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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:29"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>2</sub></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/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:29"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <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="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_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="E1:29_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:29_1"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E3:29_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:29_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T18" target="_self">Th18</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:29_1"/><i><font color="Green">E31</font></i>: 
 <a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:29_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>;<br/>
<a NAME="E7:29_1"/><i><font color="Green">E33</font></i>: 
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:29_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>;<br/>

<i><font color="Green">E34</font></i>:  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</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#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</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#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </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="taylor_1.html#E4:29_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taylor_1.html#E16"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:29_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="taylor_1.html#T15" target="_self">Th15</a></i>
;<br/>
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29"><i><font color="Green">E24</font></i></a>, <a class="txt" href="taylor_1.html#E5:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self">Th18</a></i>
;<br/>
<b>hence </b> <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="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</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="taylor_1.html#E6:29_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="taylor_1.html#E7:29_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="taylor_1.html#E9:29_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.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="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</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/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.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="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.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="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#T33">POWER:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:29"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_2"/><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27">exp_R</a> 
 
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="limfunc1.html#K4">right_open_halfline</a> 0
<b>by </b><i><a class="ref" href="taylor_1.html#T18" target="_self">Th18</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:29_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0 holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></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"><a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:29_2_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 ;<br/>

<a NAME="E2:29_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self">Th18</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:29_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:29_2_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E6:29_2_2"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E5:29_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="finseq_4.html#K4">/.</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#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="partfun2.html#T6">PARTFUN2:6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="finseq_4.html#K4">/.</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#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:29_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</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#K13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </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="taylor_1.html#E2:29_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:29_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="taylor_1.html#E16"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E6:29_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="taylor_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K12">#R</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:29_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:29_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="taylor_1.html#D3" target="_self">Def3</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_2"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K2" target="_self">log_</a> <a href="power.html#K8">number_e</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:29"><i><font color="Green">E25</font></i></a>, <a class="txt" href="taylor_1.html#E1:29_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:29"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies (  <a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:29"><i><font color="Green">E24</font></i></a>, <a class="txt" href="taylor_1.html#E3:29"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
