<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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"><a href="sin_cos.html#K15" title="SIN_COS:func.15">Conj</a> $1,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1;<br/>
<a NAME="E1:43"/>
 ex <font color="Olive" title="b1">seq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>for <font color="Olive" title="b2">k</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="Olive" title="b1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k</font>)
 
<b>from </b><i><a class="ref" href="comseq_1.html#S1" title="COMSEQ_1:sch.1">COMSEQ_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c3">seq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green" title="E37">A1</font></i>: 
for <font color="Olive" title="b1">k</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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="sin_cos.html#K15" title="SIN_COS:func.15">Conj</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 ;<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" title="c4">k</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><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> = 
<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"><a href="sin_cos.html#K15" title="SIN_COS:func.15">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="sin_cos.html#E3:43"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</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> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T16" target="_self" title="SIN_COS:th.16">Th16</a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>

<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:43"/><b>then </b><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">seq</font> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green" title="E39">A3</font></i>: 
( <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="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</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="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="comseq_2.html#T29" title="COMSEQ_2:th.29">COMSEQ_2:29</a>, <a class="ref" href="comseq_2.html#T30" title="COMSEQ_2:th.30">COMSEQ_2:30</a></i>;<br/>
<a NAME="E7:43"/>
( <font color="Maroon" title="c3">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:43"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="sin_cos.html#T23" target="_self" title="SIN_COS:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E5:43"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="sin_cos.html#E6:43"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="comseq_3.html#T10" title="COMSEQ_3:th.10">COMSEQ_3:10</a></i>;<br/>


</div>
