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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  st <font color="Olive" title="b1">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  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>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> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  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="E1:10"/><i><font color="Green" title="E9">A1</font></i>: 
( <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> &amp; <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</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> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  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="E2:10"/>
<font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <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="txt" href="taxonom1.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="taxonom1.html#T5" target="_self" title="TAXONOM1:th.5">Th5</a></i>;<br/>
<a NAME="E3:10"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> </span>)</span> <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/>
<a NAME="E4:10"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="taxonom1.html#T7" target="_self" title="TAXONOM1:th.7">Th7</a></i>;<br/>
<a NAME="E5:10"/>
<font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="taxonom1.html#T8" target="_self" title="TAXONOM1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:10"/>
<font color="Maroon" title="c2">R</font> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  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="taxonom1.html#E3:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="taxonom1.html#E4:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</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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