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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">EqR</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>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>thus </b><a NAME="E2:29"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font> implies  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">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">EqR</font>,<font color="Maroon" title="c2">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E14">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">x</font>
 ;<br/><a NAME="E2:29_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/><a NAME="E3:29_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:29_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T13" target="_self" title="EQREL_1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:29_1"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c2">y</font>
 ;<br/><a NAME="E2:29_1_2"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/><a NAME="E3:29_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:29_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T12" target="_self" title="EQREL_1:th.12">Th12</a></i>;<br/><a NAME="E4:29_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:29_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T13" target="_self" title="EQREL_1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E5:29_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:29_1"/><b>then </b>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">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">EqR</font>,<font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:29_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:29"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c4">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">EqR</font>,<font color="Maroon" title="c2">y</font>
 ;<br/>

<a NAME="E4:29"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/>


</div>
