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

<a NAME="E1:84"/><i><font color="Green">E53</font></i>: 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1 <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="84_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <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">bq</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">1 <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="E2:84_1"/><i><font color="Green">E56</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default">1 <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> 1 <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:84_1"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<a NAME="E3:84_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <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">bq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E4:84_1"/>
 <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">1 <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">bq</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">1 <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<a NAME="E5:84_1"/><b>then </b><i><font color="Green">E59</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">1 <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">bq</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> 1
 
<b>by </b><i>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E6:84_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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="84_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default">1 <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">bq</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/>
<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">1 <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">bq</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">1 <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">bq</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">1 <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/>
<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">1 <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">1 <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">1 <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">1 <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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <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> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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#D2" target="_self">Def2</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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <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> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <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> 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#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <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> 0</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <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> 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#E2:84_1_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <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> 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#E2:84_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span></span>)</span> <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> 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#E2:84_1_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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="real_1.html#K4">*</a> 1</span>)</span> <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> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="newton.html#T18">NEWTON:18</a></i>
;<br/>
<a NAME="E2:84_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:84_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_1_1_2"/><i><font color="Green">E91</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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <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">bq</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<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">1 <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">k</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/>
<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">1 <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">k</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">1 <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">k</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">1 <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">k</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">1 <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">k</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">1 <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">k</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">1 <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">bq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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">k</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">1 <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">k</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/>
<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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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">k</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">1 <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">k</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="E3:84_1_1_2"/><b>then </b><i><font color="Green">E92</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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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">k</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">1 <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">k</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">E93</font></i>: <span class="p1">(<span class="default">1 <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">k</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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">k</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#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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">k</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#E2:84_1_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
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">k</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="newton.html#T15">NEWTON:15</a></i>
;<br/>
<span class="p1">(<span class="default">1 <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">k</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">k</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="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">k</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="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">k</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#D2" target="_self">Def2</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="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">k</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="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">k</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#E2:84_1_1"><i><font color="Green">E61</font></i></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="real_1.html#K4">*</a> 1</span>)</span> <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">k</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="newton.html#T15">NEWTON:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <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">k</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="E6:84_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default">1 <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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</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">k</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#T188">XCMPLX_1:188</a></i>;<br/>
<a NAME="E7:84_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">k</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="E8:84_1_1_2"/><b>then </b>
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">k</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">k</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="E9:84_1_1_2"/><b>then </b><i><font color="Green">E95</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">k</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">k</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/>;<br/>
<a NAME="E10:84_1_1_2"/>
<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">k</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
 
<b>by </b><i/>;<br/>
<a NAME="E11:84_1_1_2"/><b>then </b>
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">k</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">k</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#D15" target="_self">Def15</a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/>
<a NAME="E12:84_1_1_2"/><b>then </b>
<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">k</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="real_1.html#K5">-</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">k</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="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E13:84_1_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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D12" target="_self">Def12</a>, <a class="ref" href="sin_cos.html#D13" target="_self">Def13</a>, <a class="ref" href="sin_cos.html#D14" target="_self">Def14</a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E14:84_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="sin_cos.html#D11" target="_self">Def11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="sin_cos.html#T3" target="_self">Th3</a>, <a class="ref" href="sin_cos.html#D10" target="_self">Def10</a>);<br/></i>
<b>hence </b><a NAME="E5:84_1_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">1 <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">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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="E7:84_1"/>
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1 <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="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="txt" href="sin_cos.html#E5:84_1"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:84"/><b>then </b><i><font color="Green">E96</font></i>: 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:84"/><i><font color="Green">E97</font></i>: 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 <a href="real_1.html#K6">/</a> 6
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<a NAME="E1:84_2"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E2:84_2"/><i><font color="Green">E99</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp; <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:84_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E4:84_2"/>
 <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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<a NAME="E5:84_2"/><b>then </b><i><font color="Green">E101</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E6:84_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 <a href="real_1.html#K6">/</a> 6
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</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> 5 <a href="real_1.html#K6">/</a> 6;<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">bq</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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/>
<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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">0 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</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"><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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</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="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</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="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="nat_1.html#K1">+</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="newton.html#T15">NEWTON:15</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</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> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 6</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T18">NEWTON:18</a>, <a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
5 <a href="real_1.html#K6">/</a> 6

;<br/>
<a NAME="E2:84_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:84_2_1"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_1_2"/><i><font color="Green">E104</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 <a href="real_1.html#K6">/</a> 6
 ;<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">bq</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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/>
<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">bq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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/>
<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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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="E3:84_2_1_2"/><b>then </b><i><font color="Green">E105</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">E106</font></i>: <span class="p1">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</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> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>
;<br/>
<span class="p1">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</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">k</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="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> <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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><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="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <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"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</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> <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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><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="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</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> <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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</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> <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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>

;<br/>
<a NAME="E6:84_2_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default">1 <a href="sin_cos.html#K25" target="_self">P_sin</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</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"><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">k</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="nat_1.html#K1">+</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="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/>
<a NAME="E7:84_2_1_2"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">k</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="E8:84_2_1_2"/><b>then </b>
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">k</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">k</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="E9:84_2_1_2"/><b>then </b>
<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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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">k</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="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:84_2_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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"><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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:84_2_1_2"/>
<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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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/>;<br/>
<a NAME="E12:84_2_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</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> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</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> <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">k</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E21"><i><font color="Green">Lemma82</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/>
<a NAME="E13:84_2_1_2"/><b>then </b>
<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"><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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</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"><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">k</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="nat_1.html#K1">+</a> 1</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="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:84_2_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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D14" target="_self">Def14</a>, <a class="ref" href="sin_cos.html#D15" target="_self">Def15</a>, <a class="ref" href="sin_cos.html#D17" target="_self">Def17</a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E15:84_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="sin_cos.html#D13" target="_self">Def13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="sin_cos.html#D11" target="_self">Def11</a>, <a class="ref" href="sin_cos.html#D12" target="_self">Def12</a>);<br/></i>
<b>hence </b><a NAME="E5:84_2_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">1 <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">bq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 <a href="real_1.html#K6">/</a> 6
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:84_2"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 5 <a href="real_1.html#K6">/</a> 6
 <b>by </b><i><a class="ref" href="sin_cos.html#T3" target="_self">Th3</a>, <a class="ref" href="sin_cos.html#D10" target="_self">Def10</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84"/><b>then </b><i><font color="Green">E109</font></i>: 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E5:84"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<a NAME="E1:84_3"/><i><font color="Green">E110</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E2:84_3"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:84_3"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 25 <a href="real_1.html#K6">/</a> 36
 
<b>by </b><i>, , <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E4:84_3"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">25 <a href="real_1.html#K6">/</a> 36</span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T3" target="_self">Th3</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E5:84_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;</a> 25 <a href="real_1.html#K6">/</a> 36
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:84_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D10" target="_self">Def10</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:84_3"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/>
<a NAME="E8:84_3"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </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> 1
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<b>hence </b><a NAME="E9:84_3"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i><a class="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="sin_cos.html#D11" target="_self">Def11</a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:84"/>
( <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 1 )
 <b>by </b><i/>;<br/>


</div>
