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<div class="add">

<b>let </b><font color="Maroon">x</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">y</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">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green">E23</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1_1"/>
( <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><b>case </b></a><a NAME="E1:80_1_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:80_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:80_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>then </b> <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E5:80_1_1_1"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2"><b>case </b></a><a NAME="E1:80_1_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:80_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>;<br/><b>then </b><i><font color="Green">E30</font></i>:  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E4:80_1_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:80_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font></span>)</span> = 
<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#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T21" target="_self">Th21</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 

;<br/><b>hence </b><a NAME="E7:80_1_1_2"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>


</div>
