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

<b>let </b><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span><br/>





<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T7" target="_self" title="SIN_COS:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K12" title="SIN_COS:func.12">Expan</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="comseq_3.html#T29" title="COMSEQ_3:th.29">COMSEQ_3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K13" title="SIN_COS:func.13">Expan_e</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="sin_cos.html#T8" target="_self" title="SIN_COS:th.8">Th8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
