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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:6"/><i><font color="Green">E3</font></i>: 
 <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T107">ZFMISC_1:107</a></i>;<br/>
<b>hence </b><a NAME="E2:6_1"/>
 <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/><i><font color="Green">E4</font></i>: 
 <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E2:6_2"/>
 <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<b>hence </b><a NAME="E4:6"/>
 <a href="eqrel_1.html#K1" target="_self">nabla</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:6"><i><font color="Green">E3</font></i></a>, <a class="txt" href="eqrel_1.html#E2:6"><i><font color="Green">E4</font></i></a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
