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

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:104"/><i><font color="Green">E52</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topgen_4.html#K1" target="_self">TotFam</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:104_1"/>
 <a href="topgen_4.html#K1" target="_self">TotFam</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:104_1"/>
 <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:104"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:104_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:104_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E4:104_2"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="prob_1.html#NM1">Field_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:104_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="topgen_4.html#T66" target="_self">Th66</a></i>;<br/><b>hence </b><a NAME="E5:104_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:104"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:104"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:104"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>5</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:104_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<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>1</sub></font><b> such that </b><br/><a NAME="E3:104_3_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E4:104_3_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:104_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="topgen_4.html#T66" target="_self">Th66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:104_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104_3"><i><font color="Green">E56</font></i></a></i>;<br/><b>hence </b><a NAME="E2:104_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104_3_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:104_3_1"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:104_3_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_3"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:104"/>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4"><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 <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:104_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:104_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/><a NAME="E2:104_4_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>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>1</sub></font><b> such that </b><br/><a NAME="E4:104_4_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104_4_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E5:104_4_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:104_4_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="topgen_4.html#T66" target="_self">Th66</a></i>;<br/><b>hence </b><a NAME="E6:104_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:104_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  as   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_4.html#E2:104"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topgen_4.html#E4:104"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topgen_4.html#E5:104"><i><font color="Green">E55</font></i></a>, <a class="ref" href="prob_1.html#T32">PROB_1:32</a></i>;<br/>
<a NAME="E8:104"/>
for <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 <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5"><b>proof </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>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:104_5"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:104_5"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5_1"><b>proof </b></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="E1:104_5_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:104_5_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:104_5_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104_5"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_4.html#E3:104_5_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="topgen_4.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_5"/>
<font color="Maroon">c<sub>6</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="topgen_4.html#E1:104"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topgen_4.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_6"><b>now </b></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="E1:104_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/><a NAME="E2:104_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:104_6"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:104"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <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="104_7"><b>proof </b></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="E1:104_7"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</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="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:104_7"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:104_7"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:104_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:104"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="prob_1.html#V1">compl-closed</a> <a href="topgen_4.html#V1" target="_self">all-open-containing</a> <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:104"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topgen_4.html#E10:104"><i><font color="Green">E56</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


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