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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="E1:44"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<a NAME="E2:44"/><i><font color="Green">E48</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0 or <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#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E1:44"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>suppose </b></a><a NAME="E1:44_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1"/><b>then </b>
<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> 0
 ;<br/><a NAME="E3:44_1_1"/><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
 ;<br/><a NAME="E4:44_1_1"/><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="E5:44_1_1"/>
<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#E3:44_1_1"><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="E6:44_1_1"/><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="E7:44_1_1"/><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#E4:44_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E8:44_1_1"/><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> 0 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> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="comptrig.html#E2:44"><i><font color="Green">E48</font></i></a>, <a class="txt" href="comptrig.html#E6:44_1_1"><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#T82">SIN_COS:82</a></i>;<br/><a NAME="E9:44_1_1"/><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> 0 or  <a href="real_1.html#K1">-</a> <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#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E10:44_1_1"/><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> 0 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="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="comptrig.html#E1:44_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E11:44_1_1"/>
 <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> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2"><b>suppose </b></a><a NAME="E1:44_1_2"/>
<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#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_2"/><b>then </b>
<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="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><b>hence </b><a NAME="E3:44_1_2"/>
 <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> 0
 <b>by </b><i><a class="ref" href="comptrig.html#D1" target="_self">Def1</a>, <a class="txt" href="comptrig.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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