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<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:53"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">f</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><a NAME="E2:53"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K31">compreal</a> 
 ;<br/>

<a NAME="E3:53"/><b>then </b><i><font color="Green">E27</font></i>: 
(  <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> &amp; <a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a>, <a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<b>then </b><i><font color="Green">E28</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T22">FDIFF_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a>, <a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a>, <a class="txt" href="sin_cos2.html#E2:53"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>

;<br/>
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a>, <a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a>, <a class="txt" href="sin_cos2.html#E2:53"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E6:53"/>
( <a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> &amp; <a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a>, <a class="ref" href="fdiff_1.html#T22">FDIFF_1:22</a></i>;<br/>


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