<?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#V5" target="_self">T_3</a>  <a href="taxonom2.html#M1" target="_self">Hierarchy</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">H</font> is <a href="taxonom2.html#V6" target="_self">lower-bounded</a> &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> )
 ;<br/>

<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>let </b><font color="Maroon">B</font> be  <a href="taxonom2.html#V4" target="_self">mutually-disjoint</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">C</font> being <a href="taxonom2.html#V4" target="_self">mutually-disjoint</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font>  st <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">C</font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">C</font>
 ;<br/>

<a NAME="E5:31"/><i><font color="Green">E47</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="E6:31"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">H</font> is <a href="taxonom2.html#V3" target="_self">hierarchic</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/><i><font color="Green">E49</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"><a NAME="E2:31_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1"/>
not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:31_1_1_1"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:31_1_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff ( <font color="Olive">h</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">h</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:31_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xx</font> &amp; <font color="Maroon">xx</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/>
<a NAME="E9:31_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E1:31"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E10:31_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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> ; <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="E1:31_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">h2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">h2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">h1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h2</font>
 ;<br/><a NAME="E2:31_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h2</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:31_1_1_1_2"/><b>then </b>
( <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h2</font> or <font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h1</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#D2" target="_self">Def2</a>, <a class="txt" href="taxonom2.html#E2:31"><i><font color="Green">E31</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E4:31_1_1_1_2"/>
<font color="Maroon">h1</font>,<font color="Maroon">h2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:31_1_1_1"/><b>then </b>
<font color="Maroon">D</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">min</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:31_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">min</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E15:31_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">min</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="txt" href="taxonom2.html#E2:31"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">min'</font> = <font color="Maroon">min</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>reconsider </b><font color="Maroon">b1'</font> = <font color="Maroon">b1</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:31_1_1_1_3"/><i><font color="Green">E70</font></i>: 
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1'</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:31_1_1_1_3"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">min'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">min'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:31_1_1_1_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:31_1_1_1_3_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:31_1_1_1_3_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b1'</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E4:31"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E6:31_1_1_1_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E7:31_1_1_1_3_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E8:31_1_1_1_3_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">min'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:31_1_1_1_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">min'</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="E11:31_1_1_1_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:31_1_1_1_3"/>
not <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">min'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1_3_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">min'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b1'</font> = <font color="Maroon">b1</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:31_1_1_1_3_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E5:31_1_1_1_3_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:31_1_1_1_3_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b1'</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E4:31"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E7:31_1_1_1_3_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">b1</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="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E8:31_1_1_1_3_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E9:31_1_1_1_3_2"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E10:31_1_1_1_3_2"/><b>then </b>
<font color="Maroon">min'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:31_1_1_1_3_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">b1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:31_1_1_1_3_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1'</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="E14:31_1_1_1_3_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:31_1_1_1_3"/>
<font color="Maroon">b1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">min'</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#D2" target="_self">Def2</a>, , <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E18:31_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/>( <font color="Maroon">min'</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E19:31_1_1_1"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span> is  <a href="taxonom2.html#V4" target="_self">mutually-disjoint</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E20:31_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</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="31_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:31_1_1_1_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:31_1_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E22:31_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E23:31_1_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E24:31_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">min'</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<b>hence </b><a NAME="E25:31_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="taxonom2.html#E4:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:31_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</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/><a NAME="E4:31_1_1"/>
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">B</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">E</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">E</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">E</font> ) ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:31_1_1"/>
<font color="Maroon">B</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="taxonom2.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_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="E1:31_1_2_1_1"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_2_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_2_1_1_1"/>
<font color="Maroon">B</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_2_1_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:31_1_2_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2_1_1_2_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:31_1_2_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_2_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_2"/>
<font color="Maroon">B</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/></div><b>end;</b></div></div><b>end;</b></div>

</div>
