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<b>thus </b><a NAME="E1:22"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun2.html#K2">"</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self">Th16</a>, <a class="ref" href="fdiff_2.html#T45">FDIFF_2:45</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun2.html#K2">"</a> </span>)</span>
 ;<br/>

<a NAME="E3:22"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <a href="sin_cos.html#K27">exp_R</a> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E5:22"/>
 <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="partfun2.html#K2">"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="taylor_1.html#T16" target="_self">Th16</a>, <a class="ref" href="fdiff_2.html#T45">FDIFF_2:45</a></i>;<br/>


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