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

<b>let </b><font color="Maroon">f</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">Z</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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">Z</font> implies for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>for <font color="Olive">g</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">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Olive">a</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">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">g</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">g</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">g</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</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="E1:45"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> 0,<font color="Maroon">Z</font>
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:45_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:45_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:45_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E5:45_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>let </b><font color="Maroon">g</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="E6:45_1"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E7:45_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 ;<br/>

<a NAME="E8:45_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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="E9:45_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E5:45_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:45_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:45_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>hence </b><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</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#E5:45_1"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</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#E5:45_1"><i><font color="Green">E40</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">y</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="E13:45_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E14:45_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">y</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E15:45_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">y</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">y</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i/>;<br/>
<a NAME="E16:45_1"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E6:45_1"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E17:45_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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="E18:45_1"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E5:45_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E19:45_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</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">a</font>,<font color="Maroon">b</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="E20:45_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">f</font>
 
<b>by </b><i><a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E21:45_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E22:45_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E9:45_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<i><font color="Green">E57</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E15:45_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5:45_1"><i><font color="Green">E40</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E24:45_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">xx</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 ;<br/><a NAME="E2:45_1_3_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">g</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">a</font>,<font color="Maroon">b</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="E3:45_1_3_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">x</font> = <font color="Maroon">xx</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E22:45_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E5:45_1_3_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E22:45_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taylor_1.html#E23:45_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font> = 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E22:45_1"><i><font color="Green">E56</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</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#E5:45_1"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="taylor_1.html#E24:45_1"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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#E5:45_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taylor_1.html#E24:45_1"><i><font color="Green">E58</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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/>.= 
<font color="Maroon">y</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E10:45_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E24:45_1"><i><font color="Green">E58</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E19:45_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="taylor_1.html#E22:45_1"><i><font color="Green">E56</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="E2:45_1_3"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E19:45_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:45_1"/><i><font color="Green">E63</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="E26:45_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">y</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E13:45_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E27:45_1"/><b>then </b>
<font color="Maroon">y</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#E9:45_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taylor_1.html#E22:45_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E28:45_1"/><b>then </b>
<font color="Maroon">y</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="E29:45_1"/><b>then </b>
<font color="Maroon">y</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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="E30:45_1"/><b>then </b>
<font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E14:45_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fcont_1.html#T26">FCONT_1:26</a></i>;<br/>
<a NAME="E31:45_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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="E32:45_1"/>
<font color="Maroon">g</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E20:45_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="fcont_1.html#T16">FCONT_1:16</a></i>;<br/>

<a NAME="E33:45_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="E34:45_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E7:45_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E35:45_1"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E36:45_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">f</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> ) )
 
<b>by </b><i>, , <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="E37:45_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</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">a</font>,<font color="Maroon">b</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="E38:45_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>( <font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</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="45_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_1_4"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E2:45_1_4"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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="E3:45_1_4"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">y</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E13:45_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E24:45_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
;<br/>
<a NAME="E5:45_1_4"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E24:45_1"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E6:45_1_4"/>
<font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="fdiff_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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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">f</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E36:45_1"><i><font color="Green">E64</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">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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">f</font> <a href="fdiff_1.html#K2">`|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E24:45_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">(<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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E13:45_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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/>
<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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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="E39:45_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E9:45_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E40:45_1"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<i><font color="Green">E130</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E23:45_1"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:45_1"><i><font color="Green">E46</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E42:45_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_6_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">xx</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 ;<br/><a NAME="E2:45_1_6_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">g</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">a</font>,<font color="Maroon">b</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="E3:45_1_6_1"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">x</font> = <font color="Maroon">xx</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E5:45_1_6_1"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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="E6:45_1_6_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="taylor_1.html#E36:45_1"><i><font color="Green">E64</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font> = 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</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#E8:45_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E38:45_1"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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#E8:45_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E38:45_1"><i><font color="Green">E66</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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">y</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E10:45_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taylor_1.html#E37:45_1"><i><font color="Green">E65</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E23:45_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="taylor_1.html#E38:45_1"><i><font color="Green">E66</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</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="E2:45_1_6"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:45_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E44:45_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E23:45_1"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E45:45_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/>
<a NAME="E46:45_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E47:45_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">g</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="txt" href="taylor_1.html#E8:45_1"><i><font color="Green">E46</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="45_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:45_1_7"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><b>thus </b><a NAME="E2:45_1_7"/>
<font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E36:45_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="taylor_1.html#E37:45_1"><i><font color="Green">E65</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">g</font>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E36:45_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="taylor_1.html#E37:45_1"><i><font color="Green">E65</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">g</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E36:45_1"><i><font color="Green">E64</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</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">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E25:45_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="taylor_1.html#E37:45_1"><i><font color="Green">E65</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</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#E37:45_1"><i><font color="Green">E65</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E49:45_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</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="E2:45"/><i><font color="Green">E139</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:45_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">Z</font>
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E3:45_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:45_2"/><i><font color="Green">E143</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:45_2"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E6:45_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E7:45_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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="E8:45_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:45_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="E10:45_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E11:45_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:45_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E13:45_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E14:45_2"/>
<font color="Maroon">k</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">k</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="E15:45_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">gk</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="E17:45_2"/><i><font color="Green">E150</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">gk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E18:45_2"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E19:45_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">gk</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">gk</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">gk</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) )
 
