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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  or <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E1:63"><i><font color="Green">E59</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">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complfld.html#T96">COMPLFLD:96</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <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">c<sub>1</sub></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="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</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">c<sub>1</sub></font></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"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E1:63_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="comptrig.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="txt" href="comptrig.html#E61"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p2"><a href="hahnban1.html#K1">[**</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" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></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" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="hahnban1.html#T11">HAHNBAN1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</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" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></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" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comptrig.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="comptrig.html#T71" target="_self">Th71</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="txt" href="comptrig.html#E2:63_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="comptrig.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><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="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T11">HAHNBAN1:11</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></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="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></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">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><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#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="comptrig.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><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#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></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="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="txt" href="comptrig.html#E1:63_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="comptrig.html#E16"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="complfld.html#T93">COMPLFLD:93</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></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="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="txt" href="comptrig.html#E1:63_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="comptrig.html#E16"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="complfld.html#T93">COMPLFLD:93</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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