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

<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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:65_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E3:65_1"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:65"/><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> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:65"/><i><font color="Green">E24</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E4:65"/><i><font color="Green">E36</font></i>: 
(  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T35" target="_self">Th35</a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span></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#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <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#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <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#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<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#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span></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#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></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"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font></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#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">TS</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font>

;<br/>
<a NAME="E6:65"/><b>then </b>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">P</font> &amp;  <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>, , , </i>;<br/>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
