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<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">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V3">symmetric</a>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R3">is_symmetric_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#D11">RELAT_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><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>assume </b><a NAME="E1:21_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</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">R</font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font></span>)</span> &amp; <font color="Maroon">y</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">R</font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font></span>)</span> &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> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font> )
 ;<br/><a NAME="E2:21_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T19">WELLORD1:19</a></i>;<br/><a NAME="E3:21_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="toler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="wellord1.html#T19">WELLORD1:19</a></i>;<br/><a NAME="E4:21_1"/>
( <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; <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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Z</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="toler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:21_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</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><a class="ref" href="toler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="toler_1.html#T4" target="_self">Th4</a>, <a class="ref" href="toler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/><a NAME="E6:21_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Z</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="toler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T107">ZFMISC_1:107</a></i>;<br/><b>hence </b><a NAME="E7:21_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</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> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:21"/><b>then </b>
<font color="Maroon">R</font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font> <a href="relat_2.html#R3">is_symmetric_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:21"/>
<font color="Maroon">R</font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">Z</font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>


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