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<b>let </b><font color="Maroon" title="c2">seq1</font>, <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="E1:44"/><i><font color="Green" title="E28">A1</font></i>: 
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="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<div><i><font color="Green" title="E29">A2</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" 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/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self" title="COMSEQ_3:def.3">Def3</a></i>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a></i>
<br/>.= 
<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="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self" title="COMSEQ_3:def.3">Def3</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self" title="COMSEQ_3:def.4">Def4</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a></i>
<br/>.= 
<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="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self" title="COMSEQ_3:def.4">Def4</a></i>
;<br/></div><b>end;</b></div>
<i><font color="Green" title="E30">A3</font></i>:  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> = 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self" title="COMSEQ_3:th.26">Th26</a></i>
<br/>.= 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<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> 1</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T23" target="_self" title="COMSEQ_3:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<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></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:44"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="series_1.html#T14" title="SERIES_1:th.14">SERIES_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T32" target="_self" title="COMSEQ_3:th.32">Th32</a></i>
<br/>.= 
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">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" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T18" target="_self" title="COMSEQ_3:th.18">Th18</a></i>
;<br/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> = 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self" title="COMSEQ_3:th.26">Th26</a></i>
<br/>.= 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<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> 1</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T23" target="_self" title="COMSEQ_3:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<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></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:44"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="series_1.html#T14" title="SERIES_1:th.14">SERIES_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T32" target="_self" title="COMSEQ_3:th.32">Th32</a></i>
<br/>.= 
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">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" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T18" target="_self" title="COMSEQ_3:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E5:44"/>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</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> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c2">seq1</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E3:44"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="comseq_3.html#T14" target="_self" title="COMSEQ_3:th.14">Th14</a></i>;<br/>


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