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

<b>let </b><font color="Maroon" title="c1">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c1">rseq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <font color="Maroon" title="c1">rseq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E3:9"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <font color="Maroon" title="c1">rseq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="seqm_3.html#T70" title="SEQM_3:th.70">SEQM_3:70</a></i>;<br/>
<a NAME="E4:9"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<a NAME="E1:9_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E2:9_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:9"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="rsspace4.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="rsspace4.html#E3:9"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="rsspace4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E6:9"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/>


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