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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:40"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:40_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:40_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="eqrel_1.html#E3:40_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="eqrel_1.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:40_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:40_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="eqrel_1.html#E3:40_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="eqrel_1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E6:40_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E4:40_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:40_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/></div><b>end;</b></div>
<a NAME="E3:40_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:40_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:40_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>thus </b><a NAME="E4:40_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:40_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="eqrel_1.html#T28" target="_self">Th28</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E5:40_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E6:40_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:40_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="eqrel_1.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_1"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:40_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="eqrel_1.html#E3:40_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:40_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:40"><i><font color="Green">E24</font></i></a></i>;<br/>


</div>
