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

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="comseq_3.html#K1">GeoSeq</a>  as   <a href="comseq_1.html#NM1">Complex_Sequence</a> ;<br/>
<a NAME="E3:36_1_1"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>then </b><i><font color="Green">E21</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T136">COMPLEX1:136</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E5:36_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>proof </b></a><div class="add">

<i><font color="Green">E23</font></i>: <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a>, <a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
;<br/>
<a NAME="E2:36_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/>
<b>hence </b><a NAME="E2:36_1_1_2_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1_1_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> 
 <b>by </b><i><a class="txt" href="clopban2.html#E1:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="clopban2.html#E4:36_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="clopban2.html#E5:36_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<a NAME="E8:36_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> 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="E9:36_1_1"/><b>then </b>
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace3.html#K3">Complex_l1_Space</a>  <b>by </b><i><a class="ref" href="csspace3.html#T8">CSSPACE3:8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
;<br/>
<a NAME="E12:36_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="csspace.html#K6">CZeroseq</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a>, <a class="ref" href="csspace.html#D6">CSSPACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E13:36_1_1"/>
not  <a href="csspace3.html#K3">Complex_l1_Space</a>  is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a>, <a class="ref" href="csspace3.html#T8">CSSPACE3:8</a></i>;<br/>


</div>
