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

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">H</font> be  <a href="abian.html#V2">covering</a>  <a href="taxonom2.html#M1" target="_self">Hierarchy</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">H</font> is <a href="taxonom2.html#V7" target="_self">with_max's</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">P'</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E2:29_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">P</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><b>take </b>
<font color="Maroon">P</font>
;<br/><b>thus </b><a NAME="E4:29_1_1"/>
( <font color="Maroon">P</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> or  not <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">P</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Y</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:29_1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Y</font></span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="t_1topsp.html#T7">T_1TOPSP:7</a></i>;<br/><a NAME="E3:29_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Y</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:29_1_2_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:29_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:29_1_2_1_1_1"/>
 ex <font color="Olive">P</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> st <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">S</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">V</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Olive">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> ) );<br/><b>consider </b><font color="Maroon">P'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">T</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font> iff ( <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">T</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E4:29_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="abian.html#T4">ABIAN:4</a></i>;<br/><a NAME="E5:29_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">P'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:29_1_2_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font>
 ;<br/>

<b>thus </b><a NAME="E2:29_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E6:29_1_2_1_1_2"/><b>then </b>
<font color="Maroon">P'</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P'</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E8:29_1_2_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_2_1_1_2_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:29_1_2_1_1_2_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:29_1_2_1_1_2_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">h'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_1_1_2_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h'</font> &amp; <font color="Maroon">h'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h'</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">T</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:29_1_2_1_1_2_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">V</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Olive">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:29_1_2_1_1_2_2_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:29_1_2_1_1_2_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:29_1_2_1_1_2_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>thus </b><a NAME="E2:29_1_2_1_1_2_3"/>
<font color="Maroon">S1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_2_1_1_2_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">S</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:29_1_2_1_1_2_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:29_1_2_1_1_2_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E5:29_1_2_1_1_2_3_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">V</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Olive">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:29_1_2_1_1_2_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>then </b><font color="Maroon">S1</font> = 
<font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="taxonom2.html#E1:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<font color="Maroon">Y</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E8:29_1_2_1_1_2_3_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:29_1_2_1_1_2_3"/>
for <font color="Olive">S2</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">S2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Olive">S2</font> or <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">S2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S2</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:29_1_2_1_1_2_3_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:29_1_2_1_1_2_3_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">S2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">T</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:29_1_2_1_1_2_3_2"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:29_1_2_1_1_2_3_2"/>
<font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E6:29_1_2_1_1_2_3_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">V</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Olive">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E7:29_1_2_1_1_2_3_2"/>
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> or <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">S2</font> )
 <b>by </b><i>, , <a class="txt" href="taxonom2.html#E1:29"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:29_1_2_1_1_2"/><b>then </b>
<font color="Maroon">P</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="E11:29_1_2_1_1_2"/>
 ex <font color="Olive">P</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> st <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:29_1_2"/>
 ex <font color="Olive">P</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> st <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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