<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">Y0</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<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">Y0</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="E2:94"/><i><font color="Green" title="E60">A1</font></i>: 
for <font color="Olive" title="b1">X0</font> being <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  holds <br/>( <font color="Maroon" title="c2">Y0</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Olive" title="b1">X0</font> or <font color="Maroon" title="c2">Y0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:94_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1_1"/>
( <font color="Maroon" title="c4">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c4">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1_1"/>
<font color="Maroon" title="c4">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:94_1_1_1_1"/>
( <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">G</font> or <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_1_2"/>
not <font color="Maroon" title="c4">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">X0</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</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="E3:94_1_1_1_2"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c4">G</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="txt" href="tex_4.html#E1:94_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="tsep_1.html#T20" title="TSEP_1:th.20">TSEP_1:20</a></i>;<br/><a NAME="E4:94_1_1_1_2"/>
( <font color="Maroon" title="c2">Y0</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c5">X0</font> or <font color="Maroon" title="c2">Y0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">X0</font> )
 <b>by </b><i><a class="txt" href="tex_4.html#E2:94"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:94_1_1_1_2"/>
( <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">G</font> or <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="tex_4.html#E3:94_1_1_1_2"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:94_1"/>
( <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">G</font> or <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E4:94"/><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#D3" target="_self" title="TEX_4:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:94"/>
<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#T69" target="_self" title="TEX_4:th.69">Th69</a></i>;<br/>


</div>
