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

<b>let </b><font color="Maroon">PT</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">PT</font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 ;<br/>

<b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">FX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font>
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">FX</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">V1</font> is <a href="pre_topc.html#V3">open</a> &amp;  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">V1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) );<br/>
<b>consider </b><font color="Maroon">HX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> holds <br/> ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">V</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">HX</font>
;<br/>

<a NAME="E6:6"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font>  st <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> holds <br/><font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a> &amp;  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:6_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6"/>
<font color="Maroon">HX</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/>

<b>thus </b><a NAME="E8:6"/>
<font color="Maroon">HX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">HX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:6_2_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">HX</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1"><b>proof </b></a><div class="add">

<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="E1:6_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 ;<br/>

<b>then 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">PT</font> ;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E4:6_2_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#T5">PCOMPS_1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_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="E1:6_2_1_1_1_1"/>
( <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <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="6_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_2_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:6_2_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><a NAME="E3:6_2_1_1_1_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/><a NAME="E4:6_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font>, <font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E6:6_2_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , , <a class="ref" href="compts_1.html#D5">COMPTS_1:def 5</a></i>;<br/><a NAME="E7:6_2_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T36">SUBSET_1:36</a></i>;<br/><a NAME="E8:6_2_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E9:6_2_1_1_1_1_1"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><a NAME="E10:6_2_1_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/><a NAME="E11:6_2_1_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:6_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E13:6_2_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">HX</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="6_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_2_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E3:6_2_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 ;<br/><i><font color="Green">E38</font></i>: <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">PT</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i/>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T27">PRE_TOPC:27</a></i>
;<br/><a NAME="E5:6_2_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<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#T53">TOPS_1:53</a></i>;<br/><a NAME="E6:6_2_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T13">PRE_TOPC:13</a></i>;<br/><a NAME="E7:6_2_1_1_1_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="tops_1.html#T27">TOPS_1:27</a></i>;<br/><a NAME="E8:6_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:6_2_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">HX</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="E7:6_2_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">HX</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:6_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">HX</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_2"><b>proof </b></a><div class="add">

<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="E1:6_2_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">HX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">PT</font> <b>by </b><i/>;<br/>
<a NAME="E5:6_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T13">PRE_TOPC:13</a></i>;<br/>
<b>hence </b><a NAME="E6:6_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 ;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_2"/>
<font color="Maroon">HX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</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>

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

<b>assume </b><a NAME="E9:6"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 ;<br/>

<a NAME="E10:6"/><b>then </b>
 ex <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Olive">V1</font> is <a href="pre_topc.html#V3">open</a> &amp;  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">V1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:6"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> )
 ;<br/>


</div>
