<?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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:82"/>
(  not  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> or  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E13</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 </span><b>by </b><i><a class="ref" href="tex_4.html#T8" target="_self">Th8</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:82_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:82_1_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:82_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">F</font> = <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> ;<br/><b>thus </b><a NAME="E1:82_1_1_1_1_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><b>thus </b><a NAME="E2:82_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tex_4.html#D5" target="_self">Def5</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><b>thus </b><a NAME="E3:82_1_1_1_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T10" target="_self">Th10</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:82_1_1_1_1"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:82_1_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:82_1_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:82_1_1_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">E</font> = <font color="Maroon">W</font> <a href="subset_1.html#K3">`</a> ;<br/><b>thus </b><a NAME="E1:82_1_1_1_2_1"/>
<font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><b>thus </b><a NAME="E2:82_1_1_1_2_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="tex_4.html#D5" target="_self">Def5</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><b>thus </b><a NAME="E3:82_1_1_1_2_1"/>
<font color="Maroon">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T10" target="_self">Th10</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:82_1_1_1_2"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


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

<b>assume </b><a NAME="E2:82"/><i><font color="Green">E29</font></i>: 
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:82"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="subset_1.html#R2">meets</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:82"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_2_1"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 </span><b>by </b><i><a class="ref" href="tex_4.html#T8" target="_self">Th8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_1"><b>suppose </b></a><a NAME="E1:82_2_1_1"/>
 ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:82_2_1_1"/>
<font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:82_2_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>and </b><br/><a NAME="E5:82_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">E</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E6:82_2_1_1"/>
<font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_4.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E7:82_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a>, <a class="ref" href="tex_4.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_2"><b>suppose </b></a><a NAME="E1:82_2_1_2"/>
 ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:82_2_1_2"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:82_2_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:82_2_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E6:82_2_1_2"/>
<font color="Maroon">F</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_4.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E7:82_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a>, <a class="ref" href="tex_4.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:82"/>
contradiction
 ;<br/>


</div>
