<?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>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="E1:7"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">PT</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">PT</font> is <a href="pcomps_1.html#V2">paracompact</a>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E24</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="E4:7"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">FX</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<a NAME="E5:7"/>
<font color="Maroon">PT</font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 
<b>by </b><i>, , <a class="ref" href="pcomps_1.html#T27">PCOMPS_1:27</a></i>;<br/>
<b>then 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="E7:7"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">HX</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">HX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font> &amp; ( 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/> 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">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) ) )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">GX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font> &amp; <font color="Maroon">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">HX</font> &amp; <font color="Maroon">GX</font> is <a href="pcomps_1.html#V1">locally_finite</a> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D4">PCOMPS_1:def 4</a></i>;<br/>
<a NAME="E10:7"/><i><font color="Green">E29</font></i>: 
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">GX</font> holds <br/> 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">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="E1:7_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<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="E3:7_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</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">PT</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E6:7_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">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">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<a NAME="E7:7_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<b>hence </b><a NAME="E8:7_1"/>
( <font color="Maroon">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="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:7"/>
 <a href="pcomps_1.html#K2">clf</a> <font color="Maroon">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">FX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2">clf</a> <font color="Maroon">GX</font>
 ;<br/>

<b>then 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">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="E4:7_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E6:7_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">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="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E7:7_2"/>
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:7"/>
 ex <font color="Olive">GX</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">GX</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Olive">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font> &amp;  <a href="pcomps_1.html#K2">clf</a> <font color="Olive">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">FX</font> &amp; <font color="Olive">GX</font> is <a href="pcomps_1.html#V1">locally_finite</a> )
 <b>by </b><i/>;<br/>


</div>
