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

<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">P1'</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> )
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27"/><i><font color="Green">E21</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:27"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:27"/>
for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span> holds <br/> ( <font color="Olive">R</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S</font> &amp; <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span>;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Q0</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Q0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q0</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:27_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">Q1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">Q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><font color="Maroon">R</font> = 
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E7:27_1_1"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E1:27_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">R'</font> = <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> ;<br/><a NAME="E3:27_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">R'</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><font color="Maroon">R'</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> = 
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E5:27_1_2"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font> is    <a href="pre_topc.html#M1" target="_self">SubSpace</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P2</font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">P1'</font>
 <b>by </b><i>, </i>;<br/>


</div>
