<?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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><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="E1:42_1"/><i><font color="Green" title="E36">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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 ;<br/><div><i><font color="Green" title="E37">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_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/><a NAME="E1:42_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:42_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:42_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<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><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="comseq_3.html#T30" title="COMSEQ_3:th.30">COMSEQ_3:30</a></i>;<br/></div><b>end;</b></div><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#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<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><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1;<br/><a NAME="E3:42_1"/>
 ex <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_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">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i><b>then consider </b><font color="Maroon" title="c4">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E5:42_1"/><i><font color="Green" title="E38">A3</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="c4">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<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><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font>
 ;<br/><div><i><font color="Green" title="E39">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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/><a NAME="E1:42_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K15" title="SIN_COS:func.15">Conj</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:42_1"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E2:42_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E5:42_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:42_1"/>
( <font color="Maroon" title="c4">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E40">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:42_1_3_1"/>
<font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E3:42_1_3_1"/><i><font color="Green" title="E41">A6</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>   st <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <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"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<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><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T22" target="_self" title="SIN_COS:th.22">Th22</a></i>;<br/><b>take </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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>assume </b><a NAME="E1:42_1_3_1_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ;<br/><a NAME="E2:42_1_3_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <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"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K15" title="SIN_COS:func.15">Conj</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E5:42_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:42_1_3_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="sin_cos.html#E1:42_1_3_1_1"><i><font color="Green" title="E42">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:42_1_3_1"/>
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>   st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">p</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_3"/>
<font color="Maroon" title="c4">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:42_1_3"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:42_1_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:42_1"/>
( <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#E6:42_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="comseq_3.html#T48" title="COMSEQ_3:th.48">COMSEQ_3:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
for <font color="Olive" title="b1">seq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st ( 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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b2">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="b2">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> ) holds <br/>( <font color="Olive" title="b1">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="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>


</div>
