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<b>let </b><font color="Maroon" title="c1">z</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<a NAME="E1:22"/><i><font color="Green" title="E22">A1</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:22_1"/><b>then </b>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T24" target="_self" title="COMPLEX2:th.24">Th24</a></i>;<br/><a NAME="E3:22_1"/><b>then </b>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a>, <a class="txt" href="complex2.html#E1:22"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:22_1"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:22"/><i><font color="Green" title="E23">A2</font></i>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>

<a NAME="E4:22"/><b>then </b>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E3:22"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="comptrig.html#T53" title="COMPTRIG:th.53">COMPTRIG:53</a></i>;<br/>


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