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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</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#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E13</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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:14"/><i><font color="Green">E14</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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_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="E1:14_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:14_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:14"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T49">PREPOWER:49</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14"/><i><font color="Green">E15</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>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</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="14_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="E1:14_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:14_2"/><i><font color="Green">E17</font></i>: 
 <a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</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="taylor_1.html#T2">TAYLOR_1:2</a></i>;<br/>
<a NAME="E3:14_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</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> <font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#D1">TAYLOR_1:def 1</a></i>;<br/>
<a NAME="E4:14_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E3:14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fdiff_6.html#E1:14_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:14_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</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#T20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E2:14_2"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E6:14"/>
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>2</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#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><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="E1:14_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:14_3"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:14"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E3:14_3"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E4:14_3"/><i><font color="Green">E20</font></i>: 
(  <a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="taylor_1.html#T2">TAYLOR_1:2</a></i>;<br/>
<a NAME="E5:14_3"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</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> <font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#D1">TAYLOR_1:def 1</a></i>;<br/>
<a NAME="E6:14_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E3:14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fdiff_6.html#E1:14_3"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>then </b> <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#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E5:14_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="taylor_1.html#T20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E4:14_3"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T53">PREPOWER:53</a>, <a class="txt" href="fdiff_6.html#E2:14_3"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T48">XCMPLX_1:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K3">|^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E3:14_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>
<b>hence </b><a NAME="E8:14_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_6.html#E5:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fdiff_6.html#E1:14_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</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>2</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#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="fdiff_6.html#E4:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fdiff_6.html#E1:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>


</div>
