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

<b>let </b><font color="Maroon">T</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">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">F</font> is <a href="topgen_4.html#V2" target="_self">all-closed-containing</a> &amp; <font color="Maroon">F</font> is <a href="prob_1.html#V1">compl-closed</a> )
 ;<br/><a NAME="E2:29_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:29_1"/>
( <font color="Maroon">F</font> is <a href="topgen_4.html#V1" target="_self">all-open-containing</a> &amp; <font color="Maroon">F</font> is <a href="prob_1.html#V1">compl-closed</a> )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:29"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">F</font> is <a href="topgen_4.html#V1" target="_self">all-open-containing</a> &amp; <font color="Maroon">F</font> is <a href="prob_1.html#V1">compl-closed</a> )
 ;<br/>

<a NAME="E3:29"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:29_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
;<br/>
<a NAME="E3:29_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E4:29_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:29_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:29_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
( <font color="Maroon">F</font> is <a href="topgen_4.html#V2" target="_self">all-closed-containing</a> &amp; <font color="Maroon">F</font> is <a href="prob_1.html#V1">compl-closed</a> )
 <b>by </b><i>, </i>;<br/>


</div>
