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<b>let </b><font color="Maroon">GX</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:104"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">R</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>thus </b><a NAME="E3:104"/>
( <font color="Maroon">R</font> is <a href="tops_1.html#V5" target="_self">closed_condensed</a> iff <font color="Maroon">R</font> is <a href="tops_1.html#V6" target="_self">open_condensed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<a NAME="E1:104_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T65" target="_self">Th65</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E2:104_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T63" target="_self">Th63</a>, <a class="ref" href="tops_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E3:104_1"/>
( <font color="Maroon">R</font> is <a href="tops_1.html#V5" target="_self">closed_condensed</a> iff <font color="Maroon">R</font> is <a href="tops_1.html#V6" target="_self">open_condensed</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T66" target="_self">Th66</a>, <a class="ref" href="tops_1.html#T54" target="_self">Th54</a>, <a class="ref" href="tops_1.html#T55" target="_self">Th55</a></i>;<br/>


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


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