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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="E1:20"/><i><font color="Green">E23</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E26</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="comptrig.html#T52">COMPTRIG:52</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/>
 <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/><a NAME="E2:20_1"/><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="complex2.html#T2" target="_self">Th2</a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:20_1"/><b>then </b>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:20_1"/>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="comptrig.html#T63">COMPTRIG:63</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E4:20"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</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="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="rcomp_1.html#T52">RCOMP_1:52</a></i>;<br/>
<b>thus </b><a NAME="E5:20"/>
(  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies  <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>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/>
 <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:20_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="comptrig.html#T66">COMPTRIG:66</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_2_1_1"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_2_1_1_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_2_1_1_1"/><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,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="comptrig.html#T59">COMPTRIG:59</a></i>;<br/><b>hence </b><a NAME="E3:20_2_1_1_1"/>
 <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> 
 <b>by </b><i><a class="txt" href="complex2.html#E1:20"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_2_1_1_2"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_2_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <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="complex1.html#T29">COMPLEX1:29</a></i>;<br/><b>hence </b><a NAME="E3:20_2_1_1_2"/>
 <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> 
 <b>by </b><i>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="comptrig.html#T55">COMPTRIG:55</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1_3"><b>suppose </b></a><a NAME="E1:20_2_1_1_3"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_2_1_1_3"/><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"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<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="comptrig.html#T60">COMPTRIG:60</a></i>;<br/><b>hence </b><a NAME="E3:20_2_1_1_3"/>
 <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> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_2"/>
 <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><b>end;</b></div>


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