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<b>let </b><font color="Maroon">z</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E23</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="txt" href="complex2.html#E1:23_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="comptrig.html#D1">COMPTRIG:def 1</a></i>;<br/><a NAME="E3:23_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#D2">COMPLEX1:def 2</a>, <a class="ref" href="complex1.html#D3">COMPLEX1:def 3</a>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/><a NAME="E4:23_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_1"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_2"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complex2.html#E1:23_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="comptrig.html#T53">COMPTRIG:53</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:23"/><i><font color="Green">E30</font></i>: 
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E3:23"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<b>hence </b><a NAME="E4:23"/>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i>, <a class="ref" href="comptrig.html#T54">COMPTRIG:54</a></i>;<br/>


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