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

<b>take </b><font color="Maroon">X</font> = <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/>

<b>thus </b><a NAME="E1:2_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</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">X</font> &amp; <font color="Olive">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E2:2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:2_1_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="coh_sp.html#D2" target="_self">COH_SP:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_1_1_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</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">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_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="E1:2_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</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#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_1_1_1"/>
 <a href="tarski.html#K3">union</a> <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#D1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_2"/>
<font color="Maroon">A</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="E2:2_1_1_2_1_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</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#T31">ZFMISC_1:31</a></i>;<br/><b>hence </b><a NAME="E3:2_1_1_2_1_1_2"/>
 <a href="tarski.html#K3">union</a> <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#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:2_1_1"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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