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

<b>let </b><font color="Maroon" title="c1">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:48"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_4.html#E1:48"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/><a NAME="E2:48_1"/><i><font color="Green" title="E16">A5</font></i>: 
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>;<br/><a NAME="E3:48_1"/>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fdiff_4.html#E2:48"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="fdiff_4.html#E1:48_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:48_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fdiff_4.html#E2:48_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="taylor_1.html#T22" title="TAYLOR_1:th.22">TAYLOR_1:22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:48"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="txt" href="fdiff_4.html#E3:48"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E6:48"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:48_2"/><i><font color="Green" title="E17">A8</font></i>: 
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>;<br/>
<a NAME="E3:48_2"/><i><font color="Green" title="E18">A9</font></i>: 
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_4.html#E2:48"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="fdiff_4.html#E1:48_2"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_4.html#E1:48"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fdiff_4.html#E5:48"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="fdiff_4.html#E1:48_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_4.html#E2:48_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="fdiff_4.html#E3:48_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="taylor_1.html#T22" title="TAYLOR_1:th.22">TAYLOR_1:22</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:48_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:48"/>
( 2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_4.html#E1:48"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fdiff_4.html#E5:48"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></i>;<br/>


</div>
