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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K9" title="PREPOWER:func.9">#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/>



<span class="kw">set </span><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="E1:30"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">p</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> holds <br/>( <a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K25" title="SIN_COS:func.25">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#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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font> implies ( <a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K25" title="SIN_COS:func.25">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#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/>

<span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K25" title="SIN_COS:func.25">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#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="E2:30_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a></span>;<br/>
<a NAME="E3:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T15" title="FDIFF_1:th.15">FDIFF_1:15</a></span>;<br/>
<a NAME="E4:30_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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/>
<a NAME="E5:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T230" title="XXREAL_1:th.230">XXREAL_1:230</a></span>;<br/>
<a NAME="E6:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<span class="lab"><font color="Green" title="E31">A8</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="fdiff_1.html#T15" title="FDIFF_1:th.15">FDIFF_1:15</a></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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="taylor_1.html#T18" target="_self" title="TAYLOR_1:th.18">Th18</a></span>
;<br/>
<a NAME="E8:30_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T65" title="SIN_COS:th.65">SIN_COS:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:30_1"/>
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K25" title="SIN_COS:func.25">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#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 href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="taylor_1.html#E16"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a>, <a class="ref" href="power.html#T55" title="POWER:th.55">POWER:55</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="taylor_1.html#T15" target="_self" title="TAYLOR_1:th.15">Th15</a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>
;<br/>
<span class="kw">hence </span> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></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> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></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"><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="power.html#T28" title="POWER:th.28">POWER:28</a></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"><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></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"><font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:30"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E26">A10</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>
<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>
<span class="kw">by</span> <span class="lab"><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></span>
;<br/>
<a NAME="E5:30"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 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#K25" title="SIN_COS:func.25">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_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K25" title="SIN_COS:func.25">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/>

<span class="kw">assume </span><a NAME="E1:30_3"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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_3"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></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></span>;<br/>
<a NAME="E3:30_3"/>
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T230" title="XXREAL_1:th.230">XXREAL_1:230</a></span>;<br/>
<a NAME="E4:30_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b1">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/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="partfun2.html#T3" title="PARTFUN2:th.3">PARTFUN2:3</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></span>
<br/>.= 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="taylor_1.html#E16"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a>, <a class="ref" href="power.html#T55" title="POWER:th.55">POWER:55</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="taylor_1.html#T15" target="_self" title="TAYLOR_1:th.15">Th15</a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="taylor_1.html#D4" target="_self" title="TAYLOR_1:def.4">Def4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:30"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#D4" target="_self" title="TAYLOR_1:def.4">Def4</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <a href="sin_cos.html#K25" title="SIN_COS:func.25">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:30"/>
( <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#K9" title="PREPOWER:func.9">#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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
