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

<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="E1:117"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:117"/><b>then </b><i><font color="Green">E98</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:117"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E4:117"/>
<font color="Maroon">c<sub>3</sub></font> is   <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/>
<a NAME="E5:117"/><b>then </b><i><font color="Green">E100</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="sin_cos.html#T39" target="_self">Th39</a>, <a class="ref" href="sin_cos.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E6:117"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:117"><i><font color="Green">E99</font></i></a>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E7:117"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:117"><i><font color="Green">E99</font></i></a>, <a class="txt" href="sin_cos.html#E5:117"><i><font color="Green">E100</font></i></a>, <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<a NAME="E8:117"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:117"><i><font color="Green">E100</font></i></a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E9:117"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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/>

<i><font color="Green">E103</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E78"><i><font color="Green">Lemma60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<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="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D25" target="_self">Def25</a></i>
<br/>.= 
<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="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </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"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D25" target="_self">Def25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </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"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 1</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"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
1 <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="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <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">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a></i>
<br/>.= 
1 <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="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T20">NEWTON:20</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>

<br/>.= 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>

;<br/>
<a NAME="E2:117_1"/><i><font color="Green">E104</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E3:117_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:117"><i><font color="Green">E98</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E4:117_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K2">*</a> 1
 
;<br/>
<a NAME="E5:117_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E6:117_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:117_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sin_cos.html#E2:117_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E7:117_1"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:117_1_2"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E78"><i><font color="Green">Lemma60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <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#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <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#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E78"><i><font color="Green">Lemma60</font></i></a></i>
<br/>.= 
<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="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <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#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/>
<a NAME="E3:117_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<i><font color="Green">E109</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D25" target="_self">Def25</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>

;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D25" target="_self">Def25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E81"><i><font color="Green">Lemma61</font></i></a></i>
;<br/>
<b>then </b><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>

;<br/>
<a NAME="E7:117_1_2"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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="E8:117_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:117_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E10:117_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:117"><i><font color="Green">E98</font></i></a>, <a class="txt" href="sin_cos.html#E8:117_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="sin_cos.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E11:117_1_2"/><i><font color="Green">E114</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T3">POWER:3</a></i>;<br/>
<a NAME="E12:117_1_2"/><i><font color="Green">E115</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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/>
<a NAME="E13:117_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E9:117_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/>
<a NAME="E14:117_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E12:117_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E15:117_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E10:117_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="sin_cos.html#E11:117_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="sin_cos.html#E13:117_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>
<i><font color="Green">E118</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>

;<br/>
<a NAME="E17:117_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</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">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E18:117_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos.html#E15:117_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="sin_cos.html#E16:117_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E19:117_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:117_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="sin_cos.html#E4:117_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sin_cos.html#E6:117_1_2"><i><font color="Green">E110</font></i></a></i>;<br/>
<a NAME="E20:117_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E18:117_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E21:117_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:117_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:117_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sin_cos.html#E6:117_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sin_cos.html#E7:117_1"><i><font color="Green">E106</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:117_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:117"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E6:117"><i><font color="Green">E101</font></i></a>, <a class="txt" href="sin_cos.html#E8:117"><i><font color="Green">E102</font></i></a>, <a class="ref" href="sin_cos.html#T41" target="_self">Th41</a></i>;<br/>


</div>
