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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/>

<b>assume </b><a NAME="E1:57"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:57"/><b>then </b>
(  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  <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="E3:57"/><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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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> &amp; (  <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#NR3" target="_self">&lt;</a> 0 ) ) )
 
<b>by </b><i><a class="ref" href="comptrig.html#T9" target="_self">Th9</a>, <a class="ref" href="comptrig.html#T64" target="_self">Th64</a></i>;<br/>
<b>hence </b><a NAME="E4:57"/>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" target="_self">Arg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="comptrig.html#T53" target="_self">Th53</a>, <a class="ref" href="comptrig.html#T54" target="_self">Th54</a>, <a class="ref" href="sin_cos.html#T34">SIN_COS:34</a>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/>


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