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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:58"/><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span> = 
<a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a>, <a class="ref" href="sin_cos3.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos3.html#E6"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos3.html#E6"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="complex1.html#K7">&lt;i&gt;</a>  <a href="complex1.html#K9">*</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos3.html#E1"><i><font color="Green">Lemma24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K1" target="_self">sin_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos3.html#K3" target="_self">sinh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos3.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos3.html#K4" target="_self">cosh_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </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"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<b>hence </b><a NAME="E3:58"/>
<a href="sin_cos3.html#K2" target="_self">cos_C</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K4">cosh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos2.html#K1">sinh</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 ;<br/>


</div>
