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

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

<b>thus </b><a NAME="E1:130"/>
 <a href="cohsp_1.html#K13" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:130_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K13" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><a NAME="E2:130_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:130_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:130_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/></div>
<b>end;</b></div>

<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="E2:130_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:130_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E4:130_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:130_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_2_1_1"><b>suppose </b></a><a NAME="E1:130_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:130_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E3:130_1_2_1_1"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_2_1_2"><b>suppose </b></a><a NAME="E1:130_1_2_1_2"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:130_1_2_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E4:130_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:130_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K13" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>hence </b><a NAME="E2:130"/>
 <a href="cohsp_1.html#K13" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K1" target="_self">FlatCoh</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T19" target="_self">Th19</a></i>;<br/>


</div>
