<?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>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E20">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="Olive" title="b1">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> 
 ;<br/>

<a NAME="E3:34"/><i><font color="Green" title="E21">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 <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="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon" title="c2">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> 
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E2:34"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b><i><font color="Green" title="E22">A4</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="c2">x</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a></i>;<br/>
<a NAME="E4:34_1"/>
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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="c2">x</font>
 <b>by </b><i><a class="txt" href="fdiff_7.html#E3:34_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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_7.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E5:34"/>
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="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:34_2"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c2">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> 
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E2:34"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<a NAME="E3:34_2"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c2">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="txt" href="fdiff_7.html#E34"><i><font color="Green" title="E18">Lm3</font></i></a></i>;<br/>
<a NAME="E4:34_2"/><i><font color="Green" title="E26">A9</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="c2">x</font>
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E2:34_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a></i>;<br/>
<a NAME="E5:34_2"/>
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>;<br/>
<b>then </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#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> </span>)</span>,<font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E4:34_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T69" title="SIN_COS:th.69">SIN_COS:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E3:34_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E7:34_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fdiff_7.html#E4:34"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="fdiff_7.html#E1:34_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <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="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="fdiff_7.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="fdiff_7.html#E3:34"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>


</div>
