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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K1">#Z</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) ) )
 ;<br/>

<a NAME="E2:21"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E3:21"/><b>then </b><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E4:21"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="txt" href="fdiff_5.html#E3:21"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E5:21"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:21"/><i><font color="Green">E10</font></i>: 
( <a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fdiff_5.html#E3:21"><i><font color="Green">E8</font></i></a>, <a class="ref" href="fdiff_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="ref" href="taylor_1.html#T2">TAYLOR_1:2</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="txt" href="fdiff_5.html#E3:21"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E9:21"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="ref" href="taylor_1.html#T2">TAYLOR_1:2</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_5.html#E8:21"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:21_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E3:21_2"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E6:21"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_5.html#E8:21"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_5.html#E8:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_5.html#E9:21"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="ref" href="taylor_1.html#D1">TAYLOR_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E6:21"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fdiff_5.html#E3:21"><i><font color="Green">E8</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fdiff_5.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T1">TAYLOR_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K7">#Z</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T103">XCMPLX_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E3:21_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E3:21_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E3:21"><i><font color="Green">E8</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a>, <a class="txt" href="fdiff_5.html#E5:21"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fdiff_5.html#E1:21_2"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:21_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:21"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_5.html#E1:21"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fdiff_5.html#E6:21"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fdiff_5.html#E8:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a></i>;<br/>


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