<?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="E1:39"/><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">G</font> = <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/>
<a NAME="E2:39"/><i><font color="Green">E17</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>thus </b><a NAME="E3:39"/>
( <font color="Maroon">A</font> <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> implies ( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <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>
 ;<br/>

<b>hence </b><a NAME="E2:39_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:39"><i><font color="Green">E16</font></i></a></i>;<br/>

<a NAME="E3:39_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E4:39_1"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:39_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">d0</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:39_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="tex_1.html#T3" target="_self">Th3</a>, , , <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E4:39_1_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</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> 
 <b>by </b><i><a class="txt" href="tex_1.html#E1:39_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E5:39_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/>


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

<b>thus </b><a NAME="E4:39"/>
( not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> implies <font color="Maroon">A</font> <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_2"/>
( not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E2:39_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<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> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="tex_1.html#E1:39"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</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/>
<a NAME="E3:39_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</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> 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>assume </b><a NAME="E5:39_2"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<a NAME="E6:39_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/>
<b>assume </b><a NAME="E7:39_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:39_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</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="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:39_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:39_2"/>
<font color="Maroon">d0</font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:39_2"/><b>then </b>
for <font color="Olive">Q</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>  holds <br/>(  not <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> or  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> or  not <font color="Olive">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:39_2"/><b>then </b>
not <font color="Maroon">Z</font> <a href="hidden.html#R2">in</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/>
<a NAME="E12:39_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:39_2"/><b>then </b>
<font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:39_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E15:39_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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


</div>
