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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</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/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">s1</font> =  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font>;<br/>
<b>set </b><font color="Maroon" title="c6">s2</font> =  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>;<br/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="clvect_1.html#T103" title="CLVECT_1:th.103">CLVECT_1:103</a></i>
;<br/>
<a NAME="E3:45"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</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="E1:45_2"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">CLVECT_1:106</a></i>;<br/><b>hence </b><a NAME="E2:45_2"/>
<span class="p1"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="clvect_1.html#D17" title="CLVECT_1:def.17">CLVECT_1:def 17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:45"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green" title="E39">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</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/>

<b>assume </b><a NAME="E1:45_3"/><i><font color="Green" title="E40">A5</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 ;<br/>

<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E3:45_3"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D11" title="CLVECT_1:def.11">CLVECT_1:def 11</a></i>;<br/>
<a NAME="E4:45_3"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban3.html#E1:45_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:45_3"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban3.html#E3:45_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:45_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="clopban3.html#E5:45"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="seqm_3.html#T11" title="SEQM_3:th.11">SEQM_3:11</a></i>;<br/>
<a NAME="E7:45_3"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:45_3"/><b>then </b><i><font color="Green" title="E44">A9</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban3.html#E5:45_3"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:45_3"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">CLVECT_1:106</a></i>;<br/>
<a NAME="E10:45_3"/><b>then </b><i><font color="Green" title="E45">A10</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="clopban3.html#E7:45_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D17" title="CLVECT_1:def.17">CLVECT_1:def 17</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p5"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E12:45_3"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban3.html#E8:45_3"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="clopban3.html#E10:45_3"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:45"/><i><font color="Green" title="E40">A11</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="clopban3.html#E3:45"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="clopban3.html#E6:45"><i><font color="Green" title="E39">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c4">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">n</font> as    <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>  <b>by </b><i><a class="txt" href="clopban3.html#E1:45"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E9:45"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
;<br/>
<b>hence </b><a NAME="E10:45"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">||.</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="clvect_1.html#K8" title="CLVECT_1:func.8">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban3.html#E7:45"><i><font color="Green" title="E40">A11</font></i></a></i>;<br/>


</div>
