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

<a NAME="E1:49"/>
<a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E25"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="relat_1.html#T104">RELAT_1:104</a></i>;<br/>
<a NAME="E2:49"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<b>thus </b><a NAME="E4:49"/>
<a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K22">cos</a> 
 <b>and </b><br/><a NAME="E4:49_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E5:49_1"/><i><font color="Green">E74</font></i>: 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="sin_cos6.html#E3:49_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E7:49_1"/><i><font color="Green">E75</font></i>: 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E8:49_1"/><i><font color="Green">E76</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:49"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sin_cos6.html#E3:49"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sin_cos6.html#E1:49_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="comptrig.html#T50">COMPTRIG:50</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>;<br/>
<a NAME="E9:49_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xcmplx_0.html#K7">/</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="real_1.html#K4">*</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xcmplx_0.html#K7">/</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E2:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>)
 <b>by </b><i><a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sin_cos6.html#E7:49_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="sin_cos6.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) &amp; <a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:49_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E4:49_1_1"/><b>then </b>
( 0 <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K5">-</a> <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:49_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:49_1_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E9:49_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/><a NAME="E7:49_1_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="int_1.html#T21">INT_1:21</a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:49_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>)
 <b>by </b><i><a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sin_cos6.html#E7:49_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="sin_cos6.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E3:49_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:49_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E4:49_1_2"/><b>then </b>
( 0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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>) <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E5:49_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E9:49_1"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E6:49_1_2"/><b>then </b>
( 0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<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> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/><b>hence </b><a NAME="E7:49_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:49_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E8:49_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sin_cos6.html#E10:49_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:49_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:49"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E7:49"/>
(  <a href="real_1.html#K1">-</a> 1 <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> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E5:49"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E8:49"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T50">COMPTRIG:50</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:49"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E11:49"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:49"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E14:49"/><i><font color="Green">E75</font></i>: 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sin_cos6.html#E11:49"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E15:49"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sin_cos6.html#E12:49"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E16:49"/><b>then </b>
( ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K33">PI</a>  ) or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="sin_cos.html#K33">PI</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E17:49"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E5:49"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sin_cos6.html#E14:49"><i><font color="Green">E75</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="sin_cos.html#T33">SIN_COS:33</a>, <a class="ref" href="sin_cos.html#T81">SIN_COS:81</a></i>;<br/>
<b>hence </b><a NAME="E18:49"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos6.html#E14:49"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sin_cos6.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div>
