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

<b>let </b><font color="Maroon" title="c1">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 2</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 2 )</span><br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 2</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font>
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="fdiff_2.html#T16" title="FDIFF_2:th.16">FDIFF_2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font>
<b>by </b><i><a class="ref" href="taylor_1.html#D5" title="TAYLOR_1:def.5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font>
<b>by </b><i><a class="ref" href="taylor_1.html#D5" title="TAYLOR_1:def.5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5" title="TAYLOR_1:def.5">TAYLOR_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E3:66"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
