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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:98"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:98"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E4:98"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )</span><br/>

<a NAME="E5:98"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></span>;<br/>
<a NAME="E6:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A5</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="comptrig.html#T53" title="COMPTRIG:th.53">COMPTRIG:53</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A6</font></span>: <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 

;<br/>
<a NAME="E8:98"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>;<br/>
<a NAME="E9:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A7</font></span>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D6" target="_self" title="COMPLEX2:def.6">Def6</a></span>;<br/>
<a NAME="E10:98"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E11:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A8</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T27" target="_self" title="COMPLEX2:th.27">Th27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">A9</font></span>: <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 

;<br/>
<span class="lab"><font color="Green" title="E78">A10</font></span>:  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E69">A1</font></span></a></span>
;<br/>
<a NAME="E14:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A11</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="complex2.html#T37" target="_self" title="COMPLEX2:th.37">Th37</a></span>;<br/>
<a NAME="E15:98"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="complex2.html#T26" target="_self" title="COMPLEX2:th.26">Th26</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<a NAME="E16:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A12</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/>
<span class="lab"><font color="Green" title="E81">A13</font></span>: <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>
<br/>.= 
 <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 

;<br/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a></span>
<br/>.= 
 <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>
;<br/>
<a NAME="E19:98"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E20:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A14</font></span>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D6" target="_self" title="COMPLEX2:def.6">Def6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  implies <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_6"/><span class="lab"><font color="Green" title="E83">A15</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:98_6"/><span class="lab"><font color="Green" title="E84">A16</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:98_6"/><span class="lab"><font color="Green" title="E85">A17</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E4:98_6"/><span class="lab"><font color="Green" title="E86">A18</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_6"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">ba</font> = <font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">B</font> =  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">BA</font> =  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">mb</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">mba</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>;<br/><a NAME="E5:98_6"/><span class="lab"><font color="Green" title="E87">A19</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>;<br/><a NAME="E6:98_6"/>
( <font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T19" target="_self" title="COMPLEX2:th.19">Th19</a>, <a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></span>;<br/><a NAME="E7:98_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A20</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">Rb</font> =  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Rba</font> =  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">Ib</font> =  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">Iba</font> =  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E89">A21</font></span>:  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_6"><span class="lab"><font color="Green" title="E83">A15</font></span></a></span>
;<br/><a NAME="E9:98_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A22</font></span>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D16" title="COMPLEX1:def.16">COMPLEX1:def 16</a></span>;<br/><a NAME="E10:98_6"/><span class="lab"><font color="Green" title="E91">A23</font></span>: 
( <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/><a NAME="E11:98_6"/><span class="lab"><font color="Green" title="E92">A24</font></span>: 
( <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/><a NAME="E12:98_6"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_6"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E13:98_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A25</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T27" target="_self" title="COMPLEX2:th.27">Th27</a></span>;<br/><a NAME="E14:98_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A26</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></span>;<br/><a NAME="E15:98_6"/><span class="lab"><font color="Green" title="E95">A27</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></span>;<br/><a NAME="E16:98_6"/><span class="lab"><font color="Green" title="E96">A28</font></span>: 
 <a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="ref" href="comptrig.html#T63" title="COMPTRIG:th.63">COMPTRIG:63</a></span>;<br/><a NAME="E17:98_6"/><span class="lab"><font color="Green" title="E97">A29</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></span>;<br/><a NAME="E18:98_6"/>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T19" target="_self" title="COMPLEX2:th.19">Th19</a>, <a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></span>;<br/><a NAME="E19:98_6"/><span class="kw">then </span>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</a></span>;<br/><a NAME="E20:98_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A30</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E7:98_6"><span class="lab"><font color="Green" title="E88">A20</font></span></a>, <a class="txt" href="#E17:98_6"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E16:98_6"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="ref" href="xcmplx_1.html#T95" title="XCMPLX_1:th.95">XCMPLX_1:95</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_6_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_1"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_1"/><span class="lab"><font color="Green" title="E99">A31</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:98_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_6"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="#E4:98_6"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></span>;<br/><a NAME="E3:98_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E4:98_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98_6"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="txt" href="#E11:98_6"><span class="lab"><font color="Green" title="E92">A24</font></span></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></span>;<br/><a NAME="E5:98_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:98_6"><span class="lab"><font color="Green" title="E95">A27</font></span></a>, <a class="ref" href="xreal_1.html#T191" title="XREAL_1:th.191">XREAL_1:191</a></span>;<br/><a NAME="E6:98_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_6"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E20:98_6"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E7:98_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A32</font></span>: 
 <a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_6"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/><a NAME="E8:98_6_2_1"/>
