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

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 ;<br/>

<a NAME="E2:10"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Olive">r</font> st <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</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">x</font> = <font color="Maroon">r</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="E4:10_1"/>
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">r</font> st <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N1</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r1</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E4:10_1_1"/>
 ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Olive">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">g</font> = <font color="Maroon">r1</font>;<br/>

<a NAME="E1:10_1_1_1"/>
<font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 ;<br/>

<a NAME="E2:10_1_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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">x</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E4:10_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</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">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">q1</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E7:10_1_1_1_1_1"/>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">q1</font>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</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="E8:10_1_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


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

<b>thus </b><a NAME="E2:10_1_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:10_1_1_1_1_2"/>
<font color="Maroon">a'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">a''</font> = <font color="Maroon">a</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</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/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a'</font>, <font color="Maroon">r1</font> = <font color="Maroon">r'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E6:10_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">r'</font>,<font color="Maroon">a''</font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">a'</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="E7:10_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E8:10_1_1_1_1_2"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">a'</font>,<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E9:10_1_1_1_1_2"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">r1</font>,<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T10">METRIC_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:10_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N1</font> as    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">r</font> <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<b>thus </b><a NAME="E6:10_1_1"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1"/>
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">r</font> st <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
<font color="Maroon">X</font> is <a href="rcomp_1.html#V3">open</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T41">RCOMP_1:41</a></i>;<br/>


</div>
