<?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/>





<a NAME="E1:25"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<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> 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a> 
 
<b>by </b><i><a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>;<br/>
<a NAME="E2:25"/><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/>
<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"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="newton.html#K2">|^</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#K11" target="_self">Expan</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/>
<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#K11" target="_self">Expan</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#K11" target="_self">Expan</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> 0</span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> 0</span>)</span>
<b>by </b><i>, </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">1 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> 0</span>)</span>
<b>by </b><i>, , </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="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<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> 
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>
;<br/>
<a NAME="E4:25"/><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="E5:25"/><i><font color="Green">E57</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="25_2"><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="E1:25_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<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> <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#K11" target="_self">Expan</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/>

<i><font color="Green">E61</font></i>: <span class="p1">(<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="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"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></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"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K4">(#)</a> <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#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></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="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="comseq_3.html#T29">COMSEQ_3:29</a></i>
;<br/>
<a NAME="E3:25_2"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1"><b>now </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>thus </b><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="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K8">+</a> <font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<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="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"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <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="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="comseq_3.html#T27">COMSEQ_3:27</a></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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</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="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/>
<a NAME="E5:25_2"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_4"><b>proof </b></a><div class="add">

<i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <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="comseq_3.html#D7">COMSEQ_3:def 7</a></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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
;<br/>
<a NAME="E2:25_2_4"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:25_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:25_2_4"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:25_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></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"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_5"><b>proof </b></a><div class="add">

<i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <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="comseq_3.html#D7">COMSEQ_3:def 7</a></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"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
;<br/>
<a NAME="E2:25_2_5"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:25_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:25_2_5"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:25_2"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:25_2_6"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <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/>
<a NAME="E2:25_2_6"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <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="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:25_2_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:25_2_6"/><b>then </b>
<font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
;<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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="complex1.html#K8">+</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i>, <a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E8:25_2"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="sin_cos.html#K11" target="_self">Expan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1"><b>now </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/><div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:25_2_7_1_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_2_7_1_1_1"/><i><font color="Green">E101</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <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="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E2:25_2_7_1_1_1"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:25_2_7_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:25_2_7_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2_7_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K5">0c</a> </span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K5">0c</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="complex1.html#K5">0c</a> 

;<br/><b>hence </b><a NAME="E4:25_2_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>thus </b><a NAME="E2:25_2_7_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_2_7_1_2_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">n</font> <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="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<i><font color="Green">E108</font></i>: <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
0

;<br/>
<i><font color="Green">E109</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</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="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
0

;<br/>
<a NAME="E4:25_2_7_1_2_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, </i>
;<br/>
<i><font color="Green">E112</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</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="comseq_1.html#K1">.</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><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" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, </i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <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="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K5">0c</a> </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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <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_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><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">n</font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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="xcmplx_0.html#K3">*</a> <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></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"><a href="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
<b>by </b><i>, , , </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_2_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_2_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:25_2_7_1_2_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:25_2_7_1_2_2_1"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0 <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="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>thus </b><a NAME="E4:25_2_7_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_2_7_1_2_2_1_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E119</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, </i>
;<br/>
<i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</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="comseq_1.html#K1">.</a> 0 = 
<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#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</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" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, </i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> 0</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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <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_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="comseq_3.html#K1">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</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="binarith.html#K3">-'</a> 0</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"><a href="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_7_1_2_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:25_2_7_1_2_2_2"/><b>then </b><i><font color="Green">E121</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E3:25_2_7_1_2_2_2"/><b>then </b><i><font color="Green">E122</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:25_2_7_1_2_2_2"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:25_2_7_1_2_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:25_2_7_1_2_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:25_2_7_1_2_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:25_2_7_1_2_2_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>thus </b><a NAME="E9:25_2_7_1_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_7_1_2_2_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E125</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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">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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<i><font color="Green">E126</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</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="real_1.html#K5">-</a> <font color="Maroon">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<i><font color="Green">E127</font></i>: <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</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="real_1.html#K5">-</a> <font color="Maroon">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/>
<a NAME="E5:25_2_7_1_2_2_2_1"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <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/>
<a NAME="E6:25_2_7_1_2_2_2_1"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="comseq_3.html#D1">COMSEQ_3:def 1</a></i>;<br/>
<a NAME="E7:25_2_7_1_2_2_2_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></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#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<i><font color="Green">E132</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></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#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></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#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E9:25_2_7_1_2_2_2_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><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="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>
 
;<br/>
<i><font color="Green">E134</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<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#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><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="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E11:25_2_7_1_2_2_2_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <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="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <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="real_1.html#K5">-</a> <font color="Maroon">k</font></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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">k</font> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>then </b><i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></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="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E13:25_2_7_1_2_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></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="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D2" target="_self">Def2</a>, , <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <font color="Maroon">n</font></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#K8" target="_self">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></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="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></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="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>

<br/>.= 
<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#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</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="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" target="_self">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <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>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</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="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_7_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:25_2_7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></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="sin_cos.html#K11" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</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> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_2_7"/>
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</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#K2">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K10" target="_self">Sift</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="sin_cos.html#K11" target="_self">Expan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<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="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"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <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#K10" target="_self">Sift</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="comseq_1.html#K4">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K11" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</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_1.html#D4">COMSEQ_1:def 4</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#K11" target="_self">Expan</a> <span class="p3">(<span class="default"><font color="Maroon">n</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_3.html#T27">COMSEQ_3:27</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:25"/>
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="E7:25"/>
<span class="p1">(<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> <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#K11" target="_self">Expan</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>
