<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; not <!--HUMANRECFAILED-->(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) implies (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:103"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:103"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E3:103"/><span class="lab"><font color="Green" title="E76">A3</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E4:103"/><span class="lab"><font color="Green" title="E77">A4</font></span>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_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="E1:103_1"/>
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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>  or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="complex2.html#T97" target="_self" title="COMPLEX2:th.97">Th97</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1"><span class="kw">suppose </span></a><a NAME="E1:103_1_1"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><a NAME="E2:103_1_1"/><span class="lab"><font color="Green" title="E79">A6</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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T84" target="_self" title="COMPLEX2:th.84">Th84</a></span>;<br/><span class="kw">thus </span><a NAME="E3:103_1_1"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="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="103_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_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="E1:103_1_1_1_1"/>
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</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>  or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:103_1_1_1_1_1"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</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"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_1_1_1_1"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="txt" href="#E1:103_1_1"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="ref" href="complex2.html#T98" target="_self" title="COMPLEX2:th.98">Th98</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:103_1_1_1_1_2"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_1_1_1_2"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="ref" href="complex2.html#T100" target="_self" title="COMPLEX2:th.100">Th100</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:103_1_1_1_1_3"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_1_1_1_3"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="complex2.html#T99" target="_self" title="COMPLEX2:th.99">Th99</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</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><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="103_1_2"><span class="kw">suppose </span></a><a NAME="E1:103_1_2"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<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="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><a NAME="E2:103_1_2"/><span class="lab"><font color="Green" title="E79">A8</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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T84" target="_self" title="COMPLEX2:th.84">Th84</a></span>;<br/><span class="kw">thus </span><a NAME="E3:103_1_2"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="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="103_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_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="E1:103_1_2_1_1"/>
(  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<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>  or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <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><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:103_1_2_1_1_1"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<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"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_2_1_1_1"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="txt" href="#E1:103_1_2"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E2:103_1_2"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="ref" href="complex2.html#T98" target="_self" title="COMPLEX2:th.98">Th98</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:103_1_2_1_1_2"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="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"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_2_1_1_2"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="complex2.html#T100" target="_self" title="COMPLEX2:th.100">Th100</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:103_1_2_1_1_3"/>
 <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:103_1_2_1_1_3"/>
( (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) or (  <a href="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <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="complex2.html#K4" title="COMPLEX2:func.4">angle</a> <font color="Maroon" title="c3">c</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="complex2.html#T99" target="_self" title="COMPLEX2:th.99">Th99</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</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><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>
