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

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

<b>assume </b><a NAME="E1:132"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">th</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>

<a NAME="E2:132"/>
 <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="sin_cos.html#E22:132"><i><font color="Green">E95</font></i></a>, <a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E3:132"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">th</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E4:132"/><i><font color="Green">E56</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">th</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:132"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">th</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:132"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</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#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</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#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T18">RCOMP_1:18</a>, , </i>;<br/>
<a NAME="E7:132"/>
0 <a href="hidden.html#R2">in</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#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> 0</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="E8:132"/><b>then </b>
 ex <font color="Olive">th2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">th2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">th</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">th2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , , , <a class="ref" href="fcont_2.html#T16">FCONT_2:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">th2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:132"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">th2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">th</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">th</font> &amp; <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th2</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E11:132"/><i><font color="Green">E62</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">th2</font> &amp; <font color="Maroon">th2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">th</font> &amp; <font color="Maroon">th</font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E12:132"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">th2</font> &amp; <font color="Maroon">th2</font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:132"/><b>then </b><i><font color="Green">E91</font></i>: 
( 0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E14:132"/>
 <a href="sin_cos.html#K35" target="_self">PI</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,4</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:132"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35" target="_self">PI</a>  &amp;  <a href="sin_cos.html#K35" target="_self">PI</a>  <a href="xxreal_0.html#R1">&lt;</a> 4 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E16:132"/><b>then </b>
<a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E17:132"/><b>then </b><i><font color="Green">E92</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
0 = 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E19:132"/><b>then </b>
( <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E20:132"/><b>then </b><i><font color="Green">E93</font></i>: 
( <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> or <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E21:132"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E22:132"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E23:132"/><b>then </b><i><font color="Green">E96</font></i>: 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="sin_cos.html#E24:132"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E24:132"/><i><font color="Green">E97</font></i>: 
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E25:132"/>
( 2 <a href="real_1.html#K4">*</a> <font color="Maroon">th2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,4</span><a href="rcomp_1.html#K2">.[</a></span> &amp; <a href="sin_cos.html#K32" target="_self">tan</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> <font color="Maroon">th2</font></span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2"><b>proof </b></a><div class="add">

<a NAME="E1:132_2"/>
( 4 <a href="nat_1.html#K2">*</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="nat_1.html#K2">*</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E2:132_2"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">th2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,4</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="E3:132_2"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">th2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:132_2"/>
<a href="sin_cos.html#K32" target="_self">tan</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> <font color="Maroon">th2</font></span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="sin_cos.html#E26:132"><i><font color="Green">E98</font></i></a>, <a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:132"/><b>then </b><i><font color="Green">E98</font></i>: 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">th2</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35" target="_self">PI</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E27:132"/>
<font color="Maroon">th2</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35" target="_self">PI</a>  <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E28:132"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
