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

<b>let </b><font color="Maroon" title="c1">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E16">A2</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 <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<a NAME="E3:27"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:27"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D7" title="SEQ_1:def.7">SEQ_1:def 7</a>, <a class="txt" href="fdiff_6.html#E1:27"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E6:27"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E8:27"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green" title="E21">A7</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 <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:27"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E10:27"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E8:27"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green" title="E23">A9</font></i>: 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T16" title="TAYLOR_1:th.16">TAYLOR_1:16</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E10:27"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="fdiff_6.html#E6:27"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green" title="E25">A11</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 <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><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> ,<font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E11:27"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="fdiff_6.html#E10:27"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="fdiff_6.html#E6:27"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:27_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a>, <a class="txt" href="fdiff_6.html#E10:27"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:27_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E8:27"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="fdiff_6.html#E9:27"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E14:27"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:27_2"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <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> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a>, <a class="txt" href="fdiff_6.html#E12:27"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/>
<a NAME="E3:27_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_2"><i><font color="Green" title="E26">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:27_2"/>
<a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T20" title="TAYLOR_1:th.20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E2:27_2"><i><font color="Green" title="E27">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b><i><font color="Green" title="E26">A15</font></i>: 
<a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E4:27"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a>, <a class="txt" href="fdiff_6.html#E3:27"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<a NAME="E17:27"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E29">A18</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a>, <a class="txt" href="fdiff_6.html#E6:27"><i><font color="Green" title="E19">A5</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a></i>
;<br/>
<a NAME="E3:27_3"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <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> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a>, <a class="txt" href="fdiff_6.html#E12:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/>
<a NAME="E4:27_3"/><i><font color="Green" title="E31">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="fdiff_6.html#E2:27"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <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> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a>, <a class="txt" href="fdiff_6.html#E15:27"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="fdiff_6.html#E3:27"><i><font color="Green" title="E17">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T20" title="TAYLOR_1:th.20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E3:27_3"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="fdiff_6.html#E4:27_3"><i><font color="Green" title="E31">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E12:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E13:27"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="fdiff_6.html#E1:27_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="fdiff_6.html#E2:27_3"><i><font color="Green" title="E29">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E6:27_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:27"/>
(  <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_6.html#E16:27"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>


</div>
