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

<a NAME="E1:20"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></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="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace.html#K20">Complex_l2_Space</a> 
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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="E3:20_1"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a>
 <b>by </b><i><a class="ref" href="csspace.html#D1">CSSPACE:def 1</a></i>;<br/>

<a NAME="E4:20_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1"/><i><font color="Green">E23</font></i>: 
<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">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="funct_2.html#R2">=</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">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E19"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E2:20_1_1"/>
<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">c<sub>1</sub></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">c<sub>1</sub></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><a class="txt" href="csspace2.html#E1:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="csspace2.html#E17"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E3:20_1_1"/><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">c<sub>1</sub></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">c<sub>1</sub></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="txt" href="csspace2.html#E1:20_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></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="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace.html#K20">Complex_l2_Space</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a>
 <b>and </b><br/><a NAME="E2:20_2"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 ;<br/>

<a NAME="E3:20_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K1">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:20_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="csspace.html#D1">CSSPACE:def 1</a></i>;<br/>
<a NAME="E4:20_2"/>
<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">c<sub>1</sub></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">c<sub>1</sub></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="txt" href="csspace2.html#E2:20_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E5:20_2"/><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">c<sub>1</sub></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">c<sub>1</sub></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><a class="ref" href="comseq_1.html#T49">COMSEQ_1:49</a></i>;<br/>
<a NAME="E6:20_2"/><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">c<sub>1</sub></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">c<sub>1</sub></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><a class="txt" href="csspace2.html#E19"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace.html#K20">Complex_l2_Space</a> 
 <b>by </b><i><a class="txt" href="csspace2.html#E3:20_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="csspace.html#D11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace.html#K20">Complex_l2_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></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="Olive">b<sub>1</sub></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="txt" href="csspace2.html#E1:20"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
