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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:43"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) implies ( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> implies <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 ;<br/>

<b>hence </b><a NAME="E2:43_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="tex_1.html#E1:43_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>thus </b><a NAME="E2:43_1_1"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tex_1.html#T24" target="_self">Th24</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:43_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_3.html#T79">TOPS_3:79</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:43_1_1"/>
( not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a> implies <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E2:43_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="tex_1.html#E1:43_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_3.html#T79">TOPS_3:79</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:43_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tex_1.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> implies <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


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

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:43"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> implies <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1"><b>case </b></a><a NAME="E1:43_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:43_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_2_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_2"><b>case </b></a><a NAME="E1:43_2_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="tex_1.html#E3:43"><i><font color="Green">E21</font></i></a>, <a class="txt" href="tex_1.html#E1:43_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tex_1.html#E1:43_2_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_2_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_2_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_2_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_2_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_1"><b>case </b></a><a NAME="E1:43_2_2_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:43_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2"><b>case </b></a><a NAME="E1:43_2_2_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_2_2_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:43_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tex_1.html#E1:43_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tex_1.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E3:43_2_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tex_1.html#E3:43"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_2_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:43_2"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tex_1.html#E2:43_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tex_1.html#E2:43"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_3.html#T73">TOPS_3:73</a></i>;<br/>


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