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

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 ;<br/><a NAME="E2:4_1"/><b>then </b><i><font color="Green">E31</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/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_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>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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#K3">field</a> <font color="Maroon">R</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:4_1_1"/><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">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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#D10">RELAT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="relat_2.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div><a NAME="E4:4_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#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T56">RELAT_1:56</a></i>;<br/><b>hence </b><a NAME="E5:4_1"/>
 <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_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="E1:4_2_1"/>
<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="E2:4_2_1"/><b>then </b>
<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>  <a href="relat_1.html#K6">id</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#D10">RELAT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:4_2_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">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:4_2"/><b>then </b>
<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/>;<br/><b>hence </b><a NAME="E4:4_2"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
( <font color="Maroon">R</font> is <a href="relat_2.html#V2" target="_self">irreflexive</a> iff  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_2.html#D4" target="_self">Def4</a></i>;<br/>


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