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





<a NAME="E1:33"/><i><font color="Green">E53</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E2:33"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</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">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</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="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">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</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">k</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="newton.html#K2">|^</a> 0</span>)</span>
<b>by </b><i>, </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">k</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</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">k</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">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <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">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">k</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">k</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">z</font> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>


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