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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <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"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E28">A2</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <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"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<a NAME="E2:48"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="hfdiff_1.html#T43" target="_self" title="HFDIFF_1:th.43">Th34</a></i>;<br/>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<i><font color="Green" title="E30">A5</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E5:48"/>
for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font> )</span><br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font></span><br/>

<a NAME="E2:48_2"/><i><font color="Green" title="E32">A9</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="hfdiff_1.html#T43" target="_self" title="HFDIFF_1:th.43">Th34</a></i>;<br/>
<a NAME="E3:48_2"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:48_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/>
<a NAME="E4:48_2"/><i><font color="Green" title="E34">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span>,<font color="Maroon" title="c2">x0</font>
<b>by </b><i><a class="txt" href="#E2:48_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="#E1:48_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hfdiff_1.html#T45" target="_self" title="HFDIFF_1:th.45">ThA</a>, <a class="txt" href="#E3:48_2"><i><font color="Green" title="E33">A11</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" title="TAYLOR_1:def.1">TAYLOR_1:def 1</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:48_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E1:48"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font>
<b>by </b><i><a class="txt" href="#E1:48_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E1:48"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E4:48_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font>
<b>by </b><i><a class="txt" href="#E1:48_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E6:48_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 1</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E3:48"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
