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

<a NAME="E1:106"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<a NAME="E1:106_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="comseq_1.html#K9">|.</a><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">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><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">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:106_1_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><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">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="comseq_3.html#T30">COMSEQ_3:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:106_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a>  is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E3:106_1"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E4:106_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E5:106_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a>  is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E6:106_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a>  is <a href="comseq_3.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E7:106_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="comseq_3.html#K7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span> is  <a href="comseq_2.html#V2">convergent</a> <a href="comseq_1.html#NM1">Complex_Sequence</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="comseq_3.html#K7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span> as  <a href="comseq_2.html#V2">convergent</a> <a href="comseq_1.html#NM1">Complex_Sequence</a> <b>by </b><i><a class="txt" href="sin_cos.html#E6:106_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>
<a NAME="E9:106_1"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
;<br/>
<a NAME="E10:106_1"/><i><font color="Green">E94</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><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">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E7:106_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="comseq_2.html#T11">COMSEQ_2:11</a></i>;<br/>
<a NAME="E11:106_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><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">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="sin_cos.html#E4:106_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sin_cos.html#E9:106_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E12:106_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E10:106_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:106_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:106"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<i><font color="Green">E90</font></i>: <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#D28" target="_self">Def28</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T37" target="_self">Th37</a></i>
;<br/>
<a NAME="E2:106_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T42" target="_self">Th42</a></i>;<br/>
<b>then </b><i><font color="Green">E91</font></i>: <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2">#N</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sin_cos.html#E111"><i><font color="Green">Lemma84</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E91"><i><font color="Green">Lemma71</font></i></a></i>
;<br/>
<i><font color="Green">E92</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<a NAME="E5:106_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T42" target="_self">Th42</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<a NAME="E6:106_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E7:106_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T3">POWER:3</a></i>;<br/>
<a NAME="E8:106_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:106_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/>
<b>hence </b><a NAME="E9:106_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:106_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sin_cos.html#E4:106_2"><i><font color="Green">E92</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E3:106"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E4:106"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:106"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 ;<br/>
<a NAME="E8:106"/>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E9:106"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:106"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green">E92</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E10:106"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:106_3"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:106_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E9:106"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:106_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
;<br/>
<a NAME="E4:106_3"/><b>then </b><i><font color="Green">E97</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<a NAME="E5:106_3"/><b>then </b><i><font color="Green">E98</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>
<a NAME="E6:106_3"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:106"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:106_3_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:106_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:106_3_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:106_3"/><b>then </b><i><font color="Green">E100</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:106_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="sin_cos.html#E6:106_3"><i><font color="Green">E99</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<a NAME="E9:106_3"/><i><font color="Green">E101</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sin_cos.html#E5:106_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/>
<a NAME="E10:106_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:106_3_3"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sin_cos.html#E1:106_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E2:106_3_3"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E3:106_3_3"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_3_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="sin_cos.html#E2:106_3_3"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:106_3_3"/><i><font color="Green">E105</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:106_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:106_3_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:106_3_3"><i><font color="Green">E104</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E6:106_3_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E4:106_3_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:106_3"/><b>then </b>
(  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sin_cos.html#E8:106_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="sin_cos.html#E9:106_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:106_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E11:106"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:106"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sin_cos.html#E4:106"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sin_cos.html#E10:106"><i><font color="Green">E94</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>


</div>
