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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 ;<br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:31"/>
for <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">D</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </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">Y</font>;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E21</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="E2:31_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E3:31_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1"/>
( <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> or <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:31_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="tex_4.html#E2:31_1"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E2:31_1_1"/><b>then </b>
( <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
<font color="Maroon">A</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:31"><i><font color="Green">E20</font></i></a>, </i>;<br/>


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