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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:13_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra7.html#K1" title="INTEGRA7:func.1">IntegralFuncs</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="integra7.html#D2" target="_self" title="INTEGRA7:def.2">Def2</a></i>;<br/><a NAME="E3:13_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">F'</font> &amp; <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">X</font> &amp; <font color="Olive" title="b1">F'</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="integra7.html#D1" target="_self" title="INTEGRA7:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:13_1"/>
( <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">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/>
( <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">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E3:13"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra7.html#K1" title="INTEGRA7:func.1">IntegralFuncs</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="integra7.html#D1" target="_self" title="INTEGRA7:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon" title="c2">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="integra7.html#D2" target="_self" title="INTEGRA7:def.2">Def2</a></i>;<br/>


</div>
