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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0
 ;<br/>

<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:44"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self">Def3</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:44"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self">Def4</a></i>
;<br/></div><b>end;</b></div>
<i><font color="Green">E30</font></i>:  <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span> = 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self">Th26</a></i>
<br/>.= 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T23" target="_self">Th23</a></i>
<br/>.= 
<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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="series_1.html#T14">SERIES_1:14</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"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T32" target="_self">Th32</a></i>
<br/>.= 
 <a href="comseq_3.html#K3" target="_self">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">c<sub>3</sub></font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span> <a href="comseq_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T18" target="_self">Th18</a></i>
;<br/>
 <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span> = 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self">Th26</a></i>
<br/>.= 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T23" target="_self">Th23</a></i>
<br/>.= 
<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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="series_1.html#T14">SERIES_1:14</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"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T32" target="_self">Th32</a></i>
<br/>.= 
 <a href="comseq_3.html#K4" target="_self">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">c<sub>3</sub></font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span> <a href="comseq_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T18" target="_self">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E5:44"/>
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span> <a href="funct_2.html#R2">=</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">c<sub>3</sub></font></span>)</span> <a href="comseq_3.html#K6" target="_self">^\</a> 1</span>)</span> <a href="comseq_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E3:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="comseq_3.html#T14" target="_self">Th14</a></i>;<br/>


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