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

<b>let </b><font color="Maroon" title="c1">vseq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a> ;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c1">vseq</font> is <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</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 <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp;  ex <font color="Olive" title="b2">seqi</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>( ( for <font color="Olive" title="b3">n</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="Olive" title="b2">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; <font color="Olive" title="b2">seqi</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b2">seqi</font> ) );<br/>
<a NAME="E2:27"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">x</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="csspace2.html#E1:27_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  = <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">seqi</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">n</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" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="comseq_1.html#S1" title="COMSEQ_1:sch.1">COMSEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:27_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c4">seqi</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:27_1_1_1"/>
 ex <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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E2:27_1_1_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> holds <br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="csspace2.html#E1:27_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="clvect_2.html#T58" title="CLVECT_2:th.58">CLVECT_2:58</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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="E1:27_1_1_1_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><b>set </b><font color="Maroon" title="c8">vm</font> = <font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font>;<br/><b>set </b><font color="Maroon" title="c9">vk</font> = <font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font>;<br/><b>set </b><font color="Maroon" title="c10">seqmk</font> =  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>;<br/><a NAME="E2:27_1_1_1_1_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E2:27_1_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="csspace2.html#E1:27_1_1_1_1_1"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/><a NAME="E3:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</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="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="ref" href="csspace.html#D15" title="CSSPACE:def.15">CSSPACE:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c5">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E5:27_1_1_1_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/><a NAME="E6:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>;<br/><a NAME="E7:27_1_1_1_1_1"/><i><font color="Green" title="E36">A11</font></i>: 
for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</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/>

<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A12</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E6:27_1_1_1_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="series_1.html#T21" title="SERIES_1:th.21">SERIES_1:21</a></i>;<br/><a NAME="E9:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E38">A13</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E10:27_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 <b>by </b><i><a class="txt" href="csspace2.html#E20"><i><font color="Green" title="E20">Lm20</font></i></a></i>;<br/><a NAME="E11:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E39">A14</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T64" title="COMSEQ_3:th.64">COMSEQ_3:64</a></i>;<br/><a NAME="E12:27_1_1_1_1_1"/>
for <font color="Olive" title="b1">j</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 <br/> ( <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="csspace2.html#E9"><i><font color="Green" title="E9">Lm9</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E11:27_1_1_1_1_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="csspace2.html#E8"><i><font color="Green" title="E8">Lm8</font></i></a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>
;<br/><a NAME="E14:27_1_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E3:27_1_1_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="csspace2.html#E18"><i><font color="Green" title="E18">Lm18</font></i></a></i>;<br/><a NAME="E15:27_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">e1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="csspace2.html#E9:27_1_1_1_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><a NAME="E16:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A15</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E8:27_1_1_1_1_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E17:27_1_1_1_1_1"/><i><font color="Green" title="E41">A16</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_3"><b>proof </b></a><div class="add">

 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> = 
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E16"><i><font color="Green" title="E16">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_1"><i><font color="Green" title="E29">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_1"><i><font color="Green" title="E29">A4</font></i></a></i>
;<br/>
<b>then </b><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
;<br/>
<a NAME="E4:27_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:27_1_1_1_1_1_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:27_1_1_1_1_1"/><i><font color="Green" title="E42">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E6:27_1_1_1_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="csspace2.html#E7:27_1_1_1_1_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="rsspace2.html#T4" title="RSSPACE2:th.4">RSSPACE2:4</a></i>;<br/><a NAME="E19:27_1_1_1_1_1"/><i><font color="Green" title="E43">A18</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E20:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></i>;<br/><a NAME="E21:27_1_1_1_1_1"/><i><font color="Green" title="E45">A20</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E16:27_1_1_1_1_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="csspace2.html#E17:27_1_1_1_1_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="txt" href="csspace2.html#E18:27_1_1_1_1_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><i><font color="Green" title="E46">A21</font></i>:  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E19:27_1_1_1_1_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>
;<br/> <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e</font></span>)</span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">e1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

<br/>.= 
<font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>
;<br/><b>hence </b><a NAME="E24:27_1_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E20:27_1_1_1_1_1"><i><font color="Green" title="E44">A19</font></i></a>, <a class="txt" href="csspace2.html#E21:27_1_1_1_1_1"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="csspace2.html#E22:27_1_1_1_1_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:27_1_1_1_1"/>
 ex <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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1"/>
<font color="Maroon" title="c4">seqi</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T46" title="COMSEQ_3:th.46">COMSEQ_3:46</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">seqi</font>
;<br/>

