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

<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:22"/>
( <font color="Maroon">X</font> is <a href="tdlat_3.html#V1" target="_self">discrete</a> implies for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> is <a href="tdlat_3.html#V1" target="_self">discrete</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E2:22_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:22"/>
for <font color="Olive">V</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> iff <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:22_2_1"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:22_2_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_2"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> iff <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:22"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V1" target="_self">discrete</a>
 <b>by </b><i/>;<br/>


</div>
