<?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>thus </b><a NAME="E1:21"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">T</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">T</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="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:21_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">B</font> is <a href="decomp_1.html#V3" target="_self">pre-semi-open</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:21_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is <a href="decomp_1.html#V3" target="_self">pre-semi-open</a>
 ;<br/>
<a NAME="E7:21_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:21_1"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<a NAME="E8:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> :  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Olive">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:21_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:21_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E11:21_1"/><i><font color="Green">E28</font></i>: 
 <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Maroon">Z</font>
 ;<br/>
<a NAME="E12:21_1"/>
<font color="Maroon">Z</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:21_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="decomp_1.html#E7:21_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="decomp_1.html#E10:21_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="decomp_1.html#E11:21_1"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<a NAME="E13:21_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">B</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E10:21_1"><i><font color="Green">E27</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:21"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">B</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">K</font> is <a href="decomp_1.html#V4" target="_self">semi-pre-open</a>
 ;<br/>
<a NAME="E7:21"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">K</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">K</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">K</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:21"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
 <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">K</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E10:21"/><b>then </b>
 <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">K</font></span>)</span></span>)</span> <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">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E11:21"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</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> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">K</font></span>)</span></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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green">E42</font></i>: 
 <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">K</font></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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T58">TOPS_1:58</a></i>;<br/>
<a NAME="E13:21"/>
 <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">K</font></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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E14:21"/><b>then </b>
<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">K</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">K</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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E15:21"/><b>then </b>
<font color="Maroon">K</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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:21"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">K</font> is <a href="decomp_1.html#V3" target="_self">pre-semi-open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E17:21"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:21"/>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">B</font> is <a href="decomp_1.html#V3" target="_self">pre-semi-open</a> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E19:21"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E20:21"/>
 <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E21:21"/><b>then </b>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> :  <a href="decomp_1.html#K5" target="_self">spInt</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K4" target="_self">psInt</a> <font color="Olive">B</font> </span>}</span> 
 
;<br/>
<a NAME="E22:21"/><b>then </b>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E23:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="decomp_1.html#E5:21"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
