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

<b>let </b><font color="Maroon">rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:8"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><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/>

<a NAME="E1:8_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:8_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">rseq</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E4:8"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:8_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<a NAME="E4:8_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:8_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8"/>
( <font color="Maroon">rseq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">rseq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T73">SEQM_3:73</a>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/>


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