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

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</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/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">F</font> is <a href="tops_2.html#V2" target="_self">closed</a> implies  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> is <a href="tops_2.html#V1" target="_self">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V2" target="_self">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" target="_self">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E2:10_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">P</font> <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="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:10_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon">P</font> 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/>


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

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> is <a href="tops_2.html#V1" target="_self">open</a>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" target="_self">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E3:10"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E4:10"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E5:10"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
