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

<b>thus </b><a NAME="E1:1_1_1"/>
( <font color="Maroon">A</font> is <a href="tsp_1.html#V1">T_0</a> implies for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> <a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">a</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> is <a href="tsp_1.html#V1">T_0</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:1_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>assume </b><a NAME="E3:1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</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; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="tsp_1.html#D8">TSP_1:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</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="E3:1_1_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:1_1_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:1_1_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:1_1_1_1_1_1_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_1_1_1"/>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="tex_4.html#T26">TEX_4:26</a></i>;<br/><a NAME="E3:1_1_1_1_1_1_1_1"/>
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></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> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><a NAME="E4:1_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T25">TEX_4:25</a></i>;<br/><a NAME="E6:1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_1_1_1"/>
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_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">MaxADSet</a> <font color="Maroon">a</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">MaxADSet</a> <font color="Maroon">b</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">MaxADSet</a> <font color="Maroon">a</font> &amp;  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_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; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</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="E3:1_1_1_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:1_1_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E5:1_1_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E1:1_1_1_1_1_1_2_1"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><a NAME="E2:1_1_1_1_1_1_2_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></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> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">W</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><a NAME="E3:1_1_1_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:1_1_1_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T25">TEX_4:25</a></i>;<br/><a NAME="E5:1_1_1_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_1_2_1"/>
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_1_1_1_2_1"/>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="tex_4.html#T26">TEX_4:26</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_1_1_1_2"/>
(  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">MaxADSet</a> <font color="Maroon">a</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">MaxADSet</a> <font color="Maroon">b</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">MaxADSet</a> <font color="Maroon">a</font> &amp;  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_1_1"/>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T55">TEX_4:55</a></i>;<br/>


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

<b>assume </b><a NAME="E2:1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> <a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">a</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:1_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>assume </b><a NAME="E2:1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E3:1_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_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">MaxADSet</a> <font color="Maroon">a</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">MaxADSet</a> <font color="Maroon">b</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">MaxADSet</a> <font color="Maroon">a</font> &amp;  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</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#T55">TEX_4:55</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_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">MaxADSet</a> <font color="Maroon">a</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">MaxADSet</a> <font color="Maroon">b</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="E3:1_1_1_2_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:1_1_1_2_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:1_1_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:1_1_1_2_1_1_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><a NAME="E2:1_1_1_2_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/><a NAME="E3:1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_2_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:1_1_1_2_1_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/><a NAME="E3:1_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_2_1_1_1_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_2_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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</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; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_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">MaxADSet</a> <font color="Maroon">a</font> &amp;  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</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="E3:1_1_1_2_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:1_1_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:1_1_1_2_1_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E1:1_1_1_2_1_1_2_1"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_2_1_1_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E2:1_1_1_2_1_1_2_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/><a NAME="E3:1_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_2_1_1_2_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E4:1_1_1_2_1_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/><a NAME="E5:1_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_2_1_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_2_1_1_2"/>
(  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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</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; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:1_1_1_2"/>
(  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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</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; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_1"/>
<font color="Maroon">A</font> is <a href="tsp_1.html#V1">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D8">TSP_1:def 8</a></i>;<br/>


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