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

<b>set </b><font color="Maroon">f</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<a NAME="E1:85"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E2:85"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:85"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:85"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 ;<br/>

<i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma42</font></i></a></i>
;<br/>
<i><font color="Green">E72</font></i>: <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma42</font></i></a></i>
;<br/>
<a NAME="E9:85"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="toprealb.html#E12"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/>
<a NAME="E10:85"/><i><font color="Green">E74</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="toprealb.html#E12"><i><font color="Green">Lemma59</font></i></a>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/>
<a NAME="E11:85"/><i><font color="Green">E75</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E12:85"/><i><font color="Green">E77</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E13:85"/><i><font color="Green">E115</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E14:85"/><i><font color="Green">E116</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E15:85"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E16:85"/><i><font color="Green">E120</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 ) or ( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) or ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 ) or ( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1"><b>suppose </b></a><a NAME="E1:85_3_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:85_3_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:85_3_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="comptrig.html#T28">COMPTRIG:28</a></i>;<br/><a NAME="E5:85_3_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/><a NAME="E6:85_3_1"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T24">COMPTRIG:24</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3_1_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 ) or ( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="real_1.html#K6">/</a> 4 ) or ( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1_1"><b>suppose </b></a><a NAME="E1:85_3_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:85_3_1_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E4:85_3_1_1_1"/><i><font color="Green">E129</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></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,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E6:85_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="toprealb.html#E9:85"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E7:85_3_1_1_1"/><b>then </b>
<font color="Maroon">x</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></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#K35">PI</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E8:85_3_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1_2"><b>suppose </b></a><a NAME="E1:85_3_1_1_2"/>
( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</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="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></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/><b>hence </b><a NAME="E4:85_3_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:85"><i><font color="Green">E61</font></i></a>, <a class="txt" href="toprealb.html#E2:85"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="comptrig.html#T29">COMPTRIG:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1_3"><b>suppose </b></a><a NAME="E1:85_3_1_1_3"/>
( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:85_3_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_1_1_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E4:85_3_1_1_3"/>
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T14">TOPREALA:14</a></i>;<br/><b>hence </b><a NAME="E5:85_3_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, , <a class="txt" href="toprealb.html#E7:85"><i><font color="Green">E71</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="comptrig.html#T25">COMPTRIG:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2"><b>suppose </b></a><a NAME="E1:85_3_2"/>
( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div class="add"><a NAME="E2:85_3_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E4:85_3_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.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="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T18">TOPREALA:18</a></i>;<br/><a NAME="E5:85_3_2"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T29">COMPTRIG:29</a></i>;<br/><a NAME="E6:85_3_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/><a NAME="E7:85_3_2"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T24">COMPTRIG:24</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3_2_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 ) or ( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) or ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2_1_1"><b>suppose </b></a><a NAME="E1:85_3_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:85_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:85_3_2_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:85"><i><font color="Green">E61</font></i></a>, <a class="txt" href="toprealb.html#E2:85"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="comptrig.html#T28">COMPTRIG:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2_1_2"><b>suppose </b></a><a NAME="E1:85_3_2_1_2"/>
( 1 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div class="add"><a NAME="E2:85_3_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_2_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span>;<br/><a NAME="E4:85_3_2_1_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/><a NAME="E5:85_3_2_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:85"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sin_cos6.html#T2">SIN_COS6:2</a></i>;<br/><a NAME="E6:85_3_2_1_2"/><i><font color="Green">E137</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E8:85_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="toprealb.html#E9:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="toprealb.html#E10:85"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E9:85_3_2_1_2"/><b>then </b>
