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

<a NAME="E1:124_1_1"/><i><font color="Green">E104</font></i>: 
1 <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"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:124_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:124_1_1_1"/>
1 <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"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E131"><i><font color="Green">Lemma103</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:124_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <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>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T13">RCOMP_1:13</a>, <a class="txt" href="sin_cos.html#E131"><i><font color="Green">Lemma103</font></i></a></i>;<br/>
<a NAME="E3:124_1_1"/><b>then </b>
1 <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"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <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="txt" href="sin_cos.html#E1:124_1_1"><i><font color="Green">E104</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:124_1_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</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> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="sin_cos.html#T76" target="_self">Th76</a>, <a class="ref" href="fcont_2.html#T16">FCONT_2:16</a></i>;<br/>
<a NAME="E6:124_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:124_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:124_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:124_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sin_cos.html#E131"><i><font color="Green">Lemma103</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> 4;<br/>

<b>thus </b><a NAME="E8:124_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="rfunct_1.html#K2">/</a> <a href="sin_cos.html#K22" target="_self">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <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#E5:124_1_1"><i><font color="Green">E105</font></i></a></i>;<br/>

<a NAME="E9:124_1_1"/>
( 0 <a href="nat_1.html#K2">*</a> 4 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> 4 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> 4 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K2">*</a> 4 )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E7:124_1_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:124_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,4</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/>


</div>
