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

<b>set </b><font color="Maroon">IT</font> =  <a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:38_1_1"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:38_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T80">ZFMISC_1:80</a></i>;<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">XX</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> =  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finsub_1.html#T26">FINSUB_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<i><font color="Green">E27</font></i>: the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">XX</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#T54">SETFAM_1:54</a></i>
<br/>.= 
<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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T55">SETFAM_1:55</a></i>
;<br/>
<a NAME="E8:38_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:38_1_1"/>
 <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E10:38_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E11:38_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E12:38_1_1"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E13:38_1_1"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:38_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<b>thus </b><a NAME="E15:38_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:38_1_1_2"/>
<font color="Maroon">a</font> <a href="subset_1.html#K8">/\</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> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:38_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="subset_1.html#K8">/\</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> <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> or <font color="Maroon">a</font> <a href="subset_1.html#K8">/\</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> <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/>
<a NAME="E6:38_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><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>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> = 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finsub_1.html#K3">/\</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K8">/\</a> <span class="p3"><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>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K8">/\</a> <span class="p3"><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>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</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="38_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:38_1_1_2_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_2_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</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="38_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:38_1_1_2_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b><i><font color="Green">E42</font></i>:  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T47">SETFAM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K3">`</a> 

;<br/><a NAME="E3:38_1_1_2_2_2"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E4:38_1_1_2_2_2"/><b>then </b>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T52">SETFAM_1:52</a></i>;<br/><a NAME="E5:38_1_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:38_1_1_2_2_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_2_2_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E16:38_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</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="38_1_1_3_1"><b>suppose </b></a><a NAME="E1:38_1_1_3_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_3_1"/>
<font color="Maroon">a</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_2"><b>suppose </b></a><a NAME="E1:38_1_1_3_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:38_1_1_3_2"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</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> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:38_1_1_3_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_8.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:38_1_1_3_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b'</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/><a NAME="E5:38_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#T10">FINSUB_1:10</a></i>;<br/><a NAME="E6:38_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">b'</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/><a NAME="E7:38_1_1_3_2"/><b>then </b>
<font color="Maroon">a</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_3_2"/>
<font color="Maroon">a</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_8.html#E7:38_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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