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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:86"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:86"/><i><font color="Green">E82</font></i>: 
( not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<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 &amp; not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not  <a href="complex2.html#K5" target="_self">angle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E5:86"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E6:86"/><i><font color="Green">E84</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E4:86"><i><font color="Green">E82</font></i></a>, <a class="ref" href="complex2.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="complex2.html#E1:86"><i><font color="Green">E79</font></i></a>, <a class="txt" href="complex2.html#E2:86"><i><font color="Green">E80</font></i></a>, <a class="txt" href="complex2.html#E3:86"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E8:86"/><i><font color="Green">E86</font></i>: 
(  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E9:86"/>
(  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<a NAME="E10:86"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex2.html#E6:86"><i><font color="Green">E84</font></i></a>, <a class="txt" href="complex2.html#E7:86"><i><font color="Green">E85</font></i></a>, <a class="txt" href="complex2.html#E8:86"><i><font color="Green">E86</font></i></a>, <a class="ref" href="complex2.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E11:86"/>
(  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  iff  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="complex2.html#E7:86"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex2.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E12:86"/>
contradiction
 <b>by </b><i><a class="txt" href="complex2.html#E4:86"><i><font color="Green">E82</font></i></a>, <a class="txt" href="complex2.html#E5:86"><i><font color="Green">E83</font></i></a>, <a class="txt" href="complex2.html#E10:86"><i><font color="Green">E87</font></i></a>, <a class="ref" href="complex2.html#T88" target="_self">Th88</a></i>;<br/>


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