<?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">w</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/>





<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
1
<b>by </b><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">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:34"/><i><font color="Green">E55</font></i>: 
0 <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> 0,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> 0,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E5:34"/><i><font color="Green">E57</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="34_3"><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="E1:34_3"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E2:34_3"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></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="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</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>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:34_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</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/>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></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"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
;<br/><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E6:34"/>
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="E7:34"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" target="_self">Alfa</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div>
