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<b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">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" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:83_1"/>
(  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E1:83"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="comseq_3.html#T54" target="_self" title="COMSEQ_3:th.54">Th54</a></i>;<br/><a NAME="E2:83_1"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 <b>by </b><i><a class="ref" href="series_1.html#T15" title="SERIES_1:th.15">SERIES_1:15</a></i>;<br/><a NAME="E3:83_1"/>
(  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n</font> &amp;  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="ref" href="comseq_3.html#T23" target="_self" title="COMSEQ_3:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c4">n</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="comseq_3.html#E2:83_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="comseq_3.html#T58" target="_self" title="COMSEQ_3:th.58">Th58</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Olive" title="b1">n</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 ;<br/>


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