<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">Z</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">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="E1:34"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:34"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:34"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:34"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, </i>;<br/>
<a NAME="E9:34"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:34_1"/>
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b>
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a>, </i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T3">TAYLOR_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:34"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E11:34"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:34_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:34_2"/><i><font color="Green">E51</font></i>: 
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<a NAME="E4:34_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a>, </i>;<br/>
<i><font color="Green">E53</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#K6">+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</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">x</font></span>)</span> <a href="real_1.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">x</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="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</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">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, , </i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>

;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<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">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</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">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, </i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <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#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span></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> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="taylor_1.html#T3">TAYLOR_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</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">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E8:34_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:34"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K6">+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>


</div>
