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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a>  <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="taxonom1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:10"/><b>then </b><i><font color="Green">E10</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a> </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#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a>  <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="taxonom1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E5:10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a>  <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="taxonom1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lang1.html#K17">[*]</a>  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="taxonom1.html#E3:10"><i><font color="Green">E10</font></i></a>, <a class="txt" href="taxonom1.html#E4:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</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>
