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

<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<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/>





<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </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/><a NAME="E1:23_1"/><i><font color="Green">E52</font></i>: 
<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> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</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"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <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></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T1">SIN_COS:1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <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></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
;<br/>
<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="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K3">Prod_real_n</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D7">SIN_COS:def 7</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 1
<b>by </b><i><a class="ref" href="sin_cos.html#D5">SIN_COS:def 5</a></i>
<br/>.= 
1

;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green">E55</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="23_4"><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="E1:23_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:23_4"/><i><font color="Green">E58</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:23_4"/>
<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> <a href="hidden.html#R1">=</a> <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>
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E4:23_4"/><b>then </b>
(  <a href="complex1.html#K3">Re</a> <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> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="complex1.html#K4">Im</a> <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> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>then </b><i><font color="Green">E60</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<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></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</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="complex1.html#T136">COMPLEX1:136</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E6:23_4"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</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="complex1.html#T153">COMPLEX1:153</a>, , <a class="ref" href="complex1.html#T134">COMPLEX1:134</a></i>;<br/>
<a NAME="E7:23_4"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:23_4"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>;<br/>
<a NAME="E9:23_4"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a>, </i>;<br/>
<a NAME="E10:23_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/>
<a NAME="E11:23_4"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T104">XCMPLX_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T2">SIN_COS:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.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="sin_cos.html#D9">SIN_COS:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E13:23_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/>
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>(, );<br/></i>
<b>hence </b><a NAME="E7:23"/>
( <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</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"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <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></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/>


</div>
