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

<b>let </b><font color="Maroon">PT</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="E1:12"/>
<font color="Maroon">PT</font> is <a href="pcomps_1.html#V3">metrizable</a>
 ;<br/>

<b>then consider </b><font color="Maroon">metr</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">PT</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PT</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">metr</font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PT</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E23</font></i>: 
 <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">PT</font>,<font color="Maroon">metr</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D9">PCOMPS_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">PM</font> being  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a><b> such that </b><br/><a NAME="E6:12"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">PM</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PT</font>,<font color="Maroon">metr</font>
 ;<br/>
<b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">FX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font>
 <b>and </b><br/><a NAME="E8:12"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FX</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E10:12"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T26">WELLORD2:26</a></i>;<br/>
<b>consider </b><font color="Maroon">Mn</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E12:12"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Mn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">FX</font>
 ;<br/>
<a NAME="E13:12"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Mn</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">UB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> ;<br/>
<a NAME="E14:12"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">UB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:12_1"/>
<font color="Maroon">UB</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">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_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="E1:12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:12_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/>
<a NAME="E5:12_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_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="E1:12_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E4:12_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E6:12_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E7:12_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:12_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_1_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 <font color="Maroon">PM</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">UB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a<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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>,  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>;<br/>
<b>consider </b><font color="Maroon">f</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">PM</font></span>)</span><b> such that </b><br/><a NAME="E17:12"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">UB</font>
 <b>and </b><br/><a NAME="E18:12"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>,<font color="Olive">n</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">V</font>,<font color="Olive">n</font>] &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">V</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="pcomps_2.html#S3" target="_self">PCOMPS_2:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>;<br/>
<b>consider </b><font color="Maroon">GX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E20:12"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">X</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">GX</font> = <font color="Maroon">GX</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PT</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">GX</font>
;<br/>

<b>thus </b><a NAME="E22:12"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_2"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> holds <br/><font color="Olive">X</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> ;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:12_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:12_2_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E2:12_2_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:12_2_1_1_1_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> ;<br/>
<a NAME="E4:12_2_1_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_2_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_2_1_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E4:12_2_1_1_1_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/>
<b>thus </b><a NAME="E5:12_2_1_1_1_1_1_1"/>
<font color="Maroon">a</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">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E6:12_2_1_1_1_1_1"/>
<font color="Maroon">NF</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_2_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_2_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E4:12_2_1_1_1_1_1_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/>
<b>thus </b><a NAME="E5:12_2_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_2_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#T36">PCOMPS_1:36</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_2_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>thus </b><a NAME="E5:12_2_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_2_1_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_2_1_1_1_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:12_2_1_1_1_2_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:12_2_1_1_1_2_1"/>
<font color="Maroon">NF</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">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_2_1_1_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_2_1_1_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:12_2_1_1_1_2_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  )
 ;<br/>
<b>thus </b><a NAME="E5:12_2_1_1_1_2_1_1"/>
<font color="Maroon">a</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">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">NF</font> = <font color="Maroon">NF</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E8:12_2_1_1_1_2_1"/>
<font color="Maroon">NF</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_2_1_1_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_2_1_1_1_2_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:12_2_1_1_1_2_1_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  )
 ;<br/>
<b>thus </b><a NAME="E5:12_2_1_1_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:12_2_1_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#T36">PCOMPS_1:36</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_2_1"/>
<font color="Maroon">X</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>
<b>hence </b><a NAME="E2:12_2"/>
<font color="Maroon">GX</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/>


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

<b>thus </b><a NAME="E23:12"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<a NAME="E1:12_3"/><i><font color="Green">E53</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_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>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:12_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 ;<br/>

<a NAME="E2:12_3_1"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#T5">PCOMPS_1:5</a></i>;<br/>
<a NAME="E3:12_3_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">G</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">G</font>] )
 
;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E6:12_3_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">X</font>]
 <b>and </b><br/><a NAME="E7:12_3_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">Y</font>] holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Olive">Y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 <b>from </b><i><a class="ref" href="pcomps_2.html#S1" target="_self">PCOMPS_2:sch 1</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:12_3_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E10:12_3_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:12_3_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E15:12_3_1"/><i><font color="Green">E101</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E3:12"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pcomps_2.html#E6:12"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>;<br/>
<a NAME="E16:12_3_1"/><i><font color="Green">E111</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E7:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="prepower.html#T106">PREPOWER:106</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:12_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E19:12_3_1"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E20:12_3_1"/><i><font color="Green">E113</font></i>: 
2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E21:12_3_1"/>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></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="E22:12_3_1"/><b>then </b>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T146">REAL_2:146</a></i>;<br/>
<a NAME="E23:12_3_1"/><b>then </b>
3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:12"><i><font color="Green">E29</font></i></a>, <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/>
<a NAME="E24:12_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E10:12"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E25:12_3_1"/><i><font color="Green">E115</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 ;<br/>

