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

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



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">B</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 ;<br/><a NAME="E2:10_1"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:10_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><a NAME="E5:10_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E6:10_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E7:10_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">B</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:10"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:10"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<a NAME="E4:10"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:10"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
<font color="Maroon">B</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 <b>by </b><i/>;<br/>


</div>
