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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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E23"><i><font color="Green">Lemma52</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>thus </b><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<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="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" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <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="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"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="comptrig.html#E24"><i><font color="Green">Lemma53</font></i></a>, </i>
<br/>.= 
<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="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>

<br/>.= 
<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="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</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">(<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" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T12">NEWTON:12</a></i>
<br/>.= 
<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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></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><a class="txt" href="comptrig.html#E22"><i><font color="Green">Lemma51</font></i></a></i>
<br/>.= 
<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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:63_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><font color="Maroon">z</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></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="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</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>
<b>by </b><i><a class="txt" href="comptrig.html#E24"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</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>
<b>by </b><i><a class="txt" href="comptrig.html#E25"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="comptrig.html#E24"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>
<br/>.= 
<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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">z</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>
<b>by </b><i><a class="txt" href="comptrig.html#E25"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="comptrig.html#E24"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
