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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E29</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<a NAME="E4:87"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T38" target="_self">Th38</a>, , <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>
<a NAME="E5:87"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E6:87"/><i><font color="Green">E32</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E7:87"/><i><font color="Green">E33</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_1"/>
not  <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<a NAME="E2:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T38" target="_self">Th38</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:87_1"/>
contradiction
 <b>by </b><i><a class="ref" href="complex2.html#T38" target="_self">Th38</a>, , <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:87"/><b>then </b>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E9:87"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E10:87"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, , <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E11:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E12:87"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></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">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K35">PI</a> 
<b>by </b><i><a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T37" target="_self">Th37</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T41" target="_self">Th41</a>, , <a class="ref" href="complex2.html#T34" target="_self">Th34</a></i>
<br/>.= 
5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 

;<br/><br/>

<a NAME="E14:87"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="hidden.html#R1">=</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 
;<br/>
<a NAME="E15:87"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a>  <a href="hidden.html#R1">=</a> 5 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 
;<br/>
<a NAME="E16:87"/><i><font color="Green">E52</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:87_3"/><i><font color="Green">E68</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/><a NAME="E2:87_3"/><i><font color="Green">E69</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:87_3"/>
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T47" target="_self">Th47</a>, <a class="ref" href="complex2.html#T49" target="_self">Th49</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:87_3"/>
contradiction
 <b>by </b><i><a class="ref" href="complex2.html#T43" target="_self">Th43</a>, <a class="ref" href="complex2.html#T44" target="_self">Th44</a>, <a class="ref" href="complex2.html#T51" target="_self">Th51</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E18:87"/><i><font color="Green">E70</font></i>: 
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<a NAME="E19:87"/>
 <a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="complex2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E20:87"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K4" target="_self">angle</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T47" target="_self">Th47</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E21:87"/>
contradiction
 <b>by </b><i><a class="ref" href="complex2.html#T43" target="_self">Th43</a>, <a class="ref" href="complex2.html#T45" target="_self">Th45</a>, <a class="ref" href="complex2.html#T49" target="_self">Th49</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


</div>
