<?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">p1</font> be   <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>;<br/><b>let </b><font color="Maroon">p2</font> be   <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>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <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>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> ;<br/>
<a NAME="E3:10"/>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<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">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="E1:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E5:10"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>take </b>
0
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a><br/>

<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_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:10_2"/><b>then </b>
 ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">r2</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">W</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>;<br/><b>assume </b><a NAME="E1:10_3"/><i><font color="Green">E48</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:10_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:10_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> 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> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E6:10_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">P</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="tops_1.html#T29">TOPS_1:29</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</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/><b>consider </b><font color="Maroon">r0</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:10_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E10:10_3"/><i><font color="Green">E74</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/><a NAME="E11:10_3"/><i><font color="Green">E75</font></i>: 
 ex <font color="Olive">r0</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</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="E2:10_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:10_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:10_3_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">s</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:10_3_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E7:10_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:10_3_1"/>
 ex <font color="Olive">r0</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_3_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_3_2_1"><b>suppose </b></a><a NAME="E1:10_3_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">p1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">p2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E4:10_3_2_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="euclid.html#T20">EUCLID:20</a></i>;<br/><a NAME="E5:10_3_2_1"/><b>then </b>
<font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <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/><b>then consider </b><font color="Maroon">r4</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:10_3_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">r4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E8:10_3_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">r4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r4</font> = <font color="Maroon">r4</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/><a NAME="E10:10_3_2_1"/>
<font color="Maroon">r4</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:10_3_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/><a NAME="E12:10_3_2_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r4</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E13:10_3_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>set </b><font color="Maroon">p3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">u3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> 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> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">v3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><i><font color="Green">E88</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T42">EUCLID:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
;<br/><a NAME="E17:10_3_2_1"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r4</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/><a NAME="E18:10_3_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">u3</font>,<font color="Maroon">u</font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, , , <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E20:10_3_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u3</font>,<font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">v1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E21:10_3_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">u3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E22:10_3_2_1"/><b>then </b>
<font color="Maroon">u3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u0</font> where <font color="Olive">u0</font> is   <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> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">u0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> </span>}</span> 
 ;<br/><a NAME="E23:10_3_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">u3</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><a NAME="E24:10_3_2_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r4</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E25:10_3_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_3_2_2"><b>suppose </b></a><a NAME="E1:10_3_2_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><a NAME="E3:10_3_2_2"/><i><font color="Green">E95</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><b>consider </b><font color="Maroon">q1</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="E5:10_3_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>thus </b><a NAME="E6:10_3_2_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E7:10"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E8:10"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
;<br/>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div>
