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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><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>assume </b><a NAME="E1:93"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> </span>}</span> 
 ;<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="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E4:93"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</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">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

<a NAME="E2:93_1"/><b>then </b><i><font color="Green">E56</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E4:93_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1"/>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><b>case </b></a><a NAME="E1:93_1_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:93_1_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2;<br/><a NAME="E3:93_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E4:93_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E5:93_1_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:93_1_1_1_1_1"/><i><font color="Green">E61</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">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E4:93_1_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E5:93_1_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_1_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:93_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E4:93_1_1_1_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T33">TOPRNS_1:33</a></i>;<br/><a NAME="E5:93_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:93_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>hence </b><a NAME="E7:93_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:93_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:93_1_1_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">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_2"><b>case </b></a><a NAME="E1:93_1_1_1_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2;<br/><a NAME="E2:93_1_1_1_2"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:93_1_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E4:93_1_1_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:93_1_1_1_2_1"/><i><font color="Green">E130</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">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E4:93_1_1_1_2_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E5:93_1_1_1_2_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_2_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:93_1_1_1_2_1_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E4:93_1_1_1_2_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T33">TOPRNS_1:33</a></i>;<br/><a NAME="E5:93_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:93_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>hence </b><a NAME="E7:93_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:93_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:93_1_1_1_2"/>
 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">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:93_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">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:93"/><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></i>;<br/>
<b>hence </b><a NAME="E6:93"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp; <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>
