<?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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:28"/><i><font color="Green">E13</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:28"/><i><font color="Green">E14</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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:28"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pre_topc.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green">E15</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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E5:28"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</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="E5:28_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:28_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pre_topc.html#E2:28"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pre_topc.html#E3:28_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="pre_topc.html#E5:28_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pre_topc.html#E1:28"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E7:28_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pre_topc.html#E5:28_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:28_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><a NAME="E3:28_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:28_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pre_topc.html#E2:28"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pre_topc.html#E1:28"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pre_topc.html#E1:28_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E5:28_1_2"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:28_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1"/>
( <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:28_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:28"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E4:28"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E7:28"/>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pre_topc.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:28"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pre_topc.html#D10" target="_self">Def10</a></i>;<br/>


</div>
