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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y0</font> being 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><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y0</font> holds <br/>( <font color="Olive" title="b1">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> iff <font color="Olive" title="b2">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> )</span><br/><b>let </b><font color="Maroon" title="c2">Y0</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"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> holds <br/>( <font color="Maroon" title="c2">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> iff <font color="Olive" title="b1">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> )</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> implies ( <font color="Maroon" title="c2">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> iff <font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> ) )</span><br/>





<b>assume </b><a NAME="E1:100"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</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">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> iff <font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> )</span><br/>

<b>thus </b><a NAME="E2:100"/>
( <font color="Maroon" title="c2">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> implies <font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> implies <font color="Maroon" title="c2">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a> )</span><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" title="E62">A2</font></i>: 
<font color="Maroon" title="c2">Y0</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"> <font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a></span><br/>

<a NAME="E2:100_1"/><b>then </b>
<font color="Maroon" title="c2">Y0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#D16" target="_self" title="TEX_4:def.16">Def16</a></i>;<br/>
<a NAME="E3:100_1"/><b>then </b><i><font color="Green" title="E63">A3</font></i>: 
<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="txt" href="tex_4.html#E1:100"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="tex_4.html#T68" target="_self" title="TEX_4:th.68">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" title="c4">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">D</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> )</span><br/><b>assume </b><a NAME="E1:100_1_1"/><i><font color="Green" title="E64">A4</font></i>: 
<font color="Maroon" title="c4">D</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> )</span><br/><b>assume </b><a NAME="E2:100_1_1"/><i><font color="Green" title="E65">A5</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font></span><br/><a NAME="E3:100_1_1"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">X0</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:100_1_1"/><i><font color="Green" title="E66">A6</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">X0</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10" title="TSEP_1:th.10">TSEP_1:10</a></i>;<br/><a NAME="E6:100_1_1"/>
<font color="Maroon" title="c5">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100_1_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="tex_4.html#E5:100_1_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="tex_4.html#T69" target="_self" title="TEX_4:th.69">Th69</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="tex_4.html#E1:100_1"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="tex_4.html#E2:100_1_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="tex_4.html#E5:100_1_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="tex_4.html#D16" target="_self" title="TEX_4:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:100_1"/>
<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#E3:100_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:100"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</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">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a></span><br/>

<a NAME="E4:100"/><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#D7" target="_self" title="TEX_4:def.7">Def7</a></i>;<br/>
<a NAME="E5:100"/><b>then </b><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon" title="c2">Y0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="tex_4.html#T69" target="_self" title="TEX_4:th.69">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" title="c4">X0</font> be    <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="c4">X0</font> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c5">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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="E2:100_2"/>
<font color="Maroon" title="c4">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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> )</span><br/><a NAME="E3:100_2"/><b>then </b><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c5">D</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/><b>assume </b><a NAME="E4:100_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font></span><br/><b>hence </b><a NAME="E5:100_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:100"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="tex_4.html#E3:100"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="tex_4.html#E3:100_2"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
<font color="Maroon" title="c2">Y0</font> is <a href="tex_4.html#V4" title="TEX_4:attr.4">maximal_anti-discrete</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E5:100"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="tex_4.html#D16" target="_self" title="TEX_4:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
