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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></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">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:34_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self">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">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:34_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#T16" target="_self">Th16</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fdiff_1.html#K2" target="_self">`|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="fdiff_1.html#K1" target="_self">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E5:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_1.html#E1:34_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" target="_self">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E2:34_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="fdiff_1.html#E3:34_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="fdiff_1.html#T21" target="_self">Th21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:34"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fdiff_1.html#K2" target="_self">`|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>


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