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<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">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="E1:17"/><i><font color="Green">E41</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">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> )
 ;<br/>

<b>then consider </b><font color="Maroon">N1</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">N1</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/>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">N2</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>, </i>;<br/>
<a NAME="E6:17"/>
<font color="Maroon">f1</font> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T32">FDIFF_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">N3</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E8:17"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E10:17"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N3</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T38">RCOMP_1:38</a></i>;<br/>
<a NAME="E11:17"/>
<font color="Maroon">N</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:17_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N3</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:17"/>
( <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> &amp;  <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> <a href="hidden.html#R1">=</a> <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/>


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