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<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="E1:84"/><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="E2:84"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/>

<b>assume </b><a NAME="E4:84"/><i><font color="Green">E30</font></i>: 
( not  <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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not  <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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not  <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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E5:84"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E6:84"/><i><font color="Green">E32</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T36" target="_self">Th36</a>, <a class="ref" href="complex2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E7:84"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="complex2.html#T34" target="_self">Th34</a>, <a class="ref" href="complex2.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E8:84"/><i><font color="Green">E48</font></i>: 
(  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> &amp;  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font> )
 
;<br/>
<a NAME="E9:84"/>
(  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font> &amp;  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font> )
 
;<br/>
<a NAME="E10:84"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K11">-</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex2.html#T38" target="_self">Th38</a>, <a class="ref" href="complex2.html#T39" target="_self">Th39</a>, <a class="ref" href="complex2.html#T40" target="_self">Th40</a>, <a class="ref" href="complex2.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E11:84"/>
(  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  iff  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K11">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="sin_cos.html#K35">PI</a>  )
 
<b>by </b><i><a class="ref" href="complex2.html#T39" target="_self">Th39</a>, </i>;<br/>
<b>hence </b><a NAME="E12:84"/>
contradiction
 <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#T41" target="_self">Th41</a>, <a class="ref" href="complex2.html#T26" target="_self">Th26</a></i>;<br/>


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