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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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>let </b><font color="Maroon">c<sub>2</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>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E23</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>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:48"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, <a class="txt" href="fdiff_6.html#E1:48"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E5:48"/><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>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <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">0 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:48"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</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> 0 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E4:48"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E7:48"/>
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T72">SIN_COS:72</a>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/>
<a NAME="E8:48"/><b>then </b><i><font color="Green">E27</font></i>: 
2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, <a class="txt" href="fdiff_6.html#E4:48"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E9:48"/>
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>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:48_1"/><i><font color="Green">E29</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T68">SIN_COS:68</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#T26">FDIFF_1:26</a>, <a class="txt" href="fdiff_6.html#E1:48"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fdiff_6.html#E8:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_6.html#E6:48"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_6.html#E1:48_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:48_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="fdiff_6.html#E6:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:48_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E5:48"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, <a class="txt" href="fdiff_6.html#E4:48"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_6.html#E1:48_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T68">SIN_COS:68</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:48_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:48"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default">2 <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</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>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_6.html#E6:48"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_6.html#E8:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_6.html#E1:48"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_1.html#T26">FDIFF_1:26</a></i>;<br/>


</div>
