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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><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="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E23</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>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="fdiff_1.html#NM2">LINEAR</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="fdiff_1.html#NM2">LINEAR</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="fdiff_1.html#NM1">REST</a><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E25</font></i>: 
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>6</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:31"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="fdiff_1.html#NM2">LINEAR</a> <b>by </b><i><a class="ref" href="fdiff_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="fdiff_1.html#NM1">REST</a> <b>by </b><i><a class="ref" href="fdiff_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</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#K13">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:31_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</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#K13">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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#K13">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E10:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="fdiff_1.html#E1:31_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E10:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="fdiff_1.html#E2:31_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E5:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fdiff_1.html#E1:31_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E8:31"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E9:31"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:31"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K13">(#)</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>5</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E10:31"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#D5" target="_self">Def5</a></i>;<br/>

<b>hence </b> <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#K13">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i><a class="txt" href="fdiff_1.html#E10:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="fdiff_1.html#E11:31"><i><font color="Green">E29</font></i></a>, <a class="ref" href="fdiff_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E8:31"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_1.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_1.html#E5:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#D6" target="_self">Def6</a></i>
;<br/><br/>


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