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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Z</font> being   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><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> <font color="Maroon" title="c1">n</font></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> <font color="Olive" title="b1">Z</font> <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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Olive" title="b1">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Z</font> be   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font> <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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font> <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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span><br/>

<a NAME="E3:51"/>
( <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a>, <a class="ref" href="zfmisc_1.html#T34" title="ZFMISC_1:th.34">ZFMISC_1:34</a></span>;<br/>
<a NAME="E4:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><font color="Maroon" title="c1">n</font> <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 class="kw">by</span> <span class="lab"><a class="ref" href="hfdiff_1.html#T3" target="_self" title="HFDIFF_1:th.3">Th3</a></span>;<br/>
<a NAME="E5:51"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 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> <font color="Maroon" title="c2">Z</font> 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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></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="51_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:51_1"/>
( <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a>, <a class="ref" href="zfmisc_1.html#T34" title="ZFMISC_1:th.34">ZFMISC_1:34</a></span>;<br/>
<a NAME="E2:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><font color="Maroon" title="c1">n</font> <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 class="kw">by</span> <span class="lab"><a class="ref" href="hfdiff_1.html#T3" target="_self" title="HFDIFF_1:th.3">Th3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> 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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font> )</span><br/>

<a NAME="E4:51_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><font color="Maroon" title="c1">n</font> <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 class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:51_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font></span><br/>

<a NAME="E6:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="prepower.html#T38" title="PREPOWER:th.38">PREPOWER:38</a></span>;<br/>
<a NAME="E7:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T36" title="PREPOWER:th.36">PREPOWER:36</a></span>;<br/>
<a NAME="E8:51_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="hfdiff_1.html#T43" target="_self" title="HFDIFF_1:th.43">Th43</a></span>;<br/>
<span class="kw">then </span><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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span>,<font color="Maroon" title="c4">x0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <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="c4">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="hfdiff_1.html#T29" target="_self" title="HFDIFF_1:th.29">Th29</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <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="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T36" title="PREPOWER:th.36">PREPOWER:36</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <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="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T36" title="PREPOWER:th.36">PREPOWER:36</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <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="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <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="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" title="XCMPLX_1:th.91">XCMPLX_1:91</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T36" title="PREPOWER:th.36">PREPOWER:36</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#D1" title="TAYLOR_1:def.1">TAYLOR_1:def 1</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="rfunct_1.html#D2" title="RFUNCT_1:def.2">RFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:51_1"/>
<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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:51"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="hfdiff_1.html#T43" target="_self" title="HFDIFF_1:th.43">Th43</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">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> <font color="Maroon" title="c1">n</font></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"><font color="Maroon" title="c1">n</font> <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></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:51"/>
<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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c2">Z</font> <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> <font color="Maroon" title="c1">n</font></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> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <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#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
