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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> ) )</span><br/>



<a NAME="E1:30"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c2">p</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font>;<br/>
<a NAME="E2:30"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self" title="TAYLOR_1:def.4">Def4</a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green" title="E26">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">x</font> holds <br/>( <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">x</font> implies ( <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> )</span><br/>

<a NAME="E2:30_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:30_1"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<a NAME="E4:30_1"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></i>;<br/>
<a NAME="E6:30_1"/><i><font color="Green" title="E32">A9</font></i>: 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E5:30_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><br/>

<i><font color="Green" title="E33">A10</font></i>:  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> = 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:30_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:30_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="taylor_1.html#E16"><i><font color="Green" title="E13">Lm4</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="taylor_1.html#T15" target="_self" title="TAYLOR_1:th.15">Th15</a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>
;<br/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>,<font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> ,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:30_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:30_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a></i>
;<br/>
<b>hence </b> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:30_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="taylor_1.html#E6:30_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="taylor_1.html#E8:30_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:30"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/><i><font color="Green" title="E27">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:30_2"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<b>assume </b><a NAME="E1:30_2_2"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span><br/>

<a NAME="E2:30_2_2"/><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<a NAME="E3:30_2_2"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:30_2_2"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="partfun2.html#T6" title="PARTFUN2:th.6">PARTFUN2:6</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:30_2_2"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:30_2_2"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:30_2_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="taylor_1.html#E16"><i><font color="Green" title="E13">Lm4</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:30_2_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="taylor_1.html#T15" target="_self" title="TAYLOR_1:th.15">Th15</a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:30_2_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30_2_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="taylor_1.html#D4" target="_self" title="TAYLOR_1:def.4">Def4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_2"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="taylor_1.html#E1:30_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="taylor_1.html#E3:30"><i><font color="Green" title="E26">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
