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

<a NAME="E1:22_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</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">PA</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">PA</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="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</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="E3:22_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">d</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">d</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:22_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:22_1_1_1"/>
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> holds <br/><font color="Olive">d</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="22_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 ;<br/>

<a NAME="E2:22_1_1_1_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font> &amp; <font color="Olive">x</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Olive">x</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> ) )
 
<b>by </b><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="E4:22_1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</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">d</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:22_1_1_1_1"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1_1_1_1"/>
<font color="Maroon">d</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="E2:22_1_1_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> implies ( <font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> &amp; <font color="Maroon">d</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="E3:22_1_1_1"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> )
 <b>by </b><i/>;<br/>


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

<a NAME="E5:22_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="E1:22_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</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="E3:22_1_1_2_1"/><i><font color="Green">E60</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="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:22_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="E5:22_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>
<a NAME="E2:22_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="22_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="E1:22_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="E3:22_1_1_2_2"/><i><font color="Green">E61</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="E4:22_1_1_2_2"/>
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:22_1_1_2_2"/><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; <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22_1_1_2_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</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">a</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:22_1_1_2_2"/>
<font color="Maroon">a</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/>
<b>hence </b><a NAME="E9:22_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>
<b>hence </b><a NAME="E3:22_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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>
<a NAME="E6:22_1_1"/><i><font color="Green">E63</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="22_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="E1:22_1_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:22_1_1_3"/><b>then </b><i><font color="Green">E64</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">PB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">A</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">A</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_1_1_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">Aa</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</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/>;<br/>
<b>consider </b><font color="Maroon">aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22_1_1_3"/><i><font color="Green">E76</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="E8:22_1_1_3"/><i><font color="Green">E77</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="E9:22_1_1_3"/><i><font color="Green">E79</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/>
<a NAME="E10:22_1_1_3"/>
 ex <font color="Olive">ax</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">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>hence </b><a NAME="E11:22_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>, </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="E12:22_1_1_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E13:22_1_1_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">B</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:22_1_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">B</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">B</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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">B</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> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1_3_1"/><i><font color="Green">E82</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="E2:22_1_1_3_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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="E3:22_1_1_3_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:22_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="E5:22_1_1_3_1"/><b>then </b><i><font color="Green">E85</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>, , , </i>;<br/><a NAME="E6:22_1_1_3_1"/><i><font color="Green">E86</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="22_1_1_3_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> ; <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="E1:22_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_3_1_1"/>
<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="E7:22_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="E8:22_1_1_3_1"/><b>then </b><i><font color="Green">E87</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>, , , </i>;<br/><a NAME="E9:22_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="22_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="E1:22_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_3_1_2"/>
<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="E10:22_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="E16:22_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="E7:22_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="22_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="E1:22_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:22_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</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">x</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> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 
<b>by </b><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="E5:22_1_1_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</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/>;<br/>
<a NAME="E6:22_1_1_4"/>
<font color="Maroon">x</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/>
<b>hence </b><a NAME="E7:22_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>
 ;<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="E9:22_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="E10:22_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">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
