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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="comseq_3.html#V1">summable</a>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E3:8"/>
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> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</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#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="csspace2.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="csspace2.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</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>  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E6:8"/>
 <a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>
<a NAME="E7:8"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="comseq_2.html#T11">COMSEQ_2:11</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="comseq_1.html#K9">.|</a></span>
 <b>by </b><i><a class="ref" href="comseq_3.html#D8">COMSEQ_3:def 8</a></i>;<br/>


</div>
