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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:64"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T11">COMSEQ_3:11</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="sin_cos3.html#D4" target="_self">Def4</a>, <a class="ref" href="sin_cos3.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E2:64"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:64"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </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>assume </b><a NAME="E1:64_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<i><font color="Green">E91</font></i>: <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<a NAME="E3:64_1"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>set </b><font color="Maroon">cn</font> = <a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">sn</font> = <a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c1</font> = <a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">s1</font> = <a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font>;<br/>
<i><font color="Green">E96</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D2">COMSEQ_3:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D2">COMSEQ_3:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:64_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos3.html#T18" target="_self">Th18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:64"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="sin_cos3.html#T14" target="_self">Th14</a>, <a class="ref" href="sin_cos3.html#T15" target="_self">Th15</a>);<br/></i>
<b>hence </b><a NAME="E5:64"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span> <a href="comseq_3.html#K2">#N</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Olive">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Olive">n</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Olive">z</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
