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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Z</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">f</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="E1:42"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:42"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:42"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fdiff_1.html#D7" target="_self">FDIFF_1:def 7</a><br/>

<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E4:42"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E5:42"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, </i>;<br/>
<b>then 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="E7:42"/><i><font color="Green">E56</font></i>: 
( <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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp;  ex <font color="Olive">L</font> being  <a href="fdiff_1.html#NM2">LINEAR</a> ex <font color="Olive">R</font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>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="E9:42"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E11:42"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T38">RCOMP_1:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">N2</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="fdiff_1.html#D5" target="_self">FDIFF_1:def 5</a><br/>

<a NAME="E12:42"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E13:42"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E14:42"/><b>then </b>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:42"/><b>then </b><i><font color="Green">E70</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">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:42"/>
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:42"/><b>then </b>
<font color="Maroon">N2</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">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E18:42"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">N2</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="fdiff_1.html#NM2">LINEAR</a>, <font color="Maroon">R</font> being   <a href="fdiff_1.html#NM1">REST</a><b> such that </b><br/><a NAME="E20:42"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E21:42"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 ;<br/>

<a NAME="E22:42"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<a NAME="E23:42"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E24:42"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E25:42"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E26:42"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E27:42"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div>
