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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d0</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<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="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> ;<br/>
<b>set </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span> ;<br/>
<b>thus </b><a NAME="E3:41"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> implies ( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:41_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T3" target="_self">Th3</a></i>;<br/>

<a NAME="E3:41_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:41_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a> )
 
<b>by </b><i><a class="ref" href="tex_1.html#T4" target="_self">Th4</a>, <a class="txt" href="tex_1.html#E1:41_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T18">TOPS_3:18</a></i>;<br/>


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

<b>thus </b><a NAME="E4:41"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a> implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:41_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_3.html#T2">TOPS_3:2</a></i>;<br/>
<b>assume </b><a NAME="E3:41_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E4:41_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</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>assume </b><a NAME="E5:41_2"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<a NAME="E6:41_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T18">TOPS_3:18</a></i>;<br/>
<a NAME="E7:41_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tex_1.html#T4" target="_self">Th4</a>, , , </i>;<br/>
<a NAME="E8:41_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<b>hence </b><a NAME="E9:41_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


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


</div>
