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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> ;<br/><b>let </b><font color="Maroon" title="c2">EqR1</font>, <font color="Maroon" title="c3">EqR2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<div><i><font color="Green" title="E20">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:37_1"/><b>then </b>
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>hence </b><a NAME="E3:37_1"/>
( <font color="Maroon" title="c2">EqR1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:37"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:37"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">EqR1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:37_2_1"/><b>then </b>
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:37"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T37" target="_self" title="EQREL_1:th.37">Th37</a></i>;<br/><b>hence </b><a NAME="E3:37_2_1"/>
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:37_2_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T34" target="_self" title="EQREL_1:th.34">Th34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:37_2"/><b>then </b>
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:37_2"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_2"/>
( <font color="Maroon" title="c2">EqR1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#T35" target="_self" title="EQREL_1:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
( <font color="Maroon" title="c2">EqR1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:37"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>


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