<font color="Maroon">x</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></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#K35">PI</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E10:85_3_2_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2_1_3"><b>suppose </b></a><a NAME="E1:85_3_2_1_3"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:85_3_2_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_2_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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/><b>hence </b><a NAME="E4:85_3_2_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, , <a class="txt" href="toprealb.html#E7:85"><i><font color="Green">E71</font></i></a>, <a class="ref" href="comptrig.html#T25">COMPTRIG:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_3"><b>suppose </b></a><a NAME="E1:85_3_3"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_3"/><b>then </b><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E4:85_3_3"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T18">TOPREALA:18</a></i>;<br/><a NAME="E5:85_3_3"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T25">COMPTRIG:25</a></i>;<br/><a NAME="E6:85_3_3"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/><a NAME="E7:85_3_3"/><b>then </b><i><font color="Green">E181</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T30">COMPTRIG:30</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3_3_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) or ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 ) or ( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_3_1_1"><b>suppose </b></a><a NAME="E1:85_3_3_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div class="add"><a NAME="E2:85_3_3_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:85_3_3_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:85_3_3_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, , , <a class="ref" href="comptrig.html#T24">COMPTRIG:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_3_1_2"><b>suppose </b></a><a NAME="E1:85_3_3_1_2"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_3_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_3_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>;<br/><a NAME="E4:85_3_3_1_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/><a NAME="E5:85_3_3_1_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:85"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sin_cos6.html#T2">SIN_COS6:2</a></i>;<br/><a NAME="E6:85_3_3_1_2"/><i><font color="Green">E192</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E8:85_3_3_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="toprealb.html#E9:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="toprealb.html#E10:85"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E9:85_3_3_1_2"/><b>then </b>
<font color="Maroon">x</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></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#K35">PI</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E10:85_3_3_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_3_1_3"><b>suppose </b></a><a NAME="E1:85_3_3_1_3"/>
( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:85_3_3_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</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"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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/><b>hence </b><a NAME="E4:85_3_3_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:85"><i><font color="Green">E61</font></i></a>, <a class="txt" href="toprealb.html#E2:85"><i><font color="Green">E66</font></i></a>, <a class="txt" href="toprealb.html#E7:85"><i><font color="Green">E71</font></i></a>, <a class="ref" href="comptrig.html#T31">COMPTRIG:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4"><b>suppose </b></a><a NAME="E1:85_3_4"/>
( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:85_3_4"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_4"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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/><a NAME="E4:85_3_4"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="jgraph_3.html#T9">JGRAPH_3:9</a></i>;<br/><a NAME="E5:85_3_4"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="sin_cos.html#K21">sin</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T25">COMPTRIG:25</a></i>;<br/><a NAME="E6:85_3_4"/><i><font color="Green">E195</font></i>: 
 <a href="sin_cos.html#K24">cos</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="comptrig.html#T31">COMPTRIG:31</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3_4_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) or ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 ) or ( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_1_1"><b>suppose </b></a><a NAME="E1:85_3_4_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div class="add"><a NAME="E2:85_3_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:85_3_4_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:85_3_4_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, , , <a class="ref" href="comptrig.html#T24">COMPTRIG:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_1_2"><b>suppose </b></a><a NAME="E1:85_3_4_1_2"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:85_3_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_4_1_2"/><b>then </b><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E4:85_3_4_1_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/><b>hence </b><a NAME="E5:85_3_4_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:85"><i><font color="Green">E61</font></i></a>, <a class="txt" href="toprealb.html#E2:85"><i><font color="Green">E66</font></i></a>, <a class="txt" href="toprealb.html#E7:85"><i><font color="Green">E71</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="comptrig.html#T30">COMPTRIG:30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_4_1_3"><b>suppose </b></a><a NAME="E1:85_3_4_1_3"/>
( 3 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:85_3_4_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:85_3_4_1_3"/><b>then </b><i><font color="Green">E197</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</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"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></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/><b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E4:85_3_4_1_3"/><i><font color="Green">E198</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K35">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K35">PI</a> </span>)</span>,<span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E6:85_3_4_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:85"><i><font color="Green">E69</font></i></a>, <a class="txt" href="toprealb.html#E8:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="toprealb.html#E9:85"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E7:85_3_4_1_3"/><b>then </b>
<font color="Maroon">x</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#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></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#K35">PI</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E8:85_3_4_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