<b>thus </b><a NAME="E6:27_1"/>
(  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">seqi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>, <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">seqi</font>] )
 <b>by </b><i><a class="txt" href="csspace2.html#E4:27_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="csspace2.html#E5:27_1"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:27"/><i><font color="Green" title="E28">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="csspace2.html#E2:27"><i><font color="Green" title="E27">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">tseq</font> = <font color="Maroon" title="c2">f</font> as   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/>
<div><i><font color="Green" title="E29">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</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>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E2:27_2"/>
 ex <font color="Olive" title="b1">i0</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 <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i0</font> &amp;  ex <font color="Olive" title="b2">seqi</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>( ( for <font color="Olive" title="b3">n</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="Olive" title="b2">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> ) &amp; <font color="Olive" title="b2">seqi</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b2">seqi</font> ) )
 <b>by </b><i><a class="txt" href="csspace2.html#E4:27"><i><font color="Green" title="E28">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_2"/>
 ex <font color="Olive" title="b1">seqi</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</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="Olive" title="b1">seqi</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font> ) &amp; <font color="Olive" title="b1">seqi</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b1">seqi</font> )
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c4">seqt</font> =  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E30">A24</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">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 <br/>for <font color="Olive" title="b3">n</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="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">k</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">e1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E31">A25</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c6">e</font> = <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/>
<a NAME="E2:27_3"/><i><font color="Green" title="E32">A26</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E3:27_3"/>
<font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E4:27_3"/><b>then </b><i><font color="Green" title="E33">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E2:27_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="xreal_1.html#T99" title="XREAL_1:th.99">XREAL_1:99</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E6:27_3"/><i><font color="Green" title="E34">A28</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace2.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="csspace2.html#E2:27_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="clvect_2.html#T58" title="CLVECT_2:th.58">CLVECT_2:58</a></i>;<br/>
<a NAME="E7:27_3"/><i><font color="Green" title="E35">A29</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; ( for <font color="Olive" title="b3">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">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/>

<b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green" title="E36">A30</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> )
 ;<br/>

<a NAME="E2:27_3_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="csspace2.html#E6:27_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="csspace2.html#E1:27_3_1"><i><font color="Green" title="E36">A30</font></i></a></i>;<br/>
<a NAME="E3:27_3_1"/><b>then </b><i><font color="Green" title="E37">A31</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</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="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="csspace.html#D15" title="CSSPACE:def.15">CSSPACE:def 15</a></i>;<br/>
<a NAME="E4:27_3_1"/><i><font color="Green" title="E38">A32</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<a NAME="E5:27_3_1"/><i><font color="Green" title="E39">A33</font></i>: 
for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</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/>

