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<b>let </b><font color="Maroon">z</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="E1:25"/><i><font color="Green">E23</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or ( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> &amp;  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  ) )
 </span><b>by </b><i>, <a class="ref" href="comptrig.html#T52">COMPTRIG:52</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/>
(  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  or  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="complex2.html#E1:25"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> &amp;  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E3:25_1_2"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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