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<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="E1:28"/><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">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">f2</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:28"/><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#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:28"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <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#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><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> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_3.html#T3">FUNCT_3:3</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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="E1:28_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><a NAME="E2:28_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f1</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><b>thus </b><a NAME="E3:28_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E4:28_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E5:28_1"/>
<font color="Maroon">f2</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><b>hence </b><a NAME="E6:28_1"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:28"/><b>then </b>
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/><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x0</font>
 
;<br/>
<b>hence </b><a NAME="E7:28"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>

<a NAME="E8:28"/><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"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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="E9:28"/>
 <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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</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="E10:28"/><b>then </b>
<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"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T4">RFUNCT_2:4</a></i>;<br/>
<b>then </b><i><font color="Green">E52</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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span> = 
<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">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<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">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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="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#D8">FDIFF_1:def 8</a></i>
<br/>.= 
 <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">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K8">(#)</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>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><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="E1:28_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:28_3"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f1</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f2</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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> = 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x0</font>
<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"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f2</font>,<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f2</font>,<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></span>)</span> <a href="real_1.html#K4">*</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#K2">.</a> <font color="Maroon">x0</font></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"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <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></span>)</span> <a href="real_1.html#K4">*</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#K2">.</a> <font color="Maroon">x0</font></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">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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="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#K2">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K8">(#)</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="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:28"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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">f2</font> <a href="fdiff_1.html#K2">`|</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K8">(#)</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>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
