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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> <a href="funct_2.html#R2">=</a>  <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> <a href="funct_2.html#R2">=</a>  <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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