<?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="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #);<br/>
<a NAME="E1:22"/><i><font color="Green">E17</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T8" target="_self">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)</span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T80">ZFMISC_1:80</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E3:22_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>thus </b><a NAME="E4:22_1"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<b>thus </b><a NAME="E4:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<a NAME="E4:22_2"/><i><font color="Green">E20</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T17">BORSUK_1:17</a></i>;<br/>
<a NAME="E5:22_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> ;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_2_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:22_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:22_2_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E1:22_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cantor_1.html#E1:22_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:22_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E3:22_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cantor_1.html#E1:22_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cantor_1.html#E4:22_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="cantor_1.html#E5:22_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E7:22_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="cantor_1.html#E5:22_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>7</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="E2:22_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:22_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:22_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)<b> such that </b><br/><a NAME="E7:22_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:22_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E3:22_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cantor_1.html#E5:22_2_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:22_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E4:22_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cantor_1.html#E7:22_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="cantor_1.html#E8:22_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 
;<br/>
<b>hence </b><a NAME="E8:22_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="txt" href="cantor_1.html#E4:22_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cantor_1.html#E5:22_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #);<br/>

<b>assume </b><a NAME="E5:22"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:22"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:22"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>assume </b><a NAME="E9:22"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E12:22"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:22"/><i><font color="Green">E22</font></i>: 
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E7:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cantor_1.html#E11:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cantor_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E14:22"/><b>then </b>
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:22"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E16:22"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E8:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cantor_1.html#E12:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setfam_1.html#T39">SETFAM_1:39</a></i>;<br/>
<b>hence </b><a NAME="E17:22"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="txt" href="cantor_1.html#E13:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cantor_1.html#E15:22"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/>


</div>
