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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">A</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="E1:3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E6:3"/>
for <font color="Olive">c</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">c</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> ;<br/>

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">c</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="E3:3_1"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:3_1"/><b>then </b><i><font color="Green">E45</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:3_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1_1"/>
(  not <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</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/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i/>;<br/><a NAME="E4:3_1_1_1_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><a NAME="E5:3_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><a NAME="E6:3_1_1_1_1_1_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">t</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E7:3_1_1_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/><a NAME="E8:3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:3_1_1_1_1_1_1"/><b>then </b>
for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> or  not <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> or  not <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:3_1_1_1_1_1_1"/><b>then </b>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/><a NAME="E11:3_1_1_1_1_1_1"/><b>then </b>
( not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>hence </b><a NAME="E12:3_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:3_1_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</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/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i/>;<br/><a NAME="E4:3_1_1_1_1_2_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><a NAME="E5:3_1_1_1_1_2_1"/><b>then </b>
 ex <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c</font>,<font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><a NAME="E6:3_1_1_1_1_2_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E7:3_1_1_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> )
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/><a NAME="E8:3_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:3_1_1_1_1_2_1"/><b>then </b>
for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> or  not <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> or  not <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E10:3_1_1_1_1_2_1"/><b>then </b>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/><a NAME="E11:3_1_1_1_1_2_1"/><b>then </b>
( not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>hence </b><a NAME="E12:3_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:3_1_1_1_1_2"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a>, <a class="ref" href="tarski.html#D3">TARSKI: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="E2:3_1_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E7:3"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:3"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
