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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> = 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E42"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="sin_cos.html#T10" target="_self">Th10</a></i>
;<br/>
<i><font color="Green">E47</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T155">COMPLEX1:155</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="sin_cos.html#E2:62"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sin_cos.html#E56"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="complex1.html#T134">COMPLEX1:134</a></i>
;<br/>
<b>thus </b><a NAME="E4:62"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><b>proof </b></a><div class="add">

<a NAME="E1:62_3"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:62"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xcmplx_1.html#T74">XCMPLX_1:74</a></i>;<br/>
<a NAME="E2:62_3"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E3:62_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E4:62_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:62_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sin_cos.html#E2:62_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xcmplx_1.html#T200">XCMPLX_1:200</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E5:62"/><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E6:62"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D22" target="_self">Def22</a></i>;<br/>
<b>hence </b><a NAME="E7:62"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:62"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sin_cos.html#E4:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sin_cos.html#E5:62"><i><font color="Green">E49</font></i></a>, <a class="ref" href="comseq_3.html#T13">COMSEQ_3:13</a></i>;<br/>


</div>
