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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Olive" title="b1">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Olive" title="b1">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> iff ( <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>



<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )
 <b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E3:12"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E4:12"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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