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<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>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><a NAME="E1:30"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; ( 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="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/><font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:30"/>
<font color="Maroon">B</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_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="30_1_1_1"><b>case </b></a><a NAME="E1:30_1_1_1"/>
<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"><b>thus </b><a NAME="E2:30_1_1_1"/>
 <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>, </i>;<br/><b>thus </b><a NAME="E3:30_1_1_1"/>
for <font color="Olive">S1</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">S1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/>( <font color="Olive">S1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( 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">B</font> or <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Olive">S2</font> or <font color="Olive">S1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">S2</font> ) ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>case </b></a><a NAME="E1:30_1_1_2"/><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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_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="E1:30_1_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="30_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_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="E2:30_1_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="30_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2_1_1_2"/><i><font color="Green">E34</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="E2:30_1_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="30_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_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="E2:30_1_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="E3:30_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_1_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="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="E3:30_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


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