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

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace.html#K20">Complex_l2_Space</a> , <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace.html#K20">Complex_l2_Space</a> , <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace.html#K20">Complex_l2_Space</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>



<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><i><font color="Green">E35</font></i>: <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
;<br/><a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 <b>by </b><i/>;<br/><a NAME="E4:21_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/><a NAME="E5:21_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="comseq_3.html#T54">COMSEQ_3:54</a></i>;<br/><a NAME="E6:21_1"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/><b>set </b><font color="Maroon">seq</font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font>;<br/><b>set </b><font color="Maroon">rseq</font> =  <a href="comseq_3.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">iseq</font> =  <a href="comseq_3.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>;<br/><a NAME="E7:21_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="comseq_3.html#T21">COMSEQ_3:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E2:21_1_2"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E8:21_1"/><i><font color="Green">E49</font></i>: 
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) &amp;  <a href="comseq_3.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:21_1_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:21_1_3_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<a NAME="E2:21_1_3_1"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:21_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_3"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) &amp;  <a href="comseq_3.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i>, <a class="ref" href="comseq_3.html#T54">COMSEQ_3:54</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:21_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:21_1_4"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<a NAME="E2:21_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="rsspace.html#T21">RSSPACE:21</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_4"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T13">COMPLEX1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:21_1"/>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1">the_set_of_ComplexSequences</a> 
 <b>by </b><i><a class="ref" href="csspace.html#D11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>;<br/><b>hence </b><a NAME="E11:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace.html#K20">Complex_l2_Space</a> 
 <b>by </b><i>, , <a class="ref" href="csspace.html#D6">CSSPACE:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace.html#K20">Complex_l2_Space</a> 
 ;<br/><a NAME="E2:21_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="csspace.html#D6">CSSPACE:def 6</a></i>
<br/>.= 
0

;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="csspace.html#T88">CSSPACE:88</a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">seqx</font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font>;<br/>
<a NAME="E3:21"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:21"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green">E62</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<i><font color="Green">E63</font></i>: <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
;<br/>
<a NAME="E2:21_3"/>
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="comseq_3.html#T54">COMSEQ_3:54</a></i>;<br/>
<a NAME="E3:21_3"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E4:21_3"/><i><font color="Green">E65</font></i>: 
(  <a href="comseq_3.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="comseq_3.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i>, <a class="ref" href="comseq_3.html#T54">COMSEQ_3:54</a></i>;<br/>
<a NAME="E5:21_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E108</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T126">COMPLEX1:126</a></i>
;<br/>
<a NAME="E2:21_3_2"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:21_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:21_3_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_3"/>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="csspace2.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>

<a NAME="E7:21_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E2:21_3_3"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T126">COMPLEX1:126</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:21_3"/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="csspace2.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="rsspace.html#T20">RSSPACE:20</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">seqy</font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font>;<br/>
<b>set </b><font color="Maroon">seqz</font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font>;<br/>
<a NAME="E6:21"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</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">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<i><font color="Green">E112</font></i>: <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#T5">COMSEQ_2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
;<br/>
<a NAME="E12:21"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E13:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E14:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E15:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E16:21"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E17:21"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E18:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E19:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E20:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E21:21"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<i><font color="Green">E115</font></i>: <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#T13">COMSEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="comseq_3.html#T55">COMSEQ_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
;<br/>
<a NAME="E23:21"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E24:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E25:21"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E26:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E27:21"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font> = 
the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#T15">COMSEQ_1:15</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_3.html#T57">COMSEQ_3:57</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default">the <a href="csspace.html#U1">scalar</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D19">CSSPACE:def 19</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D12">CSSPACE:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E29:21"/>
( ( <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace.html#K20">Complex_l2_Space</a>  ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace.html#K20">Complex_l2_Space</a>  implies <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K15">*'</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12">.|.</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, , , <a class="txt" href="csspace2.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="csspace2.html#E2"><i><font color="Green">Lemma30</font></i></a></i>;<br/>


</div>
