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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> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:100"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:100"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a> implies <font color="Maroon">c<sub>3</sub></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">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E3:100_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tex_4.html#T68" target="_self">Th68</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">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:100_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></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">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:100_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="tex_4.html#E5:100_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tex_4.html#T69" target="_self">Th69</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green">E62</font></i></a>, <a class="txt" href="tex_4.html#E1:100_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tex_4.html#E2:100_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tex_4.html#E5:100_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tex_4.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:100_1"/>
<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="txt" href="tex_4.html#E3:100_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tex_4.html#D7" target="_self">Def7</a></i>;<br/>


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

<b>assume </b><a NAME="E3:100"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:100"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tex_4.html#T69" target="_self">Th69</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">c<sub>4</sub></font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 ;<br/><a NAME="E3:100_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T68" target="_self">Th68</a></i>;<br/><b>assume </b><a NAME="E4:100_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E5:100_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green">E62</font></i></a>, <a class="txt" href="tex_4.html#E3:100"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tex_4.html#E3:100_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tex_4.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_4.html#V4" target="_self">maximal_anti-discrete</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E5:100"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tex_4.html#D16" target="_self">Def16</a></i>;<br/>


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