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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">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> implies ( <font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) )
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E3:12"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E4:12"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>


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