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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:18"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K18" target="_self">D(p,sp)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="decomp_1.html#K8" target="_self">PO</a> <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K18" target="_self">D(p,sp)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K18" target="_self">D(p,sp)</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:18_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:18_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 ;<br/>
<a NAME="E7:18_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="decomp_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:18_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:18_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:18_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:18_1"/><i><font color="Green">E25</font></i>: 
 <a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E12:18_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="decomp_1.html#V2" target="_self">pre-open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:18_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="decomp_1.html#E7:18_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="decomp_1.html#E10:18_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="decomp_1.html#E11:18_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="decomp_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E13:18_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="decomp_1.html#V2" target="_self">pre-open</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:18_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K8" target="_self">PO</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E10:18_1"><i><font color="Green">E24</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K8" target="_self">PO</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="decomp_1.html#V2" target="_self">pre-open</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:18"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V2" target="_self">pre-open</a>
 ;<br/>
<a NAME="E7:18"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="decomp_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E8:18"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E7:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E12:18"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E10:18"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E13:18"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E14:18"/>
 <a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="decomp_1.html#E11:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="decomp_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E15:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="decomp_1.html#K2" target="_self">pInt</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E16:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K18" target="_self">D(p,sp)</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E17:18"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K18" target="_self">D(p,sp)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="decomp_1.html#E13:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
