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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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>let </b><font color="Maroon">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><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"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K1">#Z</a> 2
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E38</font></i>: 
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/>( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<i><font color="Green">E39</font></i>: <font color="Maroon">f1</font> <a href="seq_1.html#K6">+</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> <font color="Maroon">f2</font></span>)</span> = 
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T40">RFUNCT_1:40</a></i>
;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E40</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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E41</font></i>: 
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/>( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <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> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E42</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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E43</font></i>: 
( <a href="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <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> <font color="Maroon">f2</font></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></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">0 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span></span>)</span> ) )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E9:19"/><b>then </b><i><font color="Green">E44</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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></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="E10:19"/>
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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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="E1:19_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f2</font></span>)</span></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>)</span>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</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">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , </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"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:19_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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:19"/>
(  <a href="seq_1.html#K16">-</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></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="taylor_1.html#K3">ln</a>  <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a>, </i>;<br/>


</div>
