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

<b>let </b><font color="Maroon">X</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">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">D</font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<a NAME="E4:4"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="tmap_1.html#K5">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tmap_1.html#D7">TMAP_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:4"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E10:4"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/>
<a NAME="E11:4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E12:4"/><b>then </b>
( <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:4"/><b>then </b>
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E14:4"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> </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/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>

;<br/>
<a NAME="E16:4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E17:4"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E18:4"/><b>then </b>
( <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E19:4"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a>, <a class="ref" href="tsep_1.html#T40">TSEP_1:40</a></i>;<br/>
<a NAME="E20:4"/><b>then </b><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</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>thus </b><a NAME="E21:4"/>
<font color="Maroon">C</font> is <a href="tops_1.html#V1">dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_2"/>
not <font color="Maroon">C</font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<a NAME="E2:4_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a></i>;<br/>
<a NAME="E3:4_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_3.html#T2">TOPS_3:2</a></i>;<br/>
<b>hence </b><a NAME="E4:4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


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

<b>thus </b><a NAME="E22:4"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T105">TMAP_1:105</a></i>;<br/>


</div>
