<?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="E1:16"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K7" target="_self">SO</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#K16" target="_self">D(alpha,s)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
  <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="16_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="E1:16_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#K7" target="_self">SO</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#K16" target="_self">D(alpha,s)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K7" target="_self">SO</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#K16" target="_self">D(alpha,s)</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="E3:16_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#V1" target="_self">semi-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="E5:16_1"/><i><font color="Green">E19</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="E6:16_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V1" target="_self">semi-open</a>
 ;<br/>
<a NAME="E7:16_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:16_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="decomp_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:16_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#K3" target="_self">alphaInt</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:16_1"><i><font color="Green">E18</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="E10:16_1"/><i><font color="Green">E22</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="E11:16_1"/><i><font color="Green">E23</font></i>: 
 <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E12:16_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:16_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="decomp_1.html#E7:16_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="decomp_1.html#E10:16_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="decomp_1.html#E11:16_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="decomp_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E13:16_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#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:16_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 <b>by </b><i><a class="txt" href="decomp_1.html#E10:16_1"><i><font color="Green">E22</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="E2:16"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 ;<br/>

<a NAME="E3:16"/><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#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font> </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="E5:16"/><i><font color="Green">E18</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="E6:16"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E7:16"/><i><font color="Green">E20</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> <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>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="decomp_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:16"/>
 <a href="tops_1.html#K1">Int</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="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E7:16"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:16"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V1" target="_self">semi-open</a>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E11:16"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:16"/><i><font color="Green">E23</font></i>: 
<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#V1" target="_self">semi-open</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E10:16"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E13:16"/>
 <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="decomp_1.html#E11:16"><i><font color="Green">E22</font></i></a>, <a class="ref" href="decomp_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E14:16"/><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#K3" target="_self">alphaInt</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K1" target="_self">sInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E15:16"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="decomp_1.html#E12:16"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:16"/>
<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#K7" target="_self">SO</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#K16" target="_self">D(alpha,s)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E5:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
