<?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">G</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>





<b>assume </b><a NAME="E1:6"/>
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green">E25</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; ( 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">g</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) ) ) )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E7:6"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">Y</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T9">PARTIT1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font> ) );<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">XX</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E10:6"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:6"/>
for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">X</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:6_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E13:6"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</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="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>take </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font>
;<br/>

<a NAME="E14:6"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:6_2"/>
for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/><font color="Olive">X1</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_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> ;<br/>

<b>assume </b><a NAME="E1:6_2_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:6_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">X</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:6_2_1"/>
<font color="Maroon">X1</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_2"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#T10">PARTIT1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:6"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> &amp; ( for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Olive">e</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font> ) holds <br/><font color="Olive">e</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">e</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> &amp; ( for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">e</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font> ) )
 ;<br/>

<a NAME="E2:6_3"/><b>then </b>
<font color="Maroon">e</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Ad</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">Ad</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">Ad</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Ad</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1">PARTIT1:def 1</a></i>;<br/>
<a NAME="E5:6_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1"><b>suppose </b></a><a NAME="E1:6_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><a NAME="E2:6_3_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:6_3_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_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> ; <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="E1:6_3_1_1_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font>
 ;<br/>

<a NAME="E2:6_3_1_1_1"/><b>then </b>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:6_3_1_1_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6"><i><font color="Green">E16</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:6_3_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_2"><b>suppose </b></a><a NAME="E1:6_3_1_2"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/><a NAME="E3:6_3_1_2"/>
<font color="Maroon">e</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E4:6_3_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:6_3_1_2"/>
<font color="Maroon">e</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><a NAME="E6:6_3_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">e</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:6_3_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E8:6_3_1_2"/>
for <font color="Olive">d</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">e</font> <a href="subset_1.html#K3">`</a>  <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_3_1_2_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:6_3_1_2_1"/><b>then </b>
<font color="Maroon">e</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:6_3_1_2_1"/>
<font color="Maroon">e</font> <a href="subset_1.html#K3">`</a>  <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#T12">PARTIT1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:6_3_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">e</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 <b>by </b><i/>;<br/><a NAME="E10:6_3_1_2"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_2_2"><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="E1:6_3_1_2_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font>
 ;<br/>

<a NAME="E2:6_3_1_2_2"/><b>then </b>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:6_3_1_2_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="bvfunc_2.html#T1" target="_self">Th1</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:6_3_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">e</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E12:6_3_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">e</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E13:6_3_1_2"/>
<font color="Maroon">e</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E14:6_3_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E15:6_3_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">Ad</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>consider </b><font color="Maroon">ad</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:6_3_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">ad</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ad</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E18:6_3_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">ad</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E19:6_3_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">ad</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>consider </b><font color="Maroon">add</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:6_3_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">add</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ad</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E22:6_3_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E5:6"><i><font color="Green">E25</font></i></a>, <a class="txt" href="bvfunc_2.html#E6:6"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E16:6"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">XX</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , , , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div>
