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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="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V4" target="_self">antisymmetric</a>
 ;<br/><a NAME="E2:23_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R4" target="_self">is_antisymmetric_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="23_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="E1:23_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> <font color="Maroon">R</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span>
 ;<br/><a NAME="E2:23_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <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> &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> <a href="relat_1.html#K4">~</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:23_1_1"/><b>then </b><i><font color="Green">E33</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> &amp; <font color="Maroon">b</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_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E4:23_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</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_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E5:23_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self">Def9</a>, <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/><a NAME="E6:23_1_1"/>
<font color="Maroon">b</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="ref" href="relat_2.html#D12" target="_self">Def12</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:23_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>  <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_2.html#D13" target="_self">Def13</a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">R</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="tarski.html#R1">c=</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#D3">RELAT_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="tarski.html#R1">c=</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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_2_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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</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> )
 ;<br/><a NAME="E2:23_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">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E3:23_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">b</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"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:23_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self">Def9</a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E3:23_2"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R4" target="_self">is_antisymmetric_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="E4:23_2"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V4" target="_self">antisymmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
( <font color="Maroon">R</font> is <a href="relat_2.html#V4" target="_self">antisymmetric</a> iff <font color="Maroon">R</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="tarski.html#R1">c=</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_2.html#D8" target="_self">Def8</a></i>;<br/>


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