<?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>;<br/>



<a NAME="E1:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">PA</font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:24"/>
<font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">PA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_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> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font>
 ;<br/>

<a NAME="E2:24_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:24_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:24_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</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">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font>;<br/>
<a NAME="E8:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:24_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:24_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:24_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:24_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:24_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E3:24_1_1_1"/>
<font color="Maroon">b</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">u</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:24_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:24_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:24_1_1_1"/>
<font color="Maroon">b</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="E14:24_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:24_1_1"/>
<font color="Maroon">b</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:24_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E17:24_1_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1"/>
<font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">PA</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>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PA</font>
 <b>by </b><i>, </i>;<br/>


</div>
