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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<b>thus </b><a NAME="E2:18"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>2</sub></font> = 1;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,1 );<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i><a NAME="E4:18_1_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">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">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">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">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,1 )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:18_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7">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">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:18_1_1_1_1"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E4:18_1_1_1_1_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,1
 ;<br/>
<a NAME="E5:18_1_1_1_1_2_1"/>
 <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E7:18_1_1_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">E15</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E2:18_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weierstr.html#E4:18_1_1_1_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="metric_1.html#T12">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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">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">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E7:18_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E9:18_1_1_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="weierstr.html#E4:18_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/><a NAME="E10:18_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E9:18_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/><a NAME="E11:18_1_1_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E12:18_1_1_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> ex <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E4:18_1_1_1_1_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>7</sub></font>,1 )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:18_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="weierstr.html#E9:18_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_3"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

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

<b>thus </b><a NAME="E5:18_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>7</sub></font>,1
 <b>by </b><i><a class="txt" href="weierstr.html#E4:18_1_1_1_1_3"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_1_1_1_1"/>
 <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> #)
 <b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="metric_1.html#K8">RealSpace</a>  ;<br/><a NAME="E15:18_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a> 
 <b>by </b><i><a class="txt" href="weierstr.html#E12:18_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="topmetr.html#D4">TOPMETR:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E17:18_1_1_1_1"/><i><font color="Green">E16</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E9:18_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="weierstr.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E19:18_1_1_1_1"/><i><font color="Green">E17</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E17:18_1_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E20:18_1_1_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E11:18_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="weierstr.html#E19:18_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E22:18_1_1_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></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">c<sub>9</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E20:18_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="weierstr.html#E1:18_1_1_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_4"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">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">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E7:18_1_1_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E8:18_1_1_1_1_4"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E23:18_1_1_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_1_5_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E22:18_1_1_1_1"><i><font color="Green">E19</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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:18_1_1_1_1"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></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">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1_1_6_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E22:18_1_1_1_1"><i><font color="Green">E19</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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E25:18_1_1_1_1"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E23:18_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="seq_4.html#D3">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">E11</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2"/><i><font color="Green">E12</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18_1_1_1_2"><i><font color="Green">E11</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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:18_1_1_1_2"><i><font color="Green">E11</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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E2:18_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="seq_4.html#D3">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" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a>
 ;<br/>


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

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


</div>
