAME="E1:33"/>
( <font color="Maroon">R</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; <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="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">R</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; <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="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:33_1"/><b>then </b>
not <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>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E3:33_1"/><b>then </b>
not <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> <a href="xboole_0.html#K3">/\</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b>
not <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>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E5:33_1"/><b>then </b><i><font color="Green">E18</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> <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="txt" href="sysrel.html#E1:33_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_1"/><i><font color="Green">E19</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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> )
 <b>by </b><i><a class="txt" href="sysrel.html#E1:33_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E8:33_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:33_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E3:33_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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> <a href="xboole_0.html#K3">/\</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:33_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E5:33_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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> <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="txt" href="sysrel.html#E7:33_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sysrel.html#E1:33_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="txt" href="sysrel.html#E1:33_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/>
<a NAME="E10:33_1"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</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/>
<b>hence </b><a NAME="E3:33_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sysrel.html#E1:33_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="sysrel.html#E5:33_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="sysrel.html#E7:33_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E12:33_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">dom</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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="txt" href="sysrel.html#E7:33_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="sysrel.html#T13" target="_self">Th13</a>, <a class="txt" href="sysrel.html#E9:33_1"><i><font color="Green">E21</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:33"/>
( <font color="Maroon">R</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">R</font> <a href="subset_1.html#K6">\</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="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> implies ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">R</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">R</font> <a href="subset_1.html#K6">\</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="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:33_2"/><b>then </b>
not <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>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E3:33_2"/><b>then </b>
not <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> <a href="xboole_0.html#K3">/\</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:33_2"/><b>then </b>
not <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>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E5:33_2"/><b>then </b><i><font color="Green">E26</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> <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_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_2"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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> )
 <b>by </b><i>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E8:33_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:33_2_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E3:33_2_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="xboole_0.html#K3">/\</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:33_2_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E5:33_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:33_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/>
<a NAME="E10:33_2"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sysrel.html#K1" target="_self">CL</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<a NAME="E2:33_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</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>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:33_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:33_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E12:33_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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="sysrel.html#T13" target="_self">Th13</a>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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


</div>
