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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> ) holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) ) );<br/>
<a NAME="E1:39"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> 0,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:39_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:39_1"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:39_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> 0 <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E5:39_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> 0 <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:39_1"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:39_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 ;<br/>

<a NAME="E8:39_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E9:39_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:39_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E7:39_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <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">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E10:39_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
0

;<br/><br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:39_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E14:39_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E15:39_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="txt" href="taylor_1.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E16:39_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E17:39_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T16">FCONT_1:16</a></i>;<br/>
<a NAME="E18:39_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E19:39_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T16">FCONT_1:16</a></i>;<br/>
<a NAME="E20:39_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E21:39_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E22:39_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E13:39_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<i><font color="Green">E49</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E20:39_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E22:39_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E6:39_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E24:39_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 ;<br/><a NAME="E2:39_1_3_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E3:39_1_3_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_3_1"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E5:39_1_3_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/><a NAME="E6:39_1_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_3_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_1_3_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_3_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E7:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_3_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <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">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E14:39_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E5:39_1_3_1"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E20:39_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E23:39_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_3_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E23:39_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:39_1"/><i><font color="Green">E51</font></i>: 
 <a href="numbers.html#K1">REAL</a>  is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E26:39_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E15:39_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E27:39_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E13:39_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taylor_1.html#E25:39_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E28:39_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E29:39_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E30:39_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E19:39_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="fcont_1.html#T26">FCONT_1:26</a></i>;<br/>
<a NAME="E31:39_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T16">FCONT_1:16</a></i>;<br/>
<b>hence </b><a NAME="E32:39_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E24:39_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="fcont_1.html#T16">FCONT_1:16</a></i>;<br/>

<a NAME="E33:39_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E34:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> 0 <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E5:39_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E35:39_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E36:39_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taylor_1.html#E20:39_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:39_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E38:39_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:39_1_4"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E2:39_1_4"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<a NAME="E3:39_1_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E15:39_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E37:39_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
;<br/>
<a NAME="E5:39_1_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E37:39_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>;<br/>

<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_1_4"><i><font color="Green">E57</font></i></a>, <a class="txt" href="taylor_1.html#E5:39_1_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#K2">`|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E37:39_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E15:39_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E4:39_1_4"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:39_1_4"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E39:39_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E13:39_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E40:39_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E36:39_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="taylor_1.html#E40:39_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E9:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E6:39_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E42:39_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_6_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 ;<br/><a NAME="E2:39_1_6_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E3:39_1_6_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_6_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E5:39_1_6_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/><a NAME="E6:39_1_6_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_6_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_1_6_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_1_6_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E7:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="taylor_1.html#E9:39_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_6_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <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">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E9:39_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_6_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E14:39_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E5:39_1_6_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E36:39_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="taylor_1.html#E6:39_1_6_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E41:39_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_6_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E41:39_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:39_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E38:39_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E44:39_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E36:39_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="taylor_1.html#E40:39_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E45:39_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i><a class="txt" href="taylor_1.html#E42:39_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/>
<a NAME="E46:39_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E47:39_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:39_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E9:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:39_1_7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><b>thus </b><a NAME="E2:39_1_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E47:39_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><b>thus </b> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E47:39_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_7"><i><font color="Green">E60</font></i></a>, <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">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E47:39_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E42:39_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="taylor_1.html#E44:39_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E44:39_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E38:39_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_1_7"><i><font color="Green">E60</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E49:39_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:39"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:39_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:39_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:39_2"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:39_2"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E6:39_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E7:39_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E8:39_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:39_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:39_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E11:39_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E12:39_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E13:39_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E14:39_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E15:39_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E17:39_2"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E18:39_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E36"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E19:39_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E9:39_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taylor_1.html#E13:39_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taylor_1.html#E15:39_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E17:39_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:39_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:39_2_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:39_2_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E5:39_2_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span>
 <b>and </b><br/><a NAME="E6:39_2_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="taylor_1.html#E35"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>thus </b><a NAME="E7:39_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:39_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_1"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
0 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E9:39_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taylor_1.html#E13:39_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taylor_1.html#E15:39_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E17:39_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>
<br/>.= 
0 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
0 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
0

;<br/><br/>


</div><b>end;</b></div><a NAME="E8:39_2_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E9:39_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E10:39_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><i><font color="Green">E55</font></i>:  <a href="relset_1.html#K4">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#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E8:39_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taylor_1.html#E10:39_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E56</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E17:39_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E11:39_2_1"><i><font color="Green">E55</font></i></a></i>
;<br/><b>thus </b><a NAME="E13:39_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:39_2_1_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:39_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_4_1"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_4_1"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:39_2_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E11:39_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_4_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E12:39_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_4_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:39_2_1_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taylor_1.html#E12:39_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<a NAME="E3:39_2_1_4"/><i><font color="Green">E58</font></i>: 
 <a href="numbers.html#K1">REAL</a>  is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E4:39_2_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E5:39_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_2_1_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E6:39_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E7:39_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E8:39_2_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E5:39_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="fcont_1.html#T20">FCONT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:39_2_1_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E19:39_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2_1_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="fcont_1.html#T20">FCONT_1:20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:39_2_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E15:39_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E16:39_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><i><font color="Green">E59</font></i>:  <a href="relset_1.html#K4">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#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E14:39_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="taylor_1.html#E16:39_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>then </b><i><font color="Green">E60</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E17:39_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E19:39_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E20:39_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/><a NAME="E21:39_2_1"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E22:39_2_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:39_2_1_7"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E17:39_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="taylor_1.html#T24" target="_self">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2_1_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E11:39_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E12:39_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_1_8"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:39_2_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E11:39_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E12:39_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E22:39_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_8"><i><font color="Green">E64</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E24:39_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E21:39_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E25:39_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/><a NAME="E26:39_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E27:39_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:39_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><a NAME="E28:39_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <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"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E17:39_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taylor_1.html#E20:39_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a></i>;<br/><a NAME="E29:39_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E19:39_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>;<br/><a NAME="E30:39_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E25:39_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/><b>hence </b><a NAME="E31:39_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E8:39_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:39_2_1_9"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><b>thus </b> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E31:39_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <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">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E31:39_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E24:39_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E29:39_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E19:39_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taylor_1.html#E28:39_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:39_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taylor_1.html#E2:39_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taylor_1.html#E3:39_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taylor_1.html#E4:39_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taylor_1.html#E9:39_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taylor_1.html#E13:39_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taylor_1.html#E15:39_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E17:39_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E18:39_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E28:39_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E17:39_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taylor_1.html#E20:39_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="taylor_1.html#E27:39_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></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">1 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:39_2_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E6:39_2_1"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E20:39_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="taylor_1.html#E1:39_2_1_9"><i><font color="Green">E70</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E33:39_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:39_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:39"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="taylor_1.html#E1:39"><i><font color="Green">E33</font></i></a>, <a class="txt" href="taylor_1.html#E2:39"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:39"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) holds <br/>( <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) )
 ;<br/>


</div>
