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

<b>reconsider </b><font color="Maroon" title="c1">a</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">x</font> = <font color="Maroon" title="c1">a</font> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a>  as   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/>
<a NAME="E3:36_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:36_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E22">A2</font></i>: <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="complex1.html#T134" title="COMPLEX1:th.134">COMPLEX1:134</a>, <a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
;<br/>
<a NAME="E2:36_1_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/>
<b>hence </b><a NAME="E2:36_1_1_1_2"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> 
 <b>by </b><i><a class="txt" href="clopban2.html#E1:36_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:36_1_1"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="clopban2.html#E3:36_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="series_1.html#T28" title="SERIES_1:th.28">SERIES_1:28</a></i>;<br/>
<a NAME="E6:36_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11" title="COMSEQ_3:def.11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b>
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">x</font> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">v</font> = <font color="Maroon" title="c2">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>  <b>by </b><i><a class="ref" href="csspace3.html#T8" title="CSSPACE3:th.8">CSSPACE3:8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0 = 
<font color="Maroon" title="c2">x</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 0
<b>by </b><i><a class="ref" href="csspace.html#T3" title="CSSPACE:th.3">CSSPACE:3</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
;<br/>
<a NAME="E10:36_1_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
 
<b>by </b><i><a class="ref" href="csspace.html#D6" title="CSSPACE:def.6">CSSPACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E11:36_1_1"/>
not  <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>  is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8" title="ANPROJ_1:def.8">ANPROJ_1:def 8</a>, <a class="ref" href="csspace3.html#T8" title="CSSPACE3:th.8">CSSPACE3:8</a></i>;<br/>


</div>
