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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:34"/>
( <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> implies 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> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</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="E2:34_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E3:34_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E1:34_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tdlat_3.html#T23" target="_self" title="TDLAT_3:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>


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

<b>assume </b><a NAME="E2:34"/><i><font color="Green" title="E21">A2</font></i>: 
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> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">A</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="E1:34_2"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><a NAME="E2:34_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E2:34"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_2"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T23" target="_self" title="TDLAT_3:th.23">Th23</a></i>;<br/>


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