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

<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>thus </b><a NAME="E2:18"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
(  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>case </b></a><a NAME="E1:18_1_1_1_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c2">r0</font> = 1;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> &amp; $1 <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="b1">x</font>,1 );<br/><b>consider </b><font color="Maroon" title="c3">R</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i><a NAME="E4:18_1_1_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c3">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">A</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="c5">x</font>,1 )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:18_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="topmetr.html#T21" title="TOPMETR:th.21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_1_1"/>
<font color="Maroon" title="c3">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:18_1_1_1_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_2"/>
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>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">x</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">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_2_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E4:18_1_1_1_1_2_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c6">A</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="c5">x</font>,1
 ;<br/>
<a NAME="E5:18_1_1_1_1_2_1"/>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E7:18_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_2_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">A</font>
;<br/>

<b>thus </b><a NAME="E2:18_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_2_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="weierstr.html#E4:18_1_1_1_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_2_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1_1_1_1_2_1"/>
<font color="Maroon" title="c4">x</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">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_1_2"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:18_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">R</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">R0</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E8:18_1_1_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c4">R0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c4">R0</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">R0</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="weierstr.html#E4:18_1_1_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></i>;<br/><a NAME="E9:18_1_1_1_1"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">R0</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E8:18_1_1_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/><a NAME="E10:18_1_1_1_1"/><i><font color="Green" title="E15">A10</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R0</font> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  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">A</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">x</font>,<font color="Olive" title="b3">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_3"/><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R0</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c7">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E4:18_1_1_1_1_3"/><i><font color="Green" title="E17">A12</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c6">A</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="c7">x</font>,1 )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="weierstr.html#E8:18_1_1_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_3"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">x</font>
;<br/>

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E5:18_1_1_1_1_3"/>
<font color="Maroon" title="c5">A</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="c7">x</font>,1
 <b>by </b><i><a class="txt" href="weierstr.html#E4:18_1_1_1_1_3"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:18_1_1_1_1"/>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> #)
 <b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">R0</font> = <font color="Maroon" title="c4">R0</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  ;<br/><a NAME="E13:18_1_1_1_1"/>
<font color="Maroon" title="c5">R0</font> <a href="topmetr.html#NR2" title="TOPMETR:NR.2">is_ball-family</a> 
 <b>by </b><i><a class="txt" href="weierstr.html#E10:18_1_1_1_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="topmetr.html#D4" title="TOPMETR:def.4">TOPMETR:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E15:18_1_1_1_1"/><i><font color="Green" title="E16">A13</font></i>: 
 ex <font color="Olive" title="b1">r1</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="c5">R0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">x1</font>,<font color="Olive" title="b1">r1</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E8:18_1_1_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="weierstr.html#T3" target="_self" title="WEIERSTR:th.3">Th3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E17:18_1_1_1_1"/><i><font color="Green" title="E17">A14</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">R0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">r1</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E15:18_1_1_1_1"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/><a NAME="E18:18_1_1_1_1"/><i><font color="Green" title="E18">A15</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">r1</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E9:18_1_1_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="weierstr.html#E17:18_1_1_1_1"><i><font color="Green" title="E17">A14</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="c8">x1</font> = <font color="Maroon" title="c6">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E20:18_1_1_1_1"/><i><font color="Green" title="E19">A16</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/>( <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_4"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_4"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r1</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E18:18_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_4"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r1</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_4"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r1</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E6:18_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r1</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E7:18_1_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">r1</font></span>)</span> &amp; <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E8:18_1_1_1_1_4"/>
( <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r1</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E21:18_1_1_1_1"/><i><font color="Green" title="E20">A18</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_5"/>
 ex <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r1</font>
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_1_5_1"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r1</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E20:18_1_1_1_1"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_1_5"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:18_1_1_1_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_6"/>
 ex <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font>
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_1_6_1"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Maroon" title="c8">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E20:18_1_1_1_1"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_1_6"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:18_1_1_1_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E21:18_1_1_1_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>case </b></a><a NAME="E1:18_1_1_1_2"/><i><font color="Green" title="E11">A19</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2"/><i><font color="Green" title="E12">A20</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_2_1_1"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18_1_1_1_2"><i><font color="Green" title="E11">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_2_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:18_1_1_1_2"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_2_2"/>
 ex <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_2_2_1"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18_1_1_1_2"><i><font color="Green" title="E11">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1_2_2"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_1_1_2"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E2:18_1_1_1_2"><i><font color="Green" title="E12">A20</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c1">P</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 ;<br/>


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

<b>thus </b><a NAME="E3:18"/>
verum
 ;<br/>


</div>
