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

<b>let </b><font color="Maroon">P</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>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 ;<br/>

<a NAME="E3:17"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">P</font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8" target="_self">Def8</a>, </i>;<br/>
<a NAME="E4:17"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self">Def9</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font></span>)</span>
 ;<br/><a NAME="E2:17_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">P</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:17_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a>, </i>;<br/><a NAME="E3:17_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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_2.html#D11" target="_self">Def11</a>, </i>;<br/><a NAME="E4:17_1_1"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:17_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D14" target="_self">Def14</a>, <a class="ref" href="relat_2.html#D15" target="_self">Def15</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:17_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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_2.html#D11" target="_self">Def11</a>, </i>;<br/><a NAME="E3:17_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> implies  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a>, </i>;<br/><a NAME="E4:17_1_2"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> implies  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:17_1_2"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D14" target="_self">Def14</a>, <a class="ref" href="relat_2.html#D15" target="_self">Def15</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:17_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D12" target="_self">Def12</a>, <a class="ref" href="relat_2.html#D13" target="_self">Def13</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:17"/><b>then </b>
<font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:17"/>
<font color="Maroon">P</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:17_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">P</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T34">RELAT_1:34</a></i>;<br/><b>assume </b><a NAME="E2:17_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font></span>)</span>
 ;<br/><a NAME="E3:17_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_2.html#D12" target="_self">Def12</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:17_2"/><b>then </b>
( not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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_2.html#D10" target="_self">Def10</a>, <a class="ref" href="relat_2.html#D11" target="_self">Def11</a>, </i>;<br/><b>hence </b><a NAME="E5:17_2"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:17"/><b>then </b>
<font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:17"/>
<font color="Maroon">P</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>by </b><i/>;<br/>


</div>
