<?xml version="1.0"?>
<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>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/>



<a NAME="E1:87"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="newton.html#K2">|^</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#D1">COMSEQ_3:def 1</a></i>;<br/>
<i><font color="Green">E55</font></i>: <font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>
<b>then </b><i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 1 = 
<font color="Maroon">z</font> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="complex1.html#K13">/</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">1,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="txt" href="sin_cos.html#E2:87"><i><font color="Green">E55</font></i></a>, <a class="ref" href="newton.html#T19">NEWTON:19</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="sin_cos.html#E1:87"><i><font color="Green">E53</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i>, <a class="ref" href="comseq_3.html#T11">COMSEQ_3:11</a></i>
;<br/>
<a NAME="E5:87"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_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">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:87_4"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a>, <a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>;<br/>
<a NAME="E2:87_4"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:87_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E3:87_4"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:87_4_1"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font>
 ;<br/>

<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></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"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#D13" target="_self">Def13</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:87_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:87_4"/>
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_cos.html#D11" target="_self">Def11</a>, <a class="ref" href="sin_cos.html#D12" target="_self">Def12</a>);<br/></i>
<b>hence </b><a NAME="E5:87_4"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:87"/>
( <font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a>  &amp; <font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="sin_cos.html#T3" target="_self">Th3</a>, <a class="ref" href="sin_cos.html#D10" target="_self">Def10</a></i>;<br/>


</div>
