<?xml version="1.0"?>
<div class="add">

<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">P</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/>



<a NAME="E1:42"/><i><font color="Green" title="E25">A1</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>  implies  ex <font color="Olive" title="b1">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">EqR</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E26">A2</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/>

<a NAME="E2:42_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</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="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> );<br/>
<a NAME="E3:42_1"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:42_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">Z</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="eqrel_1.html#E3:42_1_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A</font>
;<br/>

<b>thus </b><a NAME="E5:42_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:42_1_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:42_1"/><i><font color="Green" title="E29">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 
;<br/>
<a NAME="E5:42_1"/><i><font color="Green" title="E30">A7</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>given </b><font color="Maroon" title="c6">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E1:42_1_2"/><i><font color="Green" title="E31">A8</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c7">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E2:42_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 ;<br/>

<a NAME="E3:42_1_2"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="eqrel_1.html#E2:42_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>]
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="eqrel_1.html#E2:42_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">R</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:42_1"/><i><font color="Green" title="E31">A10</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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">R</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1" target="_self" title="EQREL_1:sch.1">EQREL_1:sch 1</a>(<a class="txt" href="eqrel_1.html#E3:42_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="eqrel_1.html#E5:42_1"><i><font color="Green" title="E30">A7</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">R</font>
;<br/>

<a NAME="E8:42_1"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><div><i><font color="Green" title="E32">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_1"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 ;<br/><a NAME="E2:42_1_3_1_1"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c4">A</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> 
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font>;<br/><a NAME="E4:42_1_3_1_1"/><i><font color="Green" title="E35">A14</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>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:42_1_3_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <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="42_1_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E36">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><a NAME="E2:42_1_3_1_1_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="c6">y</font>,<font color="Maroon" title="c5">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">R</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="eqrel_1.html#E1:42_1_3_1_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1_3_1_1"><i><font color="Green" title="E35">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">x</font>
 ;<br/><a NAME="E2:42_1_3_1_1_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="c6">y</font>,<font color="Maroon" title="c5">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">R</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:42_1_3_1_1_1_2"/><i><font color="Green" title="E37">A16</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/><a NAME="E5:42_1_3_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:42_1_3_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1_3_1_1_1_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E6:42_1_3_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_3_1_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1_3_1_1_1_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1_3_1_1_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> iff <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_3_1_1_1"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:42_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E7:42_1_3_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E4:42_1_3_1_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_3_1_2"/><i><font color="Green" title="E33">A17</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> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></i>;<br/><a NAME="E4:42_1_3_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:42_1_3_1_2"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></i>;<br/><a NAME="E5:42_1_3_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">x</font>,<font color="Maroon" title="c5">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">R</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:42_1_3_1_2"/><i><font color="Green" title="E34">A18</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/><a NAME="E8:42_1_3_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E35">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><a NAME="E2:42_1_3_1_2_1_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="c7">y</font>,<font color="Maroon" title="c5">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">R</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:42_1_3_1_2"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:42_1_3_1_2_1_1_1"/><i><font color="Green" title="E36">A20</font></i>: 
( <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/><a NAME="E5:42_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1_3_1_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1_3_1_2_1_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E6:42_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1_3_1_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="eqrel_1.html#E4:42_1_3_1_2_1_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1_2_1_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ;<br/><a NAME="E2:42_1_3_1_2_1_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="c7">y</font>,<font color="Maroon" title="c5">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">R</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="eqrel_1.html#E7:42_1_3_1_2"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_1_2_1_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:42_1_3_1_2"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="eqrel_1.html#T27" target="_self" title="EQREL_1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1_3_1_2_1_1"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_3_1_2_1_1"><i><font color="Green" title="E35">A19</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_3_1_2_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:42_1_3_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E7:42_1_3_1_2"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1_3_1"/>
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> iff <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_1_3_1"><i><font color="Green" title="E32">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_3"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T44" title="SETFAM_1:th.44">SETFAM_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:42_1"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:42"/>
( <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>  implies  ex <font color="Olive" title="b1">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">EqR</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green" title="E26">A21</font></i>: 
<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="E2:42_2"/><b>then </b><i><font color="Green" title="E27">A22</font></i>: 
<font color="Maroon" title="c2">P</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="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">EqR1</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>;<br/>
<b>take </b>
<font color="Maroon" title="c3">EqR1</font>
;<br/>

<b>thus </b><a NAME="E4:42_2"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">EqR1</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="eqrel_1.html#E2:42_2"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="eqrel_1.html#T40" target="_self" title="EQREL_1:th.40">Th40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
 ex <font color="Olive" title="b1">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">EqR</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div>
