<?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>thus </b><a NAME="E1:21"/>
<font color="Maroon">PA</font> <a href="partit1.html#NR2" target="_self">'&gt;'</a> <font color="Maroon">PA</font> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">PB</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">PB</font> holds <br/> ex <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">u</font> <a href="tarski.html#R1">c=</a> <font color="Olive">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K2" target="_self">'/\'</a> <font color="Maroon">PB</font>
 ;<br/>

<a NAME="E2:21_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
;<br/>
<a NAME="E3:21_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
;<br/>
<b>then consider </b><font color="Maroon">u1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">u2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 <b>and </b><br/><a NAME="E6:21_1_1"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 <b>and </b><br/><a NAME="E7:21_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">u2</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">u1</font>
;<br/>

<b>thus </b><a NAME="E8:21_1_1"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">u1</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1"/>
<font color="Maroon">PA</font> <a href="partit1.html#NR2" target="_self">'&gt;'</a> <font color="Maroon">PA</font> <a href="partit1.html#K2" target="_self">'/\'</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><b>end;</b></div>


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