<a NAME="E26:12_3_1"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">V</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:12_3_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:12_3_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:12_3_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:12_3_1"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds not <font color="Maroon">x</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_3_1_2"/>
<font color="Maroon">x</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_3_1_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:12_3_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">W</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> ;<br/><a NAME="E1:12_3_1_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E2:12_3_1_3_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>
<a NAME="E3:12_3_1_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E4:12_3_1_3_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E5:12_3_1_3_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E6:12_3_1_3_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_1_3_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font>
 ;<br/>

<b>then consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_3_1_3_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:12_3_1_3_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">FX</font> = <font color="Maroon">FX</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:12_3_1_3_1_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> )
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E7:12_3_1_3_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Mn</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E8:12_3_1_3_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:12_3_1_3_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">M</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E10:12_3_1_3_1_1"/>
<font color="Maroon">Mn</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:12_3_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="txt" href="pcomps_2.html#E20:12"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pcomps_2.html#E22:12"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12_3_1_3_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E3:12"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:12_3_1_3_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E13:12"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<a NAME="E9:12_3_1_3_1"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:12_3_1_3_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_1_3_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_3_1_3_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E4:12_3_1_3_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:12_3_1_3_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E7:12_3_1_3_1_2"/>
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E22:12"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:12_3_1_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E20:12"><i><font color="Green">E39</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:12_3_1_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:12_3_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </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>reconsider </b><font color="Maroon">W</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">X</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:12_3_1_3"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:12_3_1_3"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:12_3_1_3"/>
 ex <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">l</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:12_3_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12_3"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2"><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="E1:12_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_3_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:12_3_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">PT</font> <b>by </b><i>, </i>;<br/>
<a NAME="E6:12_3_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PT</font>
 
;<br/>
<b>hence </b><a NAME="E7:12_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12_3"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PT</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GX</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:12_3"/>
<font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">PT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E24:12"/>
<font color="Maroon">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">FX</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<a NAME="E1:12_4"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_4_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> ;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:12_4_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_4_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_4_1_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:12_4_1_1_1_1"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E2:12_4_1_1_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:12_4_1_1_1_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> ;<br/>
<a NAME="E4:12_4_1_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_4_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_4_1_1_1_1_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E4:12_4_1_1_1_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/>
<b>thus </b><a NAME="E5:12_4_1_1_1_1_1_1"/>
<font color="Maroon">a</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">PM</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E6:12_4_1_1_1_1_1"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font> holds <br/><font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_4_1_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_4_1_1_1_1_1_2"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:12_4_1_1_1_1_1_2"/><i><font color="Green">E143</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>
<a NAME="E5:12_4_1_1_1_1_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:12_4_1_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E6:12"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pcomps_2.html#E7:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">V</font>;<br/>

<b>thus </b><a NAME="E8:12_4_1_1_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E3:12"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:12_4_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_4_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_4_1_1_1_2"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>thus </b><a NAME="E5:12_4_1_1_1_2"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_4_1_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_4_1_1_1_2_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:12_4_1_1_1_2_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:12_4_1_1_1_2_1"/>
<font color="Maroon">NF</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">PM</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:12_4_1_1_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_4_1_1_1_2_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:12_4_1_1_1_2_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  )
 ;<br/>
<b>thus </b><a NAME="E5:12_4_1_1_1_2_1_1"/>
<font color="Maroon">a</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">PM</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">NF</font> = <font color="Maroon">NF</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E8:12_4_1_1_1_2_1"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font> holds <br/><font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1_1_2_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_4_1_1_1_2_1_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_4_1_1_1_2_1_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">V</font>,<font color="Maroon">Mn</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:12_4_1_1_1_2_1_2"/><i><font color="Green">E150</font></i>: 
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> </span>}</span>  )
 ;<br/>
<a NAME="E5:12_4_1_1_1_2_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E6:12_4_1_1_1_2_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E7:12_4_1_1_1_2_1_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:12_4_1_1_1_2_1_2"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E6:12"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pcomps_2.html#E7:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">V</font>;<br/>

<b>thus </b><a NAME="E10:12_4_1_1_1_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E3:12"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:12_4_1_1_1_2_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_4_1"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_4"/>
<font color="Maroon">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E25:12"/>
<font color="Maroon">GX</font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<a NAME="E1:12_5"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">PT</font>  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">PT</font>;<br/>

