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<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">N1</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c1">x0</font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font>
 <b>by </b><i><a class="ref" href="fdiff_2.html#T11" target="_self" title="FDIFF_2:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">N2</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x0</font><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="fdiff_2.html#T11" target="_self" title="FDIFF_2:th.11">Th11</a></i>;<br/>
<a NAME="E6:17"/>
<font color="Maroon" title="c2">f1</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">N3</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c1">x0</font><b> such that </b><br/><a NAME="E8:17"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N2</font>
 <b>by </b><i><a class="ref" href="fcont_1.html#T5" title="FCONT_1:th.5">FCONT_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c1">x0</font><b> such that </b><br/><a NAME="E10:17"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">N1</font> &amp; <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N3</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T38" title="RCOMP_1:th.38">RCOMP_1:38</a></i>;<br/>
<a NAME="E11:17"/>
<font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N1</font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E10:17"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c8">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="fdiff_2.html#E1:17_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<a NAME="E4:17_1"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">N3</font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:17"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fdiff_2.html#E10:17"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="fdiff_2.html#E1:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="fdiff_2.html#E2:17_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N2</font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E8:17"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:17"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fdiff_2.html#E5:17"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="fdiff_2.html#E2:17_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:17"/>
( <font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="fdiff_2.html#E14"><i><font color="Green" title="E13">Lm2</font></i></a></i>;<br/>


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