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

<b>let </b><font color="Maroon">X</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">Y0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:95"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">X0</font> being <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>  holds <br/>( <font color="Maroon">Y0</font> <a href="tsep_1.html#R1">misses</a> <font color="Olive">X0</font> or <font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">X0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:95_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1_1"/>
( <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_1_1_1"/>
<font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> or <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_1_1_2"/>
not <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">X0</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:95_1_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tsep_1.html#T15">TSEP_1:15</a></i>;<br/><a NAME="E4:95_1_1_1_2"/>
( <font color="Maroon">Y0</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> or <font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font> )
 <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E5:95_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> or <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a>, <a class="ref" href="tsep_1.html#D3">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="E3:95_1"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> or <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E4:95"/><b>then </b>
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:95"/>
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T12" target="_self">Th12</a></i>;<br/>


</div>
