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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:10"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E12</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="ref" href="complex2.html#T2" target="_self">Th2</a>, <a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E13</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="complex2.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="complex2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green">E14</font></i>: 
 <a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="complex2.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="complex2.html#T10" target="_self">Th10</a></i>;<br/>
<b>assume </b><a NAME="E6:10"/>
(  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E7:10"/><b>then </b>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K33">PI</a>  ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a>  ) )
 
<b>by </b><i><a class="txt" href="complex2.html#E3:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="complex2.html#E4:10"><i><font color="Green">E13</font></i></a>, <a class="txt" href="complex2.html#E5:10"><i><font color="Green">E14</font></i></a>, <a class="ref" href="comptrig.html#T33">COMPTRIG:33</a>, <a class="ref" href="comptrig.html#T34">COMPTRIG:34</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
contradiction
 <b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a></i>;<br/>


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