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<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>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>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Z</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">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f1</font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f2</font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">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">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">f1</font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:34_1"/>
<font color="Maroon">f2</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>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </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">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">f1</font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:34_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f2</font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x</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">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2" target="_self">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="fdiff_1.html#K1" target="_self">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" target="_self">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" target="_self">diff</a> <font color="Maroon">f2</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:34"/>
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">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">f2</font></span>)</span> <a href="fdiff_1.html#K2" target="_self">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" target="_self">diff</a> <font color="Maroon">f1</font>,<font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" target="_self">diff</a> <font color="Maroon">f2</font>,<font color="Olive">x</font></span>)</span>
 ;<br/>


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