<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">i</font> = 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E2:27_3_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27_3_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_3_1"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="series_1.html#T21" title="SERIES_1:th.21">SERIES_1:21</a></i>;<br/>
<a NAME="E7:27_3_1"/><b>then </b><i><font color="Green" title="E40">A34</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E8:27_3_1"/><b>then </b><i><font color="Green" title="E41">A35</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E9:27_3_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E20"><i><font color="Green" title="E20">Lm20</font></i></a></i>;<br/>
<a NAME="E10:27_3_1"/><b>then </b><i><font color="Green" title="E42">A36</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64" title="COMSEQ_3:th.64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E11:27_3_1"/>
for <font color="Olive" title="b1">j</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 <br/> ( <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="csspace2.html#E9"><i><font color="Green" title="E9">Lm9</font></i></a></i>;<br/>
<b>then </b><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E10:27_3_1"><i><font color="Green" title="E42">A36</font></i></a>, <a class="txt" href="csspace2.html#E8"><i><font color="Green" title="E8">Lm8</font></i></a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>
;<br/>
<a NAME="E13:27_3_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="csspace2.html#E3:27_3_1"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="csspace2.html#E18"><i><font color="Green" title="E18">Lm18</font></i></a></i>;<br/>
<a NAME="E14:27_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="csspace2.html#E8:27_3_1"><i><font color="Green" title="E41">A35</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E15:27_3_1"/><b>then </b>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E7:27_3_1"><i><font color="Green" title="E40">A34</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:27_3_1"/>
( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; ( for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="csspace2.html#E5:27_3_1"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a>, <a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:27_3"/><i><font color="Green" title="E36">A37</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</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> <br/> for <font color="Olive" title="b3">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b4">m</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="Olive" title="b3">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) holds <br/>( <font color="Olive" title="b3">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b3">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:27_3_2"/><i><font color="Green" title="E37">A38</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">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   <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">m</font>, <font color="Maroon" title="c10">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>thus </b> <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> = 
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E16"><i><font color="Green" title="E16">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</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>for <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">m</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="Olive" title="b1">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 ) holds <br/>( <font color="Olive" title="b1">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 );<br/>
<a NAME="E2:27_3_2"/><i><font color="Green" title="E38">A39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:27_3_2_2_1"/><i><font color="Green" title="E39">A40</font></i>: 
for <font color="Olive" title="b1">m</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" title="c9">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
 ;<br/><b>thus </b><a NAME="E2:27_3_2_2_1"/>
( <font color="Maroon" title="c9">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c10">rseq0</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E2:27_3_2_2_1_1"/><i><font color="Green" title="E40">A41</font></i>: 
( ( for <font color="Olive" title="b1">m</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" title="c10">rseq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0 ) &amp; <font color="Maroon" title="c10">rseq0</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c10">rseq0</font> )
 <b>by </b><i><a class="txt" href="csspace2.html#E6:27"><i><font color="Green" title="E29">A23</font></i></a></i>;<br/>
<div><i><font color="Green" title="E41">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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>thus </b><font color="Maroon" title="c9">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2_2_1"><i><font color="Green" title="E39">A40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2"><i><font color="Green" title="E37">A38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2"><i><font color="Green" title="E37">A38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rseq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_2_2_1_1"><i><font color="Green" title="E40">A41</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rseq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rseq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_2_2_1_1"><i><font color="Green" title="E40">A41</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">rseq</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_2_2_1_1"><i><font color="Green" title="E40">A41</font></i></a>, <a class="txt" href="csspace2.html#E25"><i><font color="Green" title="E24">Lm23</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E5:27_3_2_2_1_1"/>
( <font color="Maroon" title="c9">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 )