<b>by </b><i>, , , <a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, , , , , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">gk1</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="E1:45_2_1"/><i><font color="Green">E153</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">gk1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E2:45_2_1"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">gk1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">h</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="E4:45_2_1"/><i><font color="Green">E156</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E5:45_2_1"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">k</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="E6:45_2_1"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Olive">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E7:45_2_1"/>
<font color="Maroon">gk1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:45_2_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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/>
<b>hence </b><font color="Maroon">gk1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="taylor_1.html#T2" target="_self">Th2</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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/>
<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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">k</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">k</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="E8:45_2_1"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/><a NAME="E9:45_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E10:45_2_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E162</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">h</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">Z</font> <a href="subset_1.html#K8">/\</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i>, , , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<font color="Maroon">Z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E163</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">Z</font>
<b>by </b><i>, </i>
;<br/><b>thus </b><a NAME="E13:45_2_1"/>
<font color="Maroon">gk1</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">ghk</font> = <font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:45_2_1_4_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>thus </b><font color="Maroon">gk1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</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">k</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">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:45_2_1_4"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">gk1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<a NAME="E3:45_2_1_4"/><i><font color="Green">E167</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="E4:45_2_1_4"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:45_2_1_4"/><b>then </b>
<font color="Maroon">h</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E6:45_2_1_4"/><b>then </b>
<font color="Maroon">h</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="E7:45_2_1_4"/><b>then </b>
<font color="Maroon">h</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">a</font>,<font color="Maroon">b</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="E8:45_2_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#T20">FCONT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:45_2_1_4"/>
<font color="Maroon">gk1</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, , <a class="ref" href="fcont_1.html#T20">FCONT_1:20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:45_2_1"/><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E15:45_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</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="E16:45_2_1"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E170</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">h</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">a</font>,<font color="Maroon">b</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="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">a</font>,<font color="Maroon">b</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/><i><font color="Green">E171</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
<font color="Maroon">Z</font> <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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, <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">a</font>,<font color="Maroon">b</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/><a NAME="E19:45_2_1"/>
<font color="Maroon">k</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">k</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="E20:45_2_1"/><b>then </b><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">gfh</font> = <font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>;<br/><a NAME="E21:45_2_1"/><i><font color="Green">E173</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">gk1</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">a</font>,<font color="Maroon">b</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/><a NAME="E22:45_2_1"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_2_1_7"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="taylor_1.html#E1:45"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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="45_2_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_2_1_8"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">xx</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">xx</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</font></i></a></i>;<br/><b>thus </b><font color="Maroon">gk1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</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">k</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">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">xx</font>
<b>by </b><i>, <a class="ref" href="taylor_1.html#T4" target="_self">Th4</a>, <a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E24:45_2_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">gk1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E5"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E25:45_2_1"/><b>then </b><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">gk1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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="E26:45_2_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/><a NAME="E27:45_2_1"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><a NAME="E28:45_2_1"/><b>then </b><i><font color="Green">E181</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</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">h</font>,<font color="Olive">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a></i>;<br/><a NAME="E29:45_2_1"/><b>then </b><i><font color="Green">E182</font></i>: 
( <font color="Maroon">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">gk</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>,<font color="Olive">x</font></span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>;<br/><a NAME="E30:45_2_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">gk1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T5" target="_self">Th5</a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/><b>hence </b><a NAME="E31:45_2_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">gk1</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">a</font>,<font color="Maroon">b</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:45_2_1_9"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">gk1</font>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">gk1</font> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="taylor_1.html#T8" target="_self">Th8</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">gk1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="taylor_1.html#T8" target="_self">Th8</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E7"><i><font color="Green">Lemma69</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">gk</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="taylor_1.html#T7" target="_self">Th7</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">gk</font>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="taylor_1.html#E10"><i><font color="Green">Lemma72</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="taylor_1.html#T2" target="_self">Th2</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">h</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E10"><i><font color="Green">Lemma72</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="taylor_1.html#T6" target="_self">Th6</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p0">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">k</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , <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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">x</font></span>)</span></span>)</span>
<b>by </b><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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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="E33:45_2_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">gk1</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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="E21:45_2"/>
for <font color="Olive">g</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">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) holds <br/>( <font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">g</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">g</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">g</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">k</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">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">k</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">k</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="E3:45"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:45"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Olive">n</font>,<font color="Maroon">Z</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Olive">n</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>for <font color="Olive">g</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">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</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#K6" target="_self">Taylor</a> <font color="Maroon">f</font>,<font color="Maroon">Z</font>,<font color="Olive">x</font>,<font color="Olive">a</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) holds <br/>( <font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">g</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">g</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">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Olive">g</font>,<font color="Olive">x</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#K5" target="_self">diff</a> <font color="Maroon">f</font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</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">a</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> ) )
 ;<br/>


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