<?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> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<a NAME="E1:53"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E2:53_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:53_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:53"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:53_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><b>hence </b><a NAME="E5:53_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:53_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:53"/><i><font color="Green">E28</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>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_9.html#T23" target="_self">Th23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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>2</sub></font>;<br/><b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:53_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">a<sub>1</sub></font> <a href="subset_1.html#K3">`</a> ;<br/><i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_8.html#T16">YELLOW_8:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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>2</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="yellow_9.html#S4" target="_self">YELLOW_9:sch 4</a>()</i>
;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> ;<br/><a NAME="E7:53_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</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="E1:53_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:53_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:53_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E10:53_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="tops_2.html#V2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:53_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:53_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:53_2_3"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:53"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:53_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/><a NAME="E12:53_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<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>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="yellow_9.html#E11:53_2"><i><font color="Green">E32</font></i></a>);<br/></i><b>hence </b><a NAME="E13:53_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:53_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow_9.html#E10:53_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tops_2.html#T28">TOPS_2:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T33" target="_self">Th33</a></i>;<br/>


</div>