 <b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_2_2_1_1"><i><font color="Green" title="E40">A41</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_2_2_1_1"><i><font color="Green" title="E41">A42</font></i></a>, <a class="txt" href="csspace2.html#E25"><i><font color="Green" title="E24">Lm23</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_3_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_3_2"/><i><font color="Green" title="E39">A43</font></i>: 
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</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="E1:27_3_2_3_1"/><i><font color="Green" title="E40">A44</font></i>: 
for <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">m</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="Olive" title="b1">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> ) holds <br/>( <font color="Olive" title="b1">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> )
 ;<br/><b>thus </b><a NAME="E2:27_3_2_3_1"/>
for <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">m</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="Olive" title="b1">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) holds <br/>( <font color="Olive" title="b1">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:27_3_2_3_1_1"/><i><font color="Green" title="E41">A45</font></i>: 
for <font color="Olive" title="b1">m</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" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">rseqb</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:27_3_2_3_1_1"/><i><font color="Green" title="E42">A46</font></i>: 
for <font color="Olive" title="b1">m</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" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E4:27_3_2_3_1_1"/><i><font color="Green" title="E43">A47</font></i>: 
( <font color="Maroon" title="c11">rseqb</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> )
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2_3_1"><i><font color="Green" title="E40">A44</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_2_3_1_1"><i><font color="Green" title="E42">A46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">seq0</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E6:27_3_2_3_1_1"/><i><font color="Green" title="E44">A48</font></i>: 
( ( for <font color="Olive" title="b1">m</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" title="c12">seq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; <font color="Maroon" title="c12">seq0</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c12">seq0</font> )
 <b>by </b><i><a class="txt" href="csspace2.html#E6:27"><i><font color="Green" title="E29">A23</font></i></a></i>;<br/>
<div><i><font color="Green" title="E45">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">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>thus </b><font color="Maroon" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2_3_1_1"><i><font color="Green" title="E41">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2"><i><font color="Green" title="E37">A38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E3:27_3_2_3_1_1"><i><font color="Green" title="E42">A46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2"><i><font color="Green" title="E37">A38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">seq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E6:27_3_2_3_1_1"><i><font color="Green" title="E44">A48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">seq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">seq0</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">rseqb</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E6:27_3_2_3_1_1"><i><font color="Green" title="E44">A48</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">rseq</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c12">seq0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c12">seq0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_3_2_3_1_1"><i><font color="Green" title="E43">A47</font></i></a>, <a class="txt" href="csspace2.html#E6:27_3_2_3_1_1"><i><font color="Green" title="E44">A48</font></i></a>, <a class="txt" href="csspace2.html#E26"><i><font color="Green" title="E25">Lm24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E6:27_3_2_3_1_1"><i><font color="Green" title="E44">A48</font></i></a>, <a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseqb</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E1:27_3_2_3_1"><i><font color="Green" title="E40">A44</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_2_3_1_1"><i><font color="Green" title="E42">A46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">tseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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/>
<b>hence </b><a NAME="E9:27_3_2_3_1_1"/>
( <font color="Maroon" title="c10">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="csspace2.html#E4:27_3_2_3_1_1"><i><font color="Green" title="E43">A47</font></i></a>, <a class="txt" href="csspace2.html#E6:27_3_2_3_1_1"><i><font color="Green" title="E44">A48</font></i></a>, <a class="txt" href="csspace2.html#E7:27_3_2_3_1_1"><i><font color="Green" title="E45">A49</font></i></a>, <a class="txt" href="csspace2.html#E26"><i><font color="Green" title="E25">Lm24</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_3_2_3"/>
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_3_2"/>
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</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="csspace2.html#E2:27_3_2"><i><font color="Green" title="E38">A39</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_2"><i><font color="Green" title="E39">A43</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:27_3_2"/>
for <font color="Olive" title="b1">i</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> <br/> for <font color="Olive" title="b2">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b3">m</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="Olive" title="b2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) holds <br/>( <font color="Olive" title="b2">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b2">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:27_3"/>
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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/>

