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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>, <font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E33</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 <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">seq1</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">seq1</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">seq2</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) ) &amp; <font color="Maroon">seq2</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">seq2</font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><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">seq1</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">seq1</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">seq2</font></span><a href="clvect_1.html#K10">.||</a></span> )
 <b>by </b><i>, <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/>


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