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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_1"/>
 <a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:52_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></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">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></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">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E19"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0 or (  <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="complex2.html#E2:52_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T131">COMPLEX1:131</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2"/>
(  <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:52"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>hence </b><a NAME="E3:52"/>
 <a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/>


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