<?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> 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><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E18</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"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E19</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/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:32"/>
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"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="txt" href="fdiff_6.html#E1:32"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green">E20</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"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:32"/><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> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="xboole_0.html#K3">/\</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>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green">E21</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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:32"/><i><font color="Green">E22</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/><font color="Maroon">c<sub>2</sub></font> <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">0 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:32"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green">E23</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> &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"><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> 0 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E6:32"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E9:32"/><i><font color="Green">E24</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"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E8:32"><i><font color="Green">E23</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:32_1"/>
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b>
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T22">FDIFF_1:22</a>, <a class="txt" href="fdiff_6.html#E2:32_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:32_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T3">TAYLOR_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:32"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:32"><i><font color="Green">E18</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E11:32"/>
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"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></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">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:32_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E8:32"><i><font color="Green">E23</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:32_2"/><i><font color="Green">E28</font></i>: 
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T22">FDIFF_1:22</a>, <a class="txt" href="fdiff_6.html#E2:32_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#T22">FDIFF_1:22</a>, <a class="txt" href="fdiff_6.html#E2:32_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_6.html#E3:32_2"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, <a class="txt" href="fdiff_6.html#E1:32_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_6.html#E8:32"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E6:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fdiff_6.html#E7:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fdiff_6.html#E1:32_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>
<i><font color="Green">E31</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, <a class="txt" href="fdiff_6.html#E4:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fdiff_6.html#E1:32_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:32_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_6.html#E2:32"><i><font color="Green">E19</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="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></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>3</sub></font> = 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="fdiff_6.html#E10:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, <a class="txt" href="fdiff_6.html#E1:32_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <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#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E4:32_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="taylor_1.html#T3">TAYLOR_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E6:32_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="fdiff_6.html#E5:32_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E8:32_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></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>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:32"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></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"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font></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">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_6.html#E9:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_6.html#E1:32"><i><font color="Green">E18</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>


</div>
