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<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><a NAME="E1:7"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="comseq_3.html#V2">absolutely_summable</a>
 ;<br/>

<a NAME="E2:7"/><b>then </b><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>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green">E10</font></i>: 
 <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="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> as  <a href="comseq_2.html#V2">convergent</a> <a href="comseq_1.html#NM1">Complex_Sequence</a> <b>by </b><i><a class="txt" href="csspace2.html#E2:7"><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="E5:7"/><i><font color="Green">E11</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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"><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> )
 
;<br/>
<i><font color="Green">E12</font></i>: <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> = 
<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>
<b>by </b><i><a class="ref" href="comseq_3.html#D8">COMSEQ_3:def 8</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="txt" href="csspace2.html#E3:7"><i><font color="Green">E10</font></i></a>, <a class="ref" href="comseq_2.html#T11">COMSEQ_2:11</a></i>
;<br/>
<a NAME="E7:7"/>
<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> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E8:7"/><b>then </b><i><font color="Green">E13</font></i>: 
 <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> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E9:7"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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="xxreal_0.html#R1">&lt;=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:7_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><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> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</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="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T30">COMSEQ_3:30</a></i>;<br/>
<b>hence </b><a NAME="E2:7_2"/>
<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="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</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="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:7"/><b>then </b>
 <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="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</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>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:7"><i><font color="Green">E11</font></i></a>, <a class="txt" href="csspace2.html#E8:7"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<b>hence </b><a NAME="E11:7"/>
<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="xxreal_0.html#R1">&lt;=</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#E6:7"><i><font color="Green">E12</font></i></a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


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