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

<b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">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> ;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:41"/>
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D7" target="_self" title="FDIFF_1:def.7">Def7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a><br/>

<b>let </b><font color="Maroon" title="c5">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<a NAME="E3:41"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:41"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E5:41"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="fdiff_1.html#E3:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D7" target="_self" title="FDIFF_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T32" target="_self" title="FDIFF_1:th.32">Th32</a></i>;<br/>


</div>
