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<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:26_1"/><i><font color="Green">E31</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_1"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><a NAME="E3:26_1"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="ref" href="comptrig.html#T53">COMPTRIG:53</a>, <a class="ref" href="comptrig.html#T54">COMPTRIG:54</a></i>;<br/><span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></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="hahnban1.html#D1">HAHNBAN1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="complex2.html#E1:26_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
;<br/><b>hence </b><a NAME="E5:26_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="complex2.html#E3:26_1"><i><font color="Green">E32</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:26"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  )
 ;<br/>

<b>hence </b><a NAME="E3:26"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</a></i>;<br/>


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