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

<b>let </b><font color="Maroon">A</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>let </b><font color="Maroon">f1</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">f2</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><a NAME="E1:24"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f1</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f2</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></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">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E5:24"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E6:24"/><b>then </b>
( <font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x0</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">f1</font>,<font color="Olive">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x0</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">f2</font>,<font color="Olive">x0</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="E7:24"/><b>then </b><i><font color="Green">E48</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">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E8:24"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></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">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green">E49</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">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E11:24"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E12:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:24"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E14:24"/><b>then </b>
<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"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:24"/><b>then </b><i><font color="Green">E51</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"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</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">f2</font>,<font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <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"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</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">f2</font>,<font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <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"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <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"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:24"/>
( <font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="fdiff_1.html#T29">FDIFF_1:29</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