<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E2:12_5_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:12_5_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#T5">PCOMPS_1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:12_5_1"/><i><font color="Green">E154</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:12_5_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E8:12_5_1"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E9:12_5_1"/>
<font color="Maroon">X</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/>
<a NAME="E10:12_5_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:12_5_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E14:12_5_1"/><i><font color="Green">E158</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E3:12"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:12_5_1"/><i><font color="Green">E160</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E4:12"><i><font color="Green">E23</font></i></a>, <a class="ref" href="prepower.html#T106">PREPOWER:106</a></i>;<br/>
<a NAME="E17:12_5_1"/>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E18:12_5_1"/><b>then </b><i><font color="Green">E161</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E20:12_5_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>
<a NAME="E21:12_5_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E23:12_5_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">PT</font>
 
<b>by </b><i>, , <a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<a NAME="E24:12_5_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">W</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/>
<a NAME="E25:12_5_1"/>
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">NZ</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:12_5_1_1"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">p</font>,<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_5_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E3:12_5_1_1_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:12_5_1_1_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E5:12_5_1_1_1"/>
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 ;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:12_5_1_1_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:12_5_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">p</font>,<font color="Olive">n</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:12_5_1_1"/><i><font color="Green">E171</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:12_5_1_1"/><i><font color="Green">E172</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">y</font>,<font color="Olive">n</font>] &amp; ( for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">y</font>,<font color="Olive">t</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S4">TREES_2:sch 4</a>(<a class="txt" href="pcomps_2.html#E13:12"><i><font color="Green">E30</font></i></a>);<br/></i>
<a NAME="E5:12_5_1_1"/><i><font color="Green">E174</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:12_5_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_5_1_1_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:12_5_1_1_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:12_5_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:12_5_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E8:12_5_1_1_2"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font>
 <b>by </b><i>, , </i>;<br/>
<b>assume </b><a NAME="E9:12_5_1_1_2"/><i><font color="Green">E179</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E11:12_5_1_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E12:12_5_1_1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E13:12_5_1_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E15:12_5_1_1_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:12_5_1_1_2"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="pcomps_2.html#E22:12"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_5_1_1_2_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_5_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_5_1_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_5_1_1_2_1_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:12_5_1_1_2_1_1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">Y</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 <b>by </b><i>, , , <a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pcomps_2.html#E20:12"><i><font color="Green">E39</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E7:12_5_1_1_2_1_1_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">Y</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E8:12_5_1_1_2_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/><b>reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">Y</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:12_5_1_1_2_1_1_2"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E12:12_5_1_1_2_1_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E14:12_5_1_1_2_1_1_2"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E15:12_5_1_1_2_1_1_2"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">Y</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i/>;<br/><a NAME="E17:12_5_1_1_2_1_1_2"/><i><font color="Green">E190</font></i>: 
2 <a href="newton.html#K3">|^</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E18:12_5_1_1_2_1_1_2"/><i><font color="Green">E191</font></i>: 
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E19:12_5_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/><a NAME="E20:12_5_1_1_2_1_1_2"/>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/><b>then </b><i><font color="Green">E193</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
;<br/><a NAME="E22:12_5_1_1_2_1_1_2"/>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/><a NAME="E23:12_5_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="real_2.html#T146">REAL_2:146</a></i>;<br/><a NAME="E24:12_5_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E25:12_5_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E194</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E26:12_5_1_1_2_1_1_2"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> holds <br/> not <font color="Maroon">c</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_5_1_1_2_1_1_2_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 ;<br/>

<b>assume </b><a NAME="E2:12_5_1_1_2_1_1_2_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">c</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E3:12_5_1_1_2_1_1_2_2"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:12_5_1_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:12_5_1_1_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E28:12_5_1_1_2_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E23:12"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:12_5_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">y</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">y</font>,<font color="Maroon">x'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:12_5_1_1_2_1_1_2"/><i><font color="Green">E199</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_5_1_1_2_1_1_2_3"/>
not  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<a NAME="E2:12_5_1_1_2_1_1_2_3"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E7:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:12_5_1_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:12_5_1_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">c</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_2.html#T5" target="_self">Th5</a>, <a class="txt" href="pcomps_2.html#E6:12"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:12_5_1_1_2_1_1_2_3"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_5_1_1_2_1_1_2_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:12_5_1_1_2_1_1_2_3_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:12_5_1_1_2_1_1_2_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12_5_1_1_2_1_1_2_3"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:12_5_1_1_2_1_1_2_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:12_5_1_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:12_5_1_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:12_5_1_1_2_1_1_2_3"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E31:12_5_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">y</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">y</font>,<font color="Maroon">x'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E32:12_5_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Maroon">y</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">y</font>,<font color="Maroon">x'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E33:12_5_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E34:12_5_1_1_2_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E35:12_5_1_1_2_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E36:12_5_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:12_5_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:12_5_1_1"/>
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:12_5_1_1_3"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_5_1_1_3_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:12_5_1_1_3_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:12_5_1_1_3_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>assume </b><a NAME="E4:12_5_1_1_3_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/>

