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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">T</font> is <a href="yellow_8.html#V1">Baire</a>
 ;<br/><b>thus </b><a NAME="E2:70_1_1_1"/>
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="card_4.html#V1">countable</a> &amp; ( for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/>( <font color="Olive">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">S</font> is <a href="tops_1.html#V1">dense</a> ) ) holds <br/> ex <font color="Olive">I</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">F</font> &amp; <font color="Olive">I</font> is <a href="tops_1.html#V1">dense</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">F</font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E2:70_1_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/>( <font color="Olive">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">S</font> is <a href="tops_1.html#V1">dense</a> )
 ;<br/>

<a NAME="E3:70_1_1_1_1"/>
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">S</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:70_1_1_1_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:70_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">S</font> is <a href="tops_1.html#V1">dense</a> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:70_1_1_1_1_1"/>
<font color="Maroon">S</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T36">TOPS_3:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:70_1_1_1_1"/>
 ex <font color="Olive">I</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> &amp; <font color="Olive">I</font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a>, <a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="yellow_8.html#D3">YELLOW_8:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:70_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="card_4.html#V1">countable</a> &amp; ( for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/>( <font color="Olive">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">S</font> is <a href="tops_1.html#V1">dense</a> ) ) holds <br/> ex <font color="Olive">I</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">F</font> &amp; <font color="Olive">I</font> is <a href="tops_1.html#V1">dense</a> )
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D3">YELLOW_8:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">F</font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E4:70_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">S</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 ;<br/>

<b>set </b><font color="Maroon">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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">T</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">T</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:70_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <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">T</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E7:70_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:70_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:70_1_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E4:70_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:70_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:70_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:70_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T2">CARD_2:2</a></i>;<br/>
<a NAME="E9:70_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a>, </i>;<br/>
<a NAME="E10:70_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:70_1_1_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:70_1_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:70_1_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>
<b>thus </b><a NAME="E5:70_1_1_3"/>
<font color="Maroon">q</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">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:70_1_1_4"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:70_1_1_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:70_1_1_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E5:70_1_1_4"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="waybel12.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E6:70_1_1_4"/>
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">S</font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i>, <a class="ref" href="tops_1.html#T51">TOPS_1:51</a>, <a class="ref" href="tops_3.html#T35">TOPS_3:35</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">I</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E14:70_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">I</font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">J</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>thus </b><a NAME="E16:70_1_1"/>
<font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E17:70_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_1_5"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">q</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="E2:70_1_1_5"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:70_1_1_5"/><i><font color="Green">E113</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E5:70_1_1_5"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:70_1_1_5"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>hence </b><a NAME="E7:70_1_1_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:70_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T4">MSSUBFAM:4</a></i>;<br/>
<b>hence </b><a NAME="E19:70_1_1"/>
<font color="Maroon">J</font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T79">TOPS_1:79</a></i>;<br/>


</div>
