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<b>let </b><font color="Maroon">X0</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>reconsider </b><font color="Maroon">A</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="E2:101_3_1"/>
<font color="Maroon">X0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 ;<br/>

<a NAME="E3:101_3_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:101_3_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">X0</font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a>
 ;<br/>

<b>assume </b><a NAME="E5:101_3_1"/>
<font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a>
 ;<br/>

<a NAME="E6:101_3_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/>
<a NAME="E7:101_3_1"/><b>then </b>
<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#T15" target="_self">Th15</a>, </i>;<br/>
<b>hence </b><a NAME="E8:101_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="tex_4.html#T14" target="_self">Th14</a></i>;<br/>


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