<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:12_5_1_1_3_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>and </b><br/><a NAME="E7:12_5_1_1_3_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font>
 <b>and </b><br/><a NAME="E8:12_5_1_1_3_1"/>
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> holds <br/><font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:12_5_1_1_3_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>and </b><br/><a NAME="E11:12_5_1_1_3_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font>
 <b>and </b><br/><a NAME="E12:12_5_1_1_3_1"/>
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> holds <br/><font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E14:12_5_1_1_3_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font>
 <b>and </b><br/><a NAME="E15:12_5_1_1_3_1"/>
<font color="Maroon">V1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E16:12_5_1_1_3_1"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">V1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">w1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:12_5_1_1_3_1"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E19:12_5_1_1_3_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">V2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font><b> such that </b><br/><a NAME="E21:12_5_1_1_3_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V2</font>
 <b>and </b><br/><a NAME="E22:12_5_1_1_3_1"/>
<font color="Maroon">V2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E23:12_5_1_1_3_1"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">V2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">w2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E25:12_5_1_1_3_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E26:12_5_1_1_3_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1_3_1_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">M1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E3:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E20:12"><i><font color="Green">E39</font></i></a></i>;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k1</font>
 <b>and </b><br/><a NAME="E7:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E23:12"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E9:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E10:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span>
 <b>and </b><br/><a NAME="E11:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E236</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">M2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E13:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E14:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, , <a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pcomps_2.html#E22:12"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k2</font>
 <b>and </b><br/><a NAME="E17:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E19:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E20:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span>
 <b>and </b><br/><a NAME="E21:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E246</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/><a NAME="E22:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E247</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">c2</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">x'</font>,<font color="Maroon">w2</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">w2</font>,<font color="Maroon">c2</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E23:12_5_1_1_3_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">x'</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">c1</font>,<font color="Maroon">w1</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">w1</font>,<font color="Maroon">x'</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E24:12_5_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E248</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">x'</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">x'</font>,<font color="Maroon">c2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E25:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E249</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</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">c1</font>,<font color="Maroon">x'</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">x'</font>,<font color="Maroon">c2</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E26:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E250</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E27:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E251</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E28:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E252</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E29:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E253</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c2</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E30:12_5_1_1_3_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</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">c1</font>,<font color="Maroon">w1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E31:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E32:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E33:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E34:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E35:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E36:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E37:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E38:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E39:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E40:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E41:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E42:12_5_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E43:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E44:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E45:12_5_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E254</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><i><font color="Green">E255</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
;<br/><a NAME="E47:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E256</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><i><font color="Green">E257</font></i>: 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
;<br/><a NAME="E49:12_5_1_1_3_1_1_1_1"/>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> 1
 <b>by </b><i><a class="ref" href="pcomps_2.html#T6" target="_self">Th6</a>, <a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E50:12_5_1_1_3_1_1_1_1"/><b>then </b>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/><a NAME="E51:12_5_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/><a NAME="E52:12_5_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E53:12_5_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_2.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E54:12_5_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E55:12_5_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E56:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E57:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">Mn</font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><a NAME="E58:12_5_1_1_3_1_1_1_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pcomps_2.html#E23:12"><i><font color="Green">E51</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1_3_1_1_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">M2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M2</font>,<font color="Maroon">M1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> )
 </span><b>by </b><i>, , , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">M2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 ;<br/><div class="add"><a NAME="E2:12_5_1_1_3_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:12_5_1_1_3_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:12_5_1_1_3_1_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M2</font>,<font color="Maroon">M1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 ;<br/><div class="add"><a NAME="E2:12_5_1_1_3_1_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:12_5_1_1_3_1_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:12_5_1_1_3_1_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E60:12_5_1_1_3_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:12_5_1_1_3_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">M1</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">M1</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">M1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E20:12"><i><font color="Green">E39</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">M1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E7:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E8:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/><b>reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k1</font>
 <b>and </b><br/><a NAME="E12:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E23:12"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E14:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E15:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font></span>)</span>
 <b>and </b><br/><a NAME="E16:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E271</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font>
 <b>and </b><br/><a NAME="E17:12_5_1_1_3_1_1_1_2"/>
not <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E18:12_5_1_1_3_1_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Olive">M2</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">M2</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">M2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> : <font color="Olive">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> </span>}</span> 