<b>assume </b><a NAME="E1:27_3_3"/><i><font color="Green" title="E37">A50</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<a NAME="E2:27_3_3"/><i><font color="Green" title="E38">A51</font></i>: 
for <font color="Olive" title="b1">i</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 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</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="E1:27_3_3_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>;<br/>
<b>consider </b><font color="Maroon" title="c10">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:27_3_3_1"/><i><font color="Green" title="E39">A52</font></i>: 
for <font color="Olive" title="b1">m</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" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E4:27_3_3_1"/><i><font color="Green" title="E40">A53</font></i>: 
<font color="Maroon" title="c10">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E8:27_3"><i><font color="Green" title="E36">A37</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_3_1"><i><font color="Green" title="E39">A52</font></i></a></i>;<br/>
<a NAME="E5:27_3_3_1"/><i><font color="Green" title="E41">A54</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E8:27_3"><i><font color="Green" title="E36">A37</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_3_1"><i><font color="Green" title="E39">A52</font></i></a></i>;<br/>
<a NAME="E6:27_3_3_1"/>
for <font color="Olive" title="b1">m</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="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><font color="Maroon" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:27_3_3_1_1"/><i><font color="Green" title="E42">A55</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<a NAME="E2:27_3_3_1_1"/><i><font color="Green" title="E43">A56</font></i>: 
<font color="Maroon" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E3:27_3_3_1"><i><font color="Green" title="E39">A52</font></i></a></i>;<br/>
<a NAME="E3:27_3_3_1_1"/><i><font color="Green" title="E44">A57</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; ( for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="txt" href="csspace2.html#E7:27_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="csspace2.html#E1:27_3_3"><i><font color="Green" title="E37">A50</font></i></a>, <a class="txt" href="csspace2.html#E1:27_3_3_1_1"><i><font color="Green" title="E42">A55</font></i></a></i>;<br/>
<a NAME="E4:27_3_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace2.html#T4" title="RSSPACE2:th.4">RSSPACE2:4</a></i>;<br/>
<b>hence </b><a NAME="E5:27_3_3_1_1"/>
<font color="Maroon" title="c10">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_3_1_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="txt" href="csspace2.html#E3:27_3_3_1_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_3_3_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E4:27_3_3_1"><i><font color="Green" title="E40">A53</font></i></a>, <a class="txt" href="csspace2.html#E5:27_3_3_1"><i><font color="Green" title="E41">A54</font></i></a>, <a class="ref" href="rsspace2.html#T6" title="RSSPACE2:th.6">RSSPACE2:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_3_3"/><i><font color="Green" title="E39">A58</font></i>: 
for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</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="E1:27_3_3_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:27_3_3_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_3_3"/><i><font color="Green" title="E40">A59</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_3_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</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="E1:27_3_3_3_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E2:27_3_3_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_3"><i><font color="Green" title="E38">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_3_3_3_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E4:27_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_3_3_3"/>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:27_3_3"/><b>then </b><i><font color="Green" title="E41">A60</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E3:27_3_3"><i><font color="Green" title="E39">A58</font></i></a>, <a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></i>;<br/>
<a NAME="E6:27_3_3"/><b>then </b><i><font color="Green" title="E42">A61</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E7:27_3_3"/><i><font color="Green" title="E43">A62</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E8:27_3_3"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E2:27_3_3"><i><font color="Green" title="E38">A51</font></i></a>, <a class="txt" href="csspace2.html#E6:27_3_3"><i><font color="Green" title="E42">A61</font></i></a>, <a class="ref" href="rsspace2.html#T6" title="RSSPACE2:th.6">RSSPACE2:6</a></i>;<br/>
<a NAME="E9:27_3_3"/><b>then </b><i><font color="Green" title="E44">A63</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="csspace2.html#E7:27_3_3"><i><font color="Green" title="E43">A62</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:27_3_3"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>;<br/>
<a NAME="E11:27_3_3"/><b>then </b><i><font color="Green" title="E45">A64</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E9:27_3_3"><i><font color="Green" title="E44">A63</font></i></a>, <a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E12:27_3_3"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:27_3_3"><i><font color="Green" title="E41">A60</font></i></a>, <a class="txt" href="csspace2.html#E19"><i><font color="Green" title="E19">Lm19</font></i></a></i>;<br/>
<a NAME="E13:27_3_3"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49" title="COMSEQ_1:th.49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E14:27_3_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11" title="COMSEQ_3:def.11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E15:27_3_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E7"><i><font color="Green" title="E7">Lm7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:27_3_3"/>
( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font> )
 <b>by </b><i><a class="txt" href="csspace2.html#E3:27_3_3"><i><font color="Green" title="E39">A58</font></i></a>, <a class="txt" href="csspace2.html#E4:27_3_3"><i><font color="Green" title="E40">A59</font></i></a>, <a class="txt" href="csspace2.html#E11:27_3_3"><i><font color="Green" title="E45">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:27_3"/>
 ex <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 <br/>for <font color="Olive" title="b2">n</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="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">e1</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:27"/><i><font color="Green" title="E31">A65</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<a NAME="E1:27_4"/><i><font color="Green" title="E32">A66</font></i>: 
for <font color="Olive" title="b1">i</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</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="E1:27_4_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:27_4_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">m</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> <b> such that </b><br/><a NAME="E3:27_4"/><i><font color="Green" title="E33">A67</font></i>: 
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 )
 <b>by </b><i><a class="txt" href="csspace2.html#E7:27"><i><font color="Green" title="E30">A24</font></i></a></i>;<br/>
<a NAME="E4:27_4"/><i><font color="Green" title="E34">A68</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E3:27_4"><i><font color="Green" title="E33">A67</font></i></a></i>;<br/>
<a NAME="E5:27_4"/><i><font color="Green" title="E35">A69</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">a</font> = <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">b</font> = <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c8">d</font> = <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>;<br/>
<a NAME="E6:27_4"/><i><font color="Green" title="E36">A70</font></i>: 
2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_4"><i><font color="Green" title="E34">A68</font></i></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<a NAME="E7:27_4"/>
2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:27_4"><i><font color="Green" title="E35">A69</font></i></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<a NAME="E8:27_4"/><b>then </b><i><font color="Green" title="E37">A71</font></i>: 
<span class="p1">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E6:27_4"><i><font color="Green" title="E36">A70</font></i></a>, <a class="ref" href="series_1.html#T10" title="SERIES_1:th.10">SERIES_1:10</a></i>;<br/>
<a NAME="E9:27_4"/>
for <font color="Olive" title="b1">i</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  <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</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/>

