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

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>consider </b><font color="Maroon" title="c2">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E2:10"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">a</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E11">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 <b>by </b><i><a class="ref" href="complex2.html#T2" target="_self" title="COMPLEX2:th.2">Th2</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="complex2.html#E2:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="complex2.html#T9" target="_self" title="COMPLEX2:th.9">Th9</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green" title="E13">A4</font></i>: 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="complex2.html#E2:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="complex2.html#T10" target="_self" title="COMPLEX2:th.10">Th10</a></i>;<br/>
<b>assume </b><a NAME="E6:10"/>
(  <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</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>  &amp;  <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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>  )
 ;<br/>

<a NAME="E7:10"/><b>then </b>
( ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  ) &amp; ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  ) )
 
<b>by </b><i><a class="txt" href="complex2.html#E3:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="complex2.html#E4:10"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="complex2.html#E5:10"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="comptrig.html#T33" title="COMPTRIG:th.33">COMPTRIG:33</a>, <a class="ref" href="comptrig.html#T34" title="COMPTRIG:th.34">COMPTRIG:34</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
contradiction
 <b>by </b><i><a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a></i>;<br/>


</div>
