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

<b>assume </b><a NAME="E1:83"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:83_1"/>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E2:83_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="series_1.html#V1">summable</a> &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i><a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/><a NAME="E3:83_1"/>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon">seq</font> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">n</font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Olive">n</font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 ;<br/>


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