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

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

<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E22</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E24</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E4:32"/>
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E26"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E6:32"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  <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="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">r</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="txt" href="sin_cos6.html#E25"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:32"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos6.html#E24"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="sin_cos6.html#E25"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="sin_cos6.html#E27"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="sin_cos6.html#E21"><i><font color="Green">Lemma56</font></i></a></i>;<br/>


</div>
