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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="clopban3.html#V2" target="_self">norm_summable</a> )
 ;<br/>

<a NAME="E2:49"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:49"/>
( <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_1.html#K10">.||</a></span> )
 <b>by </b><i><a class="txt" href="clopban3.html#E1:49"><i><font color="Green">E38</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban3.html#D4" target="_self">CLOPBAN3:def 4</a><br/>


</div>
