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

<b>let </b><font color="Maroon">Z</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:57"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/>
<a NAME="E3:57"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E4:57"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="taylor_1.html#K3">ln</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E5:57"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <a href="taylor_1.html#K3">ln</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:57"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T18">TAYLOR_1:18</a>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/>
<a NAME="E7:57"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, </i>;<br/>
<a NAME="E3:57_1"/>
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <a href="taylor_1.html#K3">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T68">SIN_COS:68</a></i>;<br/>
<b>hence </b><a NAME="E4:57_1"/>
<a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:57"/><b>then </b><i><font color="Green">E43</font></i>: 
<a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, </i>;<br/>
<a NAME="E9:57"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <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">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span>,<font color="Olive">x</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, </i>;<br/>
<a NAME="E10:57"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:57_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#T18">TAYLOR_1:18</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E3:57_2"/><i><font color="Green">E49</font></i>: 
 <a href="taylor_1.html#K3">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, , </i>;<br/>
<a NAME="E4:57_2"/><i><font color="Green">E50</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <a href="taylor_1.html#K3">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T68">SIN_COS:68</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="taylor_1.html#K3">ln</a> ,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="fdiff_6.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="taylor_1.html#K3">ln</a> ,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T68">SIN_COS:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="taylor_1.html#K3">ln</a> ,<font color="Maroon">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="taylor_1.html#K3">ln</a> ,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D2">TAYLOR_1:def 2</a>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="taylor_1.html#T18">TAYLOR_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E6:57_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:57"/>
(  <a href="seq_1.html#K16">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K1">*</a> <a href="taylor_1.html#K3">ln</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>;<br/>


</div>
