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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="relat_2.html#V3">symmetric</a>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:21_1"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">E15</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green">E13</font></i></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green">E13</font></i></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">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="toler_1.html#E2:21"><i><font color="Green">E12</font></i></a>, <a class="txt" href="toler_1.html#E2:21_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="toler_1.html#E3:21_1"><i><font color="Green">E15</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green">E13</font></i></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="toler_1.html#E5:21_1"><i><font color="Green">E16</font></i></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">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="toler_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>3</sub></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/>


</div>
