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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> 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>3</sub></font>
;<br/>

<b>thus </b><a NAME="E2:57"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> as  <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="xboole_0.html#K1">{}</a>  ;<br/>
<a NAME="E5:57"/><i><font color="Green">E33</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>6</sub></font> #) as  <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#L1">TopStruct</a>  <b>by </b><i><a class="ref" href="tdlat_3.html#D1">TDLAT_3:def 1</a></i>;<br/>
<a NAME="E7:57"/><i><font color="Green">E34</font></i>: 
(  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K2">}</a></span> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T11">YELLOW_9:11</a>, <a class="ref" href="yellow_9.html#T17">YELLOW_9:17</a></i>;<br/>
<a NAME="E8:57"/><i><font color="Green">E35</font></i>: 
(  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T11">YELLOW_9:11</a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E5:57"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1"/><i><font color="Green">E38</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>thus </b><a NAME="E4:57_1"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T17">YELLOW_9:17</a></i>;<br/><b>hence </b><a NAME="E5:57_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i><a class="txt" href="topgen_2.html#E1:57_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topgen_2.html#E8:57"><i><font color="Green">E35</font></i></a>, <a class="txt" href="topgen_2.html#E7:57"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topgen_2.html#E3:57_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:57"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i><a class="ref" href="cantor_1.html#T17">CANTOR_1:17</a></i>;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</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>10</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> #);<br/><b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:57_2"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:57_2_1"/>
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>,<font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a><br/>

<a NAME="E2:57_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E1:57_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cantor_1.html#T4">CANTOR_1:4</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E3:57_2_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:57_2_1"/>
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>,<font color="Maroon">c<sub>9</sub></font> #) <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:57_2"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> #)
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E13:57"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E14:57"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> #)
 ;<br/>

<b>thus </b><a NAME="E15:57"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 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>,<font color="Maroon">c<sub>3</sub></font> #)
 <b>by </b><i><a class="txt" href="topgen_2.html#E13:57"><i><font color="Green">E37</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a><br/>

<a NAME="E16:57"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_2.html#E13:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E17:57"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:57_3"/>
<font color="Maroon">c<sub>9</sub></font> is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> #)
 <b>by </b><i><a class="txt" href="topgen_2.html#E14:57"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topgen_2.html#E1:57_3"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><a NAME="E4:57_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:57"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E20:57"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_2.html#E13:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cantor_1.html#T16">CANTOR_1:16</a></i>;<br/>
<a NAME="E21:57"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_2.html#E13:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a></i>;<br/>
<a NAME="E22:57"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  not <font color="Maroon">c<sub>9</sub></font> is <a href="struct_0.html#V3">empty</a> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E13:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E23:57"/>
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>,<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>9</sub></font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E21:57"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topgen_2.html#E17:57"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_2.html#E9:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cantor_1.html#T7">CANTOR_1:7</a>, <a class="txt" href="topgen_2.html#E7:57"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topgen_2.html#E8:57"><i><font color="Green">E35</font></i></a></i>;<br/>


</div>
