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

<b>let </b><font color="Maroon">M</font> be   <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  )
 ;<br/>

<a NAME="E2:3"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  holds <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  holds <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>;<br/>
<a NAME="E1:3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1_1"/>
( <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  )
 ;<br/>

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

<b>assume </b><a NAME="E2:3_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:3_1_1"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:3_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:3_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

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

<b>thus </b><a NAME="E5:3_1_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <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="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:3_1_1"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:3_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:3_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  )
 ;<br/>

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

<b>assume </b><a NAME="E3:3_1_2"/><i><font color="Green">E39</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
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E4:3_1_2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">F1</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E6:3_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">F1</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">F1</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  &amp;  ex <font color="Olive">p1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp;  ex <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x2</font>,<font color="Olive">r2</font></span>)</span> ) )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E8:3_1_2"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Olive">r2</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:3_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E12:3_1_2"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Olive">r</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:3_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Maroon">r1</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:3_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E17:3_1_2"/><i><font color="Green">E61</font></i>: 
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font>
 <b>by </b><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="E19:3_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></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/>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

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

<a NAME="E20:3_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x1</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E21:3_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:3_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:3_1"/>
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  holds <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:3"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<a NAME="E5:3"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font>
 <b>by </b><i>, , </i>;<br/>


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