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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>







<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c3">F</font> <a href="topmetr.html#NR2" title="TOPMETR:NR.2">is_ball-family</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/>
<a NAME="E3:2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:2"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:2"/><b>then </b><i><font color="Green" title="E4">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E7:2"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></i>;<br/>
<a NAME="E8:2"/><i><font color="Green" title="E6">A4</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">G</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c6">G</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E11:2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="topmetr.html#NR2" title="TOPMETR:NR.2">is_ball-family</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<a NAME="E1:2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">r</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="weierstr.html#E8:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="topmetr.html#D4" title="TOPMETR:def.4">TOPMETR:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1"/>
<font color="Maroon" title="c7">G</font> <a href="topmetr.html#NR2" title="TOPMETR:NR.2">is_ball-family</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#D4" title="TOPMETR:def.4">TOPMETR:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E13:2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E15:2"/><i><font color="Green" title="E8">A6</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="topmetr.html#D4" title="TOPMETR:def.4">TOPMETR:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E17:2"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E15:2"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<a NAME="E18:2"/><i><font color="Green" title="E10">A8</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<a NAME="E1:2_2"/><i><font color="Green" title="E11">A9</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_2_1"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:2_2_1"/><i><font color="Green" title="E13">A11</font></i>: 
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="weierstr.html#E3:2_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_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="E1:2_2_1_1_1"/>
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font> or <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="weierstr.html#E7:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="weierstr.html#E5:2_2_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_1_1_1"/><i><font color="Green" title="E14">A12</font></i>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_1_1_1"/>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E5:2_2_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="weierstr.html#E1:2_2_1_1_1_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E4:2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E16">A14</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:2_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="weierstr.html#E2:2_2_1_1_1_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:2_2_1_1_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_1_1_1"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E4:2_2_1_1_1_1"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_1_1_2"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E17:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_1_1_2"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:2_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="weierstr.html#E5:2_2_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2_2_1"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c8">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">r</font>
;<br/>

<b>thus </b><a NAME="E2:2_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:2_2"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c7">G</font>
;<br/>

<b>thus </b><a NAME="E19:2"/>
( <font color="Maroon" title="c7">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c7">G</font> <a href="topmetr.html#NR2" title="TOPMETR:NR.2">is_ball-family</a>  &amp;  ex <font color="Olive" title="b1">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp;  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ex <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">r</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="weierstr.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="weierstr.html#E11:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="weierstr.html#E18:2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/>


</div>
