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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:45"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:45"/><i><font color="Green">E49</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1" target="_self">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span> &amp;  <a href="comptrig.html#K1" target="_self">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="ref" href="comptrig.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E3:45"/>
<font color="Maroon">c<sub>1</sub></font> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="comptrig.html#E1:45"><i><font color="Green">E48</font></i></a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="comptrig.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E5:45"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="comptrig.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E6:45"/>
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i><a class="txt" href="comptrig.html#E4:45"><i><font color="Green">E50</font></i></a>, <a class="txt" href="comptrig.html#E61"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E7:45"/><b>then </b><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="comptrig.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:45"/><b>then </b>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="comptrig.html#E5:45"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E9:45"/><b>then </b>
(  <a href="comptrig.html#K1" target="_self">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  or <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="comptrig.html#E2:45"><i><font color="Green">E49</font></i></a>, <a class="txt" href="comptrig.html#E7:45"><i><font color="Green">E52</font></i></a>, <a class="ref" href="comptrig.html#T33" target="_self">Th33</a>, <a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>;<br/>
<b>hence </b><a NAME="E10:45"/>
 <a href="comptrig.html#K1" target="_self">Arg</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,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="txt" href="comptrig.html#E1:45"><i><font color="Green">E48</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


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