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<b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">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>let </b><font color="Maroon" title="c5">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c5">Z</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">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c5">Z</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c5">Z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self" title="FDIFF_1:th.16">Th16</a></i>;<br/><a NAME="E3:34_1"/>
<font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:34"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self" title="FDIFF_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="fdiff_1.html#T21" target="_self" title="FDIFF_1:th.21">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c5">Z</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self" title="FDIFF_1:th.16">Th16</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self" title="FDIFF_1:th.16">Th16</a></i>;<br/><a NAME="E3:34_2"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:34"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self" title="FDIFF_1:th.16">Th16</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c5">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E5:34"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="fdiff_1.html#D8" target="_self" title="FDIFF_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E2:34_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="fdiff_1.html#E3:34_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T21" target="_self" title="FDIFF_1:th.21">Th21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:34"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c5">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</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">f1</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>


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