 <b>by </b><i>, , <a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pcomps_2.html#E22:12"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">M2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E20:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E21:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/><b>reconsider </b><font color="Maroon">NF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k2</font>
 <b>and </b><br/><a NAME="E25:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E27:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E28:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font></span>)</span>
 <b>and </b><br/><a NAME="E29:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E278</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M2</font>
 <b>and </b><br/><a NAME="E30:12_5_1_1_3_1_1_1_2"/>
not <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E31:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E279</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">c2</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">x'</font>,<font color="Maroon">w2</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">w2</font>,<font color="Maroon">c2</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E32:12_5_1_1_3_1_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">x'</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">c1</font>,<font color="Maroon">w1</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">w1</font>,<font color="Maroon">x'</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E33:12_5_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E280</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">x'</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">x'</font>,<font color="Maroon">c2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E34:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E281</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</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">c1</font>,<font color="Maroon">x'</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">x'</font>,<font color="Maroon">c2</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E35:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E282</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E36:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E283</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E37:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E284</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E8:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E38:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E285</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c2</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E39:12_5_1_1_3_1_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</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">c1</font>,<font color="Maroon">w1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E40:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E41:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E42:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</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"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E43:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E44:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E45:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E46:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E47:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E48:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E49:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E50:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E51:12_5_1_1_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E52:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w2</font>,<font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E53:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E54:12_5_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E286</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><i><font color="Green">E287</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</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">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
;<br/><a NAME="E56:12_5_1_1_3_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="txt" href="pcomps_2.html#E18:12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E57:12_5_1_1_3_1_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E59:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E288</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E61:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E290</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="pcomps_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E62:12_5_1_1_3_1_1_1_2"/>
<font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="pcomps_2.html#T6" target="_self">Th6</a>, </i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E64:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E65:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E292</font></i>: 
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><i><font color="Green">E293</font></i>: 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="pcomps_2.html#T8" target="_self">Th8</a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
;<br/><a NAME="E67:12_5_1_1_3_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E68:12_5_1_1_3_1_1_1_2"/><b>then </b>
2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E69:12_5_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E294</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <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="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/><a NAME="E70:12_5_1_1_3_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E71:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</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="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E72:12_5_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/><a NAME="E73:12_5_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E295</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E74:12_5_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E75:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c2</font>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E76:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">Mn</font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><a NAME="E77:12_5_1_1_3_1_1_1_2"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pcomps_2.html#E23:12"><i><font color="Green">E51</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1_3_1_1_1_2_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">M2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M2</font>,<font color="Maroon">M1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> )
 </span><b>by </b><i>, , , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_2_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">M2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 ;<br/><div class="add"><a NAME="E2:12_5_1_1_3_1_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:12_5_1_1_3_1_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c2</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M2</font>,<font color="Maroon">Mn</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:12_5_1_1_3_1_1_1_2_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_3_1_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:12_5_1_1_3_1_1_1_2_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M2</font>,<font color="Maroon">M1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font>
 ;<br/><div class="add"><a NAME="E2:12_5_1_1_3_1_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Mn</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:12_5_1_1_3_1_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_2.html#K1" target="_self">PartUnion</a> <font color="Maroon">M1</font>,<font color="Maroon">Mn</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:12_5_1_1_3_1_1_1_2_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E79:12_5_1_1_3_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E29:12_5_1_1_3_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_5_1_1_3"/>
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12_5_1_1"/><b>then </b><i><font color="Green">E300</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> , <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E8:12_5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:12_5_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_4_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="E1:12_5_1_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:12_5_1_1_4_1"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:12_5_1_1_4_1"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1_1_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1_4_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_5_1_1_4_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_5_1_1_4_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_5_1_1_4_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_5_1_1_4_1_1_1_2"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E4:12_5_1_1_4_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="pcomps_2.html#E17:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E5:12_5_1_1_4_1_1_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_5_1_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_5_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:12_5_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:12_5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E26:12_5_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PT</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="pcomps_2.html#E10:12"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pcomps_2.html#E12:12"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_5"/>
<font color="Maroon">GX</font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D2">PCOMPS_1:def 2</a></i>;<br/>


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


</div>
