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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">T</font> is <a href="pcomps_1.html#V2" target="_self">paracompact</a>
 <b>and </b><br/><a NAME="E2:27"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:27"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> )
 ;<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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  or  ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> ) );<br/>
<b>consider </b><font color="Maroon">GX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<a NAME="E8:27"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:27_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:27_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 </span><b>by </b><i><a class="txt" href="pcomps_1.html#E3:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1_1"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:27_1_1_1_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:27_1_1_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:27_1_1_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:27_1_1_1_1_1_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="txt" href="pcomps_1.html#E5:27"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">X</font> = <font color="Maroon">W</font>;<br/>

<b>thus </b><a NAME="E6:27_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:27_1_1_1_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="txt" href="pcomps_1.html#E5:27"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a>, <a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1_2"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">X</font> = <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> ;<br/>

<b>thus </b><a NAME="E1:27_1_1_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:27"><i><font color="Green">E28</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:27_1_1_1_1_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:27_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T14">PRE_TOPC:14</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1"/>
<font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:27"/>
<font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_2"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> holds <br/><font color="Olive">X</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:27_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  or  ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> ) )
 </span><b>by </b><i>, <a class="txt" href="pcomps_1.html#E1:27"><i><font color="Green">E26</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_2"/>
 ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_2"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_2_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_2"/>
<font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">FX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">FX</font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E12:27"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">FX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E13:27"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">FX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E14:27"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">FX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>set </b><font color="Maroon">HX</font> = <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> ) </span>}</span> ;<br/>
<a NAME="E15:27"/><i><font color="Green">E86</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">FX</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">HX</font> = <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<a NAME="E17:27"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> holds <br/><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:27_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:27_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:27_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E5:27_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:27_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E8:27_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pcomps_1.html#E8:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E10:27_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  or  ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_3_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_2"/>
 ex <font color="Olive">V</font>, <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">W</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:27_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:27_3_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E5:27_3_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:27_3_1_1_2"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:27_3_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:27_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E9:27_3_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E10:27_3_1_1_2"/>
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E11:27_3_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:27_3_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E13:27_3_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="tops_1.html#T41">TOPS_1:41</a></i>;<br/><a NAME="E14:27_3_1_1_2"/><b>then </b>
<font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E15:27_3_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E16:27_3_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E17:27_3_1_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:27_3"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>

<b>take </b><font color="Maroon">Z</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">HX</font>;<br/>

<a NAME="E18:27"/>
<font color="Maroon">HX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:27"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">HX</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E20:27"/>
<font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>

<a NAME="E21:27"/>
<font color="Maroon">HX</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E1:27"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/>
<b>hence </b><a NAME="E22:27"/>
<font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>

<a NAME="E23:27"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_4"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span>
 ;<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="E3:27_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:27_4"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font></span>)</span>
 ;<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="E3:27_4_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:27_4_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">HX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E6:27_4_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:27_4_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E8:27_4_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E9:27_4_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pcomps_1.html#E8:27"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:27"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>

<b>thus </b><a NAME="E25:27"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:27_5_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:27_5_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:27_5_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E3:27_5_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:27"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E4:27_5_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E7:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pcomps_1.html#E8:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E5:27_5_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E8:27"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:27_5_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:27_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_5"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


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

<b>thus </b><a NAME="E26:27"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6"><b>proof </b></a><div class="add">

<a NAME="E1:27_6"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:27_6"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>


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


</div>
