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

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



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">s</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="complex2.html#T19">COMPLEX2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</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"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</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"><span class="p5"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</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"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</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>then </b><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"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="complex2.html#T19">COMPLEX2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</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"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</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"><span class="p5"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</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"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</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/>
<a NAME="E5:53"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/>
<a NAME="E6:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</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"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T68">SQUARE_1:68</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>;<br/>
<a NAME="E9:53"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E11:53"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E12:53"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E13:53"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>;<br/>
<a NAME="E14:53"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>;<br/>
<a NAME="E15:53"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="power.html#T5">POWER:5</a></i>;<br/>
<b>hence </b><a NAME="E16:53"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">s</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, , <a class="ref" href="power.html#T5">POWER:5</a></i>;<br/>


</div>