<i><font color="Green" title="E38">A72</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
;<br/>
<i><font color="Green" title="E39">A73</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
;<br/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> = 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p3">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>

;<br/>
<a NAME="E4:27_4_2"/><b>then </b><i><font color="Green" title="E40">A74</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">x</font> = <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>;<br/>
<b>set </b><font color="Maroon" title="c11">y</font> = <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>;<br/>
<a NAME="E5:27_4_2"/><i><font color="Green" title="E41">A75</font></i>: 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E4:27_4_2"><i><font color="Green" title="E40">A74</font></i></a></i>;<br/>
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> = 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>

;<br/>
<b>hence </b><a NAME="E7:27_4_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E1:27_4_2"><i><font color="Green" title="E38">A72</font></i></a>, <a class="txt" href="csspace2.html#E2:27_4_2"><i><font color="Green" title="E39">A73</font></i></a>, <a class="txt" href="csspace2.html#E5:27_4_2"><i><font color="Green" title="E41">A75</font></i></a>, <a class="txt" href="csspace2.html#E24"><i><font color="Green" title="E23">Lm22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:27_4"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="csspace2.html#E1:27_4"><i><font color="Green" title="E32">A66</font></i></a>, <a class="txt" href="csspace2.html#E8:27_4"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:27"/>
<font color="Maroon" title="c3">tseq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="ref" href="csspace.html#D1" title="CSSPACE:def.1">CSSPACE:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">tv</font> = <font color="Maroon" title="c3">tseq</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a>  <b>by </b><i><a class="txt" href="csspace2.html#E8:27"><i><font color="Green" title="E31">A65</font></i></a>, <a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<a NAME="E11:27"/>
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">m</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 <br/>for <font color="Olive" title="b3">n</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="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">tv</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27_5"/><i><font color="Green" title="E32">A76</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">m</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> <b> such that </b><br/><a NAME="E3:27_5"/><i><font color="Green" title="E33">A77</font></i>: 
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">e</font> )
 <b>by </b><i><a class="txt" href="csspace2.html#E7:27"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="csspace2.html#E1:27_5"><i><font color="Green" title="E32">A76</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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/><b>assume </b><a NAME="E1:27_5_1"/><i><font color="Green" title="E34">A78</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><b>set </b><font color="Maroon" title="c9">seqvn</font> =  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span>;<br/><a NAME="E2:27_5_1"/><i><font color="Green" title="E35">A79</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E3:27_5"><i><font color="Green" title="E33">A77</font></i></a>, <a class="txt" href="csspace2.html#E1:27_5_1"><i><font color="Green" title="E34">A78</font></i></a></i>;<br/><a NAME="E3:27_5_1"/>
<font color="Maroon" title="c3">tseq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 <b>by </b><i><a class="ref" href="csspace.html#D1" title="CSSPACE:def.1">CSSPACE:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c3">tseq</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a>  <b>by </b><i><a class="txt" href="csspace2.html#E8:27"><i><font color="Green" title="E31">A65</font></i></a>, <a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a>  ;<br/><a NAME="E6:27_5_1"/>
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E12"><i><font color="Green" title="E12">Lm12</font></i></a></i>;<br/><a NAME="E7:27_5_1"/><b>then </b>
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">tseq</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E16"><i><font color="Green" title="E16">Lm16</font></i></a></i>;<br/><a NAME="E8:27_5_1"/><b>then </b><i><font color="Green" title="E36">A80</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E2:27_5_1"><i><font color="Green" title="E35">A79</font></i></a>, <a class="txt" href="csspace2.html#E18"><i><font color="Green" title="E18">Lm18</font></i></a></i>;<br/><a NAME="E9:27_5_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#D13" title="CSSPACE:def.13">CSSPACE:def 13</a></i>;<br/><a NAME="E10:27_5_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T36" title="CSSPACE:th.36">CSSPACE:36</a></i>;<br/><a NAME="E11:27_5_1"/><b>then </b><i><font color="Green" title="E37">A81</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="csspace2.html#E8:27_5_1"><i><font color="Green" title="E36">A80</font></i></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></i>;<br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">tv</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tv</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">tv</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="csspace.html#T49" title="CSSPACE:th.49">CSSPACE:49</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="csspace.html#K12" title="CSSPACE:func.12">.|.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">v</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="csspace.html#D15" title="CSSPACE:def.15">CSSPACE:def 15</a></i>
;<br/><b>hence </b><a NAME="E13:27_5_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">tv</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace2.html#E1:27_5"><i><font color="Green" title="E32">A76</font></i></a>, <a class="txt" href="csspace2.html#E11:27_5_1"><i><font color="Green" title="E37">A81</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:27_5"/>
 ex <font color="Olive" title="b1">m</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 <br/>for <font color="Olive" title="b2">n</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="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">vseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">tv</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:27"/>
<font color="Maroon" title="c1">vseq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_2.html#T9" title="CLVECT_2:th.9">CLVECT_2:9</a></i>;<br/>


</div>
