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<b>let </b><font color="Maroon">x0</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><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/><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><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:31"/><i><font color="Green">E36</font></i>: 
 <a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:31"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x0</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>hence </b><a NAME="E5:31"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>;<br/>

<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="prepower.html#K12">#R</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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