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<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">Y0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:100"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 ;<br/>

<b>thus </b><a NAME="E2:100"/>
( <font color="Maroon">Y0</font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a> implies <font color="Maroon">A</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">Y0</font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a>
 ;<br/>

<a NAME="E2:100_1"/><b>then </b>
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:100_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:100_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">D</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 ;<br/><b>assume </b><a NAME="E2:100_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E3:100_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">X0</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:100_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10">TSEP_1:10</a></i>;<br/><a NAME="E6:100_1_1"/>
<font color="Maroon">X0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:100_1"/>
<font color="Maroon">A</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, </i>;<br/>


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

<b>assume </b><a NAME="E3:100"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a>
 ;<br/>

<a NAME="E4:100"/><b>then </b>
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:100"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="tex_4.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>reconsider </b><font color="Maroon">D</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><b>assume </b><a NAME="E2:100_2"/>
<font color="Maroon">X0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 ;<br/><a NAME="E3:100_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">D</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E4:100_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/><b>hence </b><a NAME="E5:100_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
<font color="Maroon">Y0</font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T13" target="_self">Th13</a></i>;<br/>


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