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

<b>let </b><font color="Maroon">Y</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">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PB</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>



<b>set </b><font color="Maroon">RA</font> =  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font>;<br/>
<b>set </b><font color="Maroon">RB</font> =  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">PA</font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">PB</font>
 ;<br/><a NAME="E2:30_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:30_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_1"/>
 <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:30"/><i><font color="Green">E45</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 ;<br/>

<a NAME="E3:30"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 ;<br/>

<a NAME="E2:30_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> ;<br/>
<a NAME="E4:30_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:30_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:30_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x</font>;<br/>
<a NAME="E6:30_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:30_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:30_2"/>
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_2_2_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:30_2_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> )
 ;<br/>
<b>consider </b><font color="Maroon">B2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:30_2_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:30_2_2_1"/>
<font color="Maroon">B2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E7:30_2_2_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_2_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_2_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:30_2_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:30_2_2_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:30_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30"/>
<font color="Maroon">PA</font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">PB</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>


</div>
