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<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>



<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> implies  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:66"/><i><font color="Green">E21</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span>
 ;<br/>

<a NAME="E3:66"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E4:66"/>
 <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:66"/><b>then </b>
<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#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a> <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#K9">/\</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E6:66"/><b>then </b><i><font color="Green">E36</font></i>: 
<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#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">TS</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:66"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> = 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tops_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">TS</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>
<b>by </b><i>, <a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E9:66"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


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