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

<b>let </b><font color="Maroon" title="c3">z</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>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/>



<i><font color="Green" title="E71">A1</font></i>: <font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a> </span>)</span> <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> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
<br/>.= 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E72">A2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c3">z</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self" title="SIN_COS:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c3">z</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T37" target="_self" title="SIN_COS:th.37">Th37</a>, <a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</a></i>
<br/>.= 
<font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="txt" href="sin_cos.html#E69"><i><font color="Green" title="E52">Lm9</font></i></a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
;<br/>
<i><font color="Green" title="E73">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self" title="SIN_COS:def.8">Def8</a></i>
<br/>.= 
 <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a>, <a class="ref" href="comseq_3.html#T11" title="COMSEQ_3:th.11">COMSEQ_3:11</a></i>
;<br/>
<a NAME="E4:85"/>
<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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</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"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<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>means </b><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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1</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"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1;<br/>
<a NAME="E1:85_4"/>
<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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="complex1.html#T134" title="COMPLEX1:th.134">COMPLEX1:134</a>, <a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>;<br/>
<a NAME="E2:85_4"/><b>then </b><i><font color="Green" title="E74">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="sin_cos.html#E80"><i><font color="Green" title="E60">Lm11</font></i></a></i>;<br/>
<a NAME="E3:85_4"/><i><font color="Green" title="E75">A5</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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_1"><b>proof </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/>

<b>assume </b><a NAME="E1:85_4_1"/><i><font color="Green" title="E76">A6</font></i>: 
<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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">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"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">k</font>
 ;<br/>

<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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<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">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="comseq_3.html#K1" title="COMSEQ_3:func.1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="sin_cos.html#E1:85_4_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:85_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:85_4"/>
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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="sin_cos.html#E2:85_4"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="sin_cos.html#E3:85_4"><i><font color="Green" title="E75">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:85_4"/>
<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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</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"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:85"/>
( <font color="Maroon" title="c3">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <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">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</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"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:85"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="sin_cos.html#E2:85"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="sin_cos.html#E3:85"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="comseq_3.html#D1" title="COMSEQ_3:def.1">COMSEQ_3:def 1</a></i>;<br/>


</div>
