<?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>let </b><font color="Maroon">Z1</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1,<font color="Maroon">Z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </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="E1:50_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E2:50_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:50_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:50_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">k</font>,<font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:50_1"/>
<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="E6:50_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>,<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">Z1</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><a NAME="E7:50_1"/><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> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z1</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/><b>hence </b><a NAME="E8:50_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">Z1</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">Z1</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<font color="Maroon">f</font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1,<font color="Maroon">Z1</font>
 <b>by </b><i/>;<br/>


</div>
