<?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" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="yellow_8.html#V1" title="YELLOW_8:attr.1">Baire</a>
 ;<br/><b>thus </b><a NAME="E2:70_1_1_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/>( <font color="Olive" title="b2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) ) holds <br/> ex <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">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" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>and </b><br/><a NAME="E2:70_1_1_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 ;<br/>

<a NAME="E3:70_1_1_1_1"/>
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">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" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:70_1_1_1_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E2:70_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E2:70_1_1_1_1"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:70_1_1_1_1_1"/>
<font color="Maroon" title="c3">S</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">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" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="waybel12.html#E1:70_1_1_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="waybel12.html#E1:70_1_1_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="yellow_8.html#D3" title="YELLOW_8:def.3">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" title="E43">A4</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/>( <font color="Olive" title="b2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) ) holds <br/> ex <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D3" title="YELLOW_8:def.3">YELLOW_8:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>and </b><br/><a NAME="E4:70_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:70_1_1"/><i><font color="Green" title="E46">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">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" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">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" title="c6">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1_2"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:70_1_1_2"/><i><font color="Green" title="E47">A8</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 ;<br/>
<a NAME="E4:70_1_1_2"/><i><font color="Green" title="E48">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:70_1_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E6:70_1_1"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:70_1_1_2"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:70_1_1_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="waybel12.html#E4:70_1_1_2"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">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" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T2" title="CARD_2:th.2">CARD_2:2</a></i>;<br/>
<a NAME="E9:70_1_1"/><b>then </b><i><font color="Green" title="E47">A10</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:70_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="waybel12.html#T2" target="_self" title="WAYBEL12:th.2">Th2</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" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="c6">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1_3"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:70_1_1_3"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E4:70_1_1_3"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>
<b>thus </b><a NAME="E5:70_1_1_3"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:70_1_1_3"><i><font color="Green" title="E48">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">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" title="c7">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:70_1_1_4"/>
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:70_1_1_4"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">A</font>
 <b>and </b><br/><a NAME="E4:70_1_1_4"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E5:70_1_1_4"/>
<font color="Maroon" title="c8">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:70_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="waybel12.html#E4:70_1_1_4"><i><font color="Green" title="E49">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:70_1_1_4"/>
( <font color="Maroon" title="c7">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="waybel12.html#E3:70_1_1_4"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a>, <a class="ref" href="tops_3.html#T35" title="TOPS_3:th.35">TOPS_3:35</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c7">I</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E14:70_1_1"/><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Maroon" title="c7">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="waybel12.html#E2:70_1_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="waybel12.html#E9:70_1_1"><i><font color="Green" title="E47">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">J</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">J</font>
;<br/>

<b>thus </b><a NAME="E16:70_1_1"/>
<font color="Maroon" title="c8">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E17:70_1_1"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c6">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">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" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_1_5"/><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:70_1_1_5"/><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c6">F1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">X1</font> = <font color="Maroon" title="c9">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel12.html#E1:70_1_1_5"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<a NAME="E4:70_1_1_5"/><i><font color="Green" title="E51">A17</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E5:70_1_1_5"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:70_1_1_5"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<a NAME="E6:70_1_1_5"/><b>then </b>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E2:70_1_1_5"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<b>hence </b><a NAME="E7:70_1_1_5"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:70_1_1_5"><i><font color="Green" title="E51">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:70_1_1"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c6">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T4" title="MSSUBFAM:th.4">MSSUBFAM:4</a></i>;<br/>
<b>hence </b><a NAME="E19:70_1_1"/>
<font color="Maroon" title="c8">J</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E14:70_1_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="tops_1.html#T79" title="TOPS_1:th.79">TOPS_1:79</a></i>;<br/>


</div>
