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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<a NAME="E1:33"/><i><font color="Green">E27</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E2:33"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K12" target="_self">Expan_e</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" target="_self">Coef_e</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E2:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="sin_cos.html#D14" target="_self">Def14</a></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#K9" target="_self">Coef_e</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#T11">COMSEQ_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E3:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="sin_cos.html#D11" target="_self">Def11</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self">Def8</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>


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