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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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">ab2</font> =  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>);<br/>
<a NAME="E1:71"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T34" title="COMPTRIG:th.34">COMPTRIG:34</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_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="E1:71_1"/>
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_1"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><a NAME="E2:71_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A3</font></span>: 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D3" target="_self" title="COMPLEX2:def.3">Def5</a></span>;<br/><span class="kw">thus </span><a NAME="E3:71_1_1"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_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="E1:71_1_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="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>  or <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_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="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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><a NAME="E2:71_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A4</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="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="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="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#D4" target="_self" title="COMPLEX2:def.4">Def6</a></span>;<br/><a NAME="E3:71_1_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A5</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" target="_self" title="COMPLEX2:th.70">Th84</a></span>;<br/><a NAME="E4:71_1_1_1_1_1"/>
(  ex <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">i</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"><a href="real_1.html#K1" title="REAL_1:func.1">-</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> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="complex2.html#T54" target="_self" title="COMPLEX2:th.54">Th68</a>, <a class="ref" href="complex2.html#T70" target="_self" title="COMPLEX2:th.70">Th84</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1_1_1_1_1"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:71_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E2:71_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E3:71_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="complex2.html#T2" target="_self" title="COMPLEX2:th.2">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="71_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A6</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="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#NR3" title="XXREAL_0:NR.3">&lt;</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:71_1_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">i</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"><a href="real_1.html#K1" title="REAL_1:func.1">-</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> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="complex2.html#T54" target="_self" title="COMPLEX2:th.54">Th68</a></span>;<br/><a NAME="E4:71_1_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<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#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">i</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"><a href="real_1.html#K1" title="REAL_1:func.1">-</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> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</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="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E5:71_1_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A9</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</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="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> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_1_1_2"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="complex2.html#D4" target="_self" title="COMPLEX2:def.4">Def6</a></span>;<br/><a NAME="E6:71_1_1_1_1_2"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</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="real_1.html#K1" title="REAL_1:func.1">-</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> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</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="real_1.html#K1" title="REAL_1:func.1">-</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> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" target="_self" title="COMPLEX2:th.70">Th84</a></span>;<br/><span class="kw">hence </span><a NAME="E7:71_1_1_1_1_2"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:71_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E5:71_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E3:71_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E4:71_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="complex2.html#T2" target="_self" title="COMPLEX2:th.2">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_2"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:71_1_2"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_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="E1:71_1_2_1_1"/>
(  <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>  or  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A11</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><a NAME="E2:71_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A12</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="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="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="#E1:71_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="comptrig.html#T35" title="COMPTRIG:th.35">COMPTRIG:35</a></span>;<br/> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) = 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="complex2.html#D3" target="_self" title="COMPLEX2:def.3">Def5</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="complex2.html#E13"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a>, <a class="ref" href="complex2.html#T52" target="_self" title="COMPLEX2:th.52">Th66</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:71_1_2_1_1_1"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="complex2.html#D4" target="_self" title="COMPLEX2:def.4">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="71_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A13</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)</span><br/><div class="add"><a NAME="E2:71_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</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="real_1.html#K1" title="REAL_1:func.1">-</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="c1">a</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T34" title="COMPTRIG:th.34">COMPTRIG:34</a></span>;<br/><a NAME="E3:71_1_2_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="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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="complex2.html#E13"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a></span>;<br/><a NAME="E4:71_1_2_1_1_2"/><span class="kw">then </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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">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="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="complex2.html#D4" target="_self" title="COMPLEX2:def.4">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1_2_1_1_2"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E1:71_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="complex2.html#D3" target="_self" title="COMPLEX2:def.3">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>

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