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<b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> &amp; not <font color="Maroon" title="c2">X0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> implies  not <font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:101_2_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> or  not <font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a> )</span><br/>

<a NAME="E3:101_2_1"/><b>then </b><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/>
<b>assume </b><a NAME="E4:101_2_1"/><i><font color="Green" title="E62">A3</font></i>: 
not <font color="Maroon" title="c2">X0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a></span><br/>

<b>assume </b><a NAME="E5:101_2_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:101_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#T68" target="_self" title="TEX_4:th.68">Th68</a></i>;<br/>
<a NAME="E7:101_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E3:101_2_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="tex_4.html#T18" target="_self" title="TEX_4:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E8:101_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_4.html#E4:101_2_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="tex_4.html#T74" target="_self" title="TEX_4:th.74">Th74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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