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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <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#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:49"/>
<span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1"/><i><font color="Green">E31</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E2:49_1"/><i><font color="Green">E32</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E3:49_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E4:49_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E5:49_1"/>
 <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E6:49_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:49_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </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/>

<b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="partit1.html#E1:49_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:49_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="partit1.html#E2:49_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="partit1.html#E6:49_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:49_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:49_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:49_1_1"/>
 <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="partit1.html#E2:49_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:49_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="partit1.html#E1:49_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="partit1.html#E3:49_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="partit1.html#E4:49_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="partit1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E9:49_1"/>
 <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E10:49_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E11:49_1"/>
<span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="partit1.html#E8:49_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="partit1.html#T5" target="_self">Th5</a></i>;<br/>


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

<b>thus </b><a NAME="E2:49"/>
<span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="eqrel_1.html#K4">/\</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T38" target="_self">Th38</a></i>
<br/>.= 
 <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="eqrel_1.html#T17">EQREL_1:17</a></i>
<br/>.= 
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T38" target="_self">Th38</a></i>
;<br/>
<b>hence </b><a NAME="E2:49_2"/>
<span class="p1">(<span class="default"><a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit1.html#T29" target="_self">Th29</a></i>;<br/>


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


</div>