(  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E5:98_6"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="#E4:98_6"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="txt" href="#E1:98_6_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="ref" href="comptrig.html#T59" title="COMPTRIG:th.59">COMPTRIG:59</a></span>;<br/><a NAME="E9:98_6_2_1"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98_6_2_1"><span class="lab"><font color="Green" title="E100">A32</font></span></a>, <a class="ref" href="complex2.html#T7" target="_self" title="COMPLEX2:th.7">Th7</a></span>;<br/><a NAME="E10:98_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E11:98_6_2_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_2"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_2"/><span class="lab"><font color="Green" title="E99">A33</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:98_6_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></span>;<br/><a NAME="E3:98_6_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A34</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="ref" href="comptrig.html#T55" title="COMPTRIG:th.55">COMPTRIG:55</a></span>;<br/><a NAME="E4:98_6_2_2"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_6"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E5:98_6"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="#E1:98_6_2_2"><span class="lab"><font color="Green" title="E99">A33</font></span></a>, <a class="ref" href="comptrig.html#T59" title="COMPTRIG:th.59">COMPTRIG:59</a></span>;<br/><a NAME="E5:98_6_2_2"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_6_2_2"><span class="lab"><font color="Green" title="E100">A34</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E6:98_6_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E7:98_6_2_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_3"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_3"/><span class="lab"><font color="Green" title="E99">A35</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:98_6_2_3"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_6_2_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_6_2_3_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_3_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:98_6_2_3_1_1_1"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="ref" href="comptrig.html#T59" title="COMPTRIG:th.59">COMPTRIG:59</a></span>;<br/><a NAME="E3:98_6_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A36</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E4:98_6_2_3_1_1_1"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E1:98_6_2_3"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="comptrig.html#T60" title="COMPTRIG:th.60">COMPTRIG:60</a></span>;<br/><a NAME="E5:98_6_2_3_1_1_1"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E6:98_6_2_3_1_1_1"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_6_2_3_1_1_1"><span class="lab"><font color="Green" title="E100">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:98_6_2_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:98_6_2_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_3_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:98_6_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></span>;<br/><a NAME="E3:98_6_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A37</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="ref" href="comptrig.html#T55" title="COMPTRIG:th.55">COMPTRIG:55</a></span>;<br/><a NAME="E4:98_6_2_3_1_1_2"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E1:98_6_2_3"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="comptrig.html#T60" title="COMPTRIG:th.60">COMPTRIG:60</a></span>;<br/><a NAME="E5:98_6_2_3_1_1_2"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_6_2_3_1_1_2"><span class="lab"><font color="Green" title="E100">A37</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E6:98_6_2_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E7:98_6_2_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:98_6_2_3_1_1_3"/><span class="lab"><font color="Green" title="E100">A38</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_6"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="#E4:98_6"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="square_1.html#T114" title="SQUARE_1:th.114">SQUARE_1:114</a></span>;<br/><a NAME="E3:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E4:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98_6"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="txt" href="#E10:98_6"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></span>;<br/><a NAME="E5:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98_6"><span class="lab"><font color="Green" title="E94">A26</font></span></a>, <a class="txt" href="#E17:98_6"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/><a NAME="E6:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_6"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E20:98_6"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E7:98_6_2_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A39</font></span>: 
 <a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_6"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/><a NAME="E8:98_6_2_3_1_1_3"/><span class="lab"><font color="Green" title="E102">A40</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E1:98_6_2_3_1_1_3"><span class="lab"><font color="Green" title="E100">A38</font></span></a>, <a class="ref" href="comptrig.html#T60" title="COMPTRIG:th.60">COMPTRIG:60</a></span>;<br/><a NAME="E9:98_6_2_3_1_1_3"/>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span>,<a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_6"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:98_6"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E1:98_6_2_3"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="comptrig.html#T60" title="COMPTRIG:th.60">COMPTRIG:60</a></span>;<br/><a NAME="E10:98_6_2_3_1_1_3"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98_6_2_3_1_1_3"><span class="lab"><font color="Green" title="E102">A40</font></span></a>, <a class="txt" href="#E7:98_6_2_3_1_1_3"><span class="lab"><font color="Green" title="E101">A39</font></span></a>, <a class="ref" href="complex2.html#T8" target="_self" title="COMPLEX2:th.8">Th8</a></span>;<br/><a NAME="E11:98_6_2_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E12:98_6_2_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E22:98"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E72">A4</font></span></a></span>;<br/>
<a NAME="E23:98"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E24:98"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<a NAME="E25:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A41</font></span>: 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E26:98"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<a NAME="E27:98"/><span class="kw">then </span>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D6" target="_self" title="COMPLEX2:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="txt" href="#E20:98"><span class="lab"><font color="Green" title="E82">A14</font></span></a></span>
<br/>.= 
 <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E81">A13</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )</span><br/>

<a NAME="E29:98"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E83">A41</font></span></a>, <a class="ref" href="complex2.html#T22" target="_self" title="COMPLEX2:th.22">Th22</a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:98"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D6" target="_self" title="COMPLEX2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span><br/>

<a NAME="E31:98"/><span class="lab"><font color="Green" title="E84">A42</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T52" title="COMPTRIG:th.52">COMPTRIG:52</a></span>;<br/>
<a NAME="E32:98"/>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T52" title="COMPTRIG:th.52">COMPTRIG:52</a></span>;<br/>
<a NAME="E33:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A43</font></span>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E34:98"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E74">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:98"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>,<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:98"><span class="lab"><font color="Green" title="E82">A14</font></span></a>, <a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E81">A13</font></span></a>, <a class="txt" href="#E33:98"><span class="lab"><font color="Green" title="E85">A43</font></span></a>, <a class="txt" href="#E31:98"><span class="lab"><font color="Green" title="E84">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
