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

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K3">RAT</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  st <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">a</font></span><a href="realset1.html#K5">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:72"/>
 ex <font color="Olive">F</font> being   <a href="hidden.html#M1">set</a>  st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 
<b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">F</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 ;<br/>
<a NAME="E5:72"/><i><font color="Green">E50</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="numbers.html#T12">NUMBERS:12</a>, <a class="ref" href="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/>
<a NAME="E6:72"/>
<font color="Maroon">F</font> <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 <a href="topmetr.html#K3">R^1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:72_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:72_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:72"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E8:72"/>
 ex <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> &amp; <font color="Olive">F</font> is <a href="card_4.html#V1">countable</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E1:72_2"/>
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">B</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:72_2_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a> <b> such that </b><br/><a NAME="E3:72_2_1"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">a</font></span><a href="realset1.html#K5">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:72_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="numbers.html#T12">NUMBERS:12</a></i>;<br/>
<a NAME="E6:72_2_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:72_2_1"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_2"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>

<a NAME="E3:72_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">x</font></span><a href="realset1.html#K5">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:72_2_2"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">x</font></span><a href="realset1.html#K5">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:72_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:72_2_2_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:72_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">x</font></span><a href="realset1.html#K5">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">y</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="E2:72_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">x</font></span><a href="realset1.html#K5">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a> <b> such that </b><br/><a NAME="E4:72_2_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">z</font></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">z</font>] )
 ;<br/>
<a NAME="E5:72_2_2"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">z</font></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:72_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:72_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">x</font></span><a href="realset1.html#K5">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span>  is <a href="card_4.html#V1">countable</a>
 
<b>from </b><i><a class="ref" href="topgen_4.html#S1" target="_self">TOPGEN_4:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E5:72_2"/>
<font color="Maroon">F</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:72_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:72_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  <b>by </b><i><a class="txt" href="topgen_4.html#E1:72"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E3:72_2_3"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">x</font></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:72_2_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">x</font></span><a href="realset1.html#K5">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:72_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">x</font></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:72_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E7:72_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:72_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a> <b> such that </b><br/><a NAME="E11:72_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">a</font></span><a href="realset1.html#K5">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:72_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:72_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:72"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:72"/>
<font color="Maroon">A</font> is <a href="topgen_4.html#V5" target="_self">F_sigma</a>
 <b>by </b><i/>;<br/>


</div>
