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

<b>thus </b><a NAME="E1:7_1_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">X</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> iff <font color="Olive">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E16</font></i>: 
<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="E3:7_1_1_1"/><i><font color="Green">E25</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="E5:7_1_1_1"/><i><font color="Green">E26</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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:7_1_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:7_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:7_1_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1"><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="E1:7_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:7_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:7_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1">PARTIT1:def 1</a></i>;<br/>
<a NAME="E5:7_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> implies ( <font color="Maroon">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<a NAME="E9:7_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_1_1_1_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:7_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<b>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="E4:7_1_1_1_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:7_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:7_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:7_1_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:7_1_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_1_2_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:7_1_1_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:7_1_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E10:7_1_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:7_1_1_1_3"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:7_1_1_1_3"/><b>then </b>
<font color="Maroon">A</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:7_1_1_1_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Aa</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">Aa</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">Aa</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1">PARTIT1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:7_1_1_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aa</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:7_1_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">aa</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="E9:7_1_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">aa</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Aa</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">ax</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:7_1_1_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">ax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">aa</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="E12:7_1_1_1_3"/>
<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="txt" href="bvfunc_2.html#E1:7_1_1_1"><i><font color="Green">E16</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E13:7_1_1_1_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E14:7_1_1_1_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">B</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1_1_3_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:7_1_1_1_3_1"/><i><font color="Green">E54</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">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</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="7_1_1_1_3_1_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="E1:7_1_1_1_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:7_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:7_1_1_1_3_1_1"/>
<font color="Maroon">B</font> <a href="partit1.html#R1">is_a_dependent_set_of</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_3_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</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#T11">PARTIT1:11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:7_1_1_1_3_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E4:7_1_1_1_3_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="bvfunc_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E5:7_1_1_1_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:7_1_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:7_1_1_1_3_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:7_1_1_1_3_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="bvfunc_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E8:7_1_1_1_3_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:7_1_1_1_3_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1_1_1_3_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:7_1_1_1_3_1"/>
<font color="Maroon">A</font> <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="E16:7_1_1_1_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:7_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:7_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E13:7_1_1_1"/>
<font color="Maroon">X</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E14:7_1_1_1"/>
( <font color="Maroon">X</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> &amp; <font color="Maroon">X</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">x</font> <a href="bvfunc_2.html#R1" target="_self">is_upper_min_depend_of</a> <font color="Maroon">G</font> ) ) )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:7_1_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">X</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> st <font color="Olive">X</font> <a href="hidden.html#R1">=</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font> )
 ;<br/>


</div>
