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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/>





<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> implies  <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> = 
<span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T20">EUCLID_4:20</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="square_1.html#T82">SQUARE_1:82</a>, <a class="ref" href="euclid_4.html#T22">EUCLID_4:22</a></i>
;<br/>
<a NAME="E3:66_1"/><b>then </b><i><font color="Green">E66</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E4:66_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</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="hidden.html#R2">in</a>  <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E5:66_1"/><i><font color="Green">E89</font></i>: 
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>  <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><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="E1:66_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x0</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:66_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 ;<br/>
<b>thus </b><a NAME="E4:66_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T10" target="_self">Th10</a>, <a class="ref" href="euclid_2.html#T14">EUCLID_2:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:66_1"/><b>then </b>
 <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<a NAME="E7:66_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a>, , <a class="ref" href="euclidlp.html#T9" target="_self">Th9</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:66_1"/>
 <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E105</font></i>: 
 <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon">x0</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:66_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><a NAME="E4:66_2"/>
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="euclidlp.html#T8" target="_self">Th8</a>, <a class="ref" href="euclid_4.html#T21">EUCLID_4:21</a></i>;<br/><a NAME="E5:66_2"/><b>then </b>
<font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid_4.html#T22">EUCLID_4:22</a></i>;<br/><b>hence </b><a NAME="E6:66_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
(  <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 ;<br/>


</div>
