<?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>assume </b><a NAME="E1:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pcomps_1.html#D9">PCOMPS_1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E4:27"/><i><font color="Green">E28</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nagata_2.html#E4:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">a<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> ;<br/>
<a NAME="E6:27"/><i><font color="Green">E30</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </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="E1:27_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:27_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></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 <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_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> ; <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="E1:27_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font>)
 ;<br/>

<a NAME="E2:27_1_1"/>
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>6</sub></font>,<a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:27_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:27_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_1"/>
<font color="Maroon">c<sub>7</sub></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 <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:27"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E31</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="nagata_2.html#E6:27"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Olive">b<sub>1</sub></font> is <a href="tops_2.html#V1">open</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> );<br/>
<a NAME="E9:27"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:27_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>7</sub></font>,<a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E2:27_2"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:27_2_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></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="27_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:27_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E4:27"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E3:27_2_1_1"/>
2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E4:27_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E5:27_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:27_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E6:27_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:27"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:27_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:27_2_1"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:27_2_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_2_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:27_2_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nagata_2.html#E3:27_2_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:27_2_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E7:27_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nagata_2.html#E3:27_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E6:27_2_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:27_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:27_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E5:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:27_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:27_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>)
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:27"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E3:27_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_2"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:27_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/><b>hence </b><a NAME="E5:27_2_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:27_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:27_2"/><i><font color="Green">E35</font></i>: 
(  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> is <a href="tops_2.html#V1">open</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nagata_2.html#E3:27_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nagata_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E8:27_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E9:27_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:27_2"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="nagata_2.html#E9:27"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>;<br/>
<a NAME="E12:27"/><i><font color="Green">E34</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:27_3"/><i><font color="Green">E35</font></i>: 
(  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_2.html#V2">onto</a> &amp;  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E4:27_3"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:27_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:27_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:27_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E9:27_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:27_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green">E38</font></i>: 
<a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:27_3_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:27_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:27_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E6:27_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nagata_2.html#E8:27_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nagata_2.html#E1:27_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/><a NAME="E3:27_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_3_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:27_3"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><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></span>)</span><b> such that </b><br/><a NAME="E14:27"/><i><font color="Green">E35</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="nagata_2.html#E12:27"><i><font color="Green">E34</font></i></a>);<br/></i>
<a NAME="E15:27"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<a NAME="E1:27_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V1">discrete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:27_4_1"/>
 <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E2:27_4_1"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_4_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:27_4_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:27_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:27_4_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>13</sub></font> is <a href="tops_2.html#V1">open</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>13</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>13</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_2.html#E6:27_4_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E9:27_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="nagata_1.html#V1">discrete</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:27_4_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_2.html#E8:27_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nagata_1.html#D2">NAGATA_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:27_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="nagata_1.html#V1">discrete</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E4:27_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nagata_2.html#E6:27_4_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_2.html#E8:27_4_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_4"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a>
 <b>by </b><i><a class="ref" href="nagata_1.html#D2">NAGATA_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:27"/><i><font color="Green">E37</font></i>: 
 <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:27_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:27_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E3:27_5"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E5:27_5"/>
 <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E6:27_5"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:27_5"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:27_5"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:27_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:27_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:27_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:27_5"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>16</sub></font> is <a href="tops_2.html#V1">open</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>16</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>16</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_2.html#E10:27_5"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E15:27_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E14:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E3:27_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_2.html#E8:27_5"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nagata_2.html#E10:27_5"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E16:27_5"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>15</sub></font> is <a href="tops_2.html#V1">open</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:27_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nagata_2.html#E14:27_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E17:27_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:27_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:27"/>
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/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</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="27_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:27_6"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:27_6"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E3:27_6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nagata_2.html#E1:27_6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E2:27_6"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:27_6"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:27_6"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nagata_2.html#E3:27_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:27_6"/><i><font color="Green">E42</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:27_6"><i><font color="Green">E41</font></i></a>, <a class="ref" href="prepower.html#T106">PREPOWER:106</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:27_6"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font> is <a href="tops_2.html#V1">open</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:27"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E11:27_6"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:27_6"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E12:27_6"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:27_6"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E3:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nagata_2.html#E14:27_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:27_6"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="cantor_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E10:27_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nagata_2.html#E14:27_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<a NAME="E18:27_6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K1">In</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E19:27_6"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1)
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nagata_2.html#E17:27_6"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E21:27_6"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E18:27_6"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E23:27_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:27_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E24:27_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="cantor_1.html#K3">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E14:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E10:27_6"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E25:27_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:27_6_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:27_6_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E17:27_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nagata_2.html#E21:27_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E3:27_6_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E4:27_6_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:27_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nagata_2.html#E17:27_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nagata_2.html#E21:27_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E5:27_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:27_6_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:27_6_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:27_6_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:27_6_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E8:27_6_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/><a NAME="E9:27_6_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:27_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:27_6_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_6_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:27_6"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:27_6"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nagata_2.html#E14:27_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nagata_2.html#E23:27_6"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nagata_2.html#E24:27_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E27:27_6"/>
ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:27"/><b>then </b>
 <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E16:27"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>
<a NAME="E19:27"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="nagata_1.html#V5">Basis_sigma_discrete</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nagata_1.html#D5">NAGATA_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E20:27"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V5">Basis_sigma_discrete</a>
 ;<br/>


</div>
