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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:65"/><i><font color="Green">E67</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</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> <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>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E2:65"><i><font color="Green">E66</font></i></a>, <a class="ref" href="complex2.html#T68" target="_self">Th68</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green">E68</font></i>: 
 <a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="complex2.html#E2:65"><i><font color="Green">E66</font></i></a>, <a class="ref" href="complex2.html#T66" target="_self">Th66</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:65"/><i><font color="Green">E69</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex2.html#T68" target="_self">Th68</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:65"/><i><font color="Green">E70</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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>
 <b>by </b><i><a class="txt" href="complex2.html#E1:65"><i><font color="Green">E65</font></i></a>, <a class="ref" href="complex2.html#T68" target="_self">Th68</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E11:65"/><i><font color="Green">E71</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E2:65"><i><font color="Green">E66</font></i></a>, <a class="ref" href="complex2.html#T68" target="_self">Th68</a></i>;<br/>
<a NAME="E12:65"/><i><font color="Green">E72</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E4:65"><i><font color="Green">E67</font></i></a>, <a class="txt" href="complex2.html#E7:65"><i><font color="Green">E69</font></i></a>, <a class="txt" href="complex2.html#E9:65"><i><font color="Green">E70</font></i></a>, <a class="txt" href="complex2.html#E11:65"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E13:65"/><i><font color="Green">E73</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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> &amp;  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="complex2.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<b>thus </b> <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</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>
<b>by </b><i><a class="txt" href="complex2.html#E2:65"><i><font color="Green">E66</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <span class="p2">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E12:65"><i><font color="Green">E72</font></i></a>, <a class="txt" href="complex2.html#E13:65"><i><font color="Green">E73</font></i></a>, <a class="ref" href="complex2.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="complex2.html#K4" target="_self">angle</a> <span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="complex2.html#K3" target="_self">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="complex2.html#E5:65"><i><font color="Green">E68</font></i></a>, <a class="ref" href="complex2.html#D5" target="_self">Def5</a></i>
;<br/>


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