<?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">b</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><a NAME="E1:24"/><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="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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span></span>)</span> &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/>( <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="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) )
 ;<br/>

<a NAME="E2:24"/>
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> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E3:24"/><b>then </b>
<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"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E5:24"/><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> <font color="Maroon">f1</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E39</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">1 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">f2</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">1 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:24_1"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:24_1"/>
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:24"/><b>then </b><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">1 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span>
 
;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f1</font> <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"><font color="Maroon">f1</font> <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> 1 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, </i>;<br/>
<a NAME="E10:24"/><i><font color="Green">E43</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">f2</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">1 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">f2</font> <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"><font color="Maroon">f2</font> <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> 1 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, </i>;<br/>
<a NAME="E12:24"/>
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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E13:24"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , <a class="ref" href="fdiff_2.html#T21">FDIFF_2:21</a></i>;<br/>
<a NAME="E14:24"/><i><font color="Green">E46</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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:24_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">f1</font> <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="E3:24_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f2</font> <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="E4:24_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>,<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="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f2</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T14">FDIFF_2:14</a>, , </i>
<br/>.= 
<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">f1</font> <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> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f2</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <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> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f2</font> <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> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </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"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f2</font> <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> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></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">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:24_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:24"/><i><font color="Green">E57</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/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</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
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><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="E1:24_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:24_3"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E3:24_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E5:24_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T141">XREAL_1:141</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:24"/><i><font color="Green">E61</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/><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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <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="24_4"><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="E1:24_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:24_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font> <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="E3:24_4"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:24_4"/>
<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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T20">TAYLOR_1:20</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:24"/><b>then </b><i><font color="Green">E64</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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <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="E18:24"/>
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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_5"><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="E1:24_5"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:24_5"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font> <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="E3:24_5"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:24_5"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, , </i>;<br/>
<a NAME="E5:24_5"/><i><font color="Green">E70</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , </i>;<br/>
 <a href="fdiff_1.html#K1">diff</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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>,<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">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</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="taylor_1.html#T20">TAYLOR_1:20</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </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">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="square_1.html#K5">^2</a> </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">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<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="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<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="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T48">XCMPLX_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T55">XCMPLX_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E7:24_5"/>
<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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</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">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:24"/>
( <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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></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"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, </i>;<br/>


</div>
