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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:76"/>
(  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> implies  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="tex_4.html#E65"><i><font color="Green">Lemma49</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:76"/><i><font color="Green">E51</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:76"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_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="E1:76_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="tex_4.html#E1:76_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:76_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tex_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E3:76_1_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E65"><i><font color="Green">Lemma49</font></i></a></i>;<br/><a NAME="E4:76_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:76_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E5:76_1_1_1_1_1"/>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:76"><i><font color="Green">E51</font></i></a>, <a class="txt" href="tex_4.html#E3:76_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E6:76_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E4:76_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:76_1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tex_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E3:76_1_1_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E65"><i><font color="Green">Lemma49</font></i></a></i>;<br/><a NAME="E4:76_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:76_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E5:76_1_1_1_1_2"/>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:76"><i><font color="Green">E51</font></i></a>, <a class="txt" href="tex_4.html#E3:76_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E6:76_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E4:76_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:76_1_1"/>
<span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:76_1"/>
<span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:76"/><i><font color="Green">E53</font></i>: 
(  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a> &amp;  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></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#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:76"/>
(  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:76"/><b>then </b>
(  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="tex_4.html#E3:76"><i><font color="Green">E52</font></i></a>, <a class="txt" href="tex_4.html#E4:76"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tex_4.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E7:76"/>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
