<?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="E1:45"/><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="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:45"/><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 &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/>

<a NAME="E3:45"/><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"><span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a>, </i>;<br/>
<a NAME="E4:45"/>
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"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, </i>;<br/>
<a NAME="E5:45"/><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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:45"/><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#K22">cos</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="E7:45"/><b>then </b><i><font color="Green">E39</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="E8:45"/><i><font color="Green">E43</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="E9:45"/><b>then </b><i><font color="Green">E46</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="E10:45"/><i><font color="Green">E47</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</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="E11:45"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a>  <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#T26">FDIFF_1:26</a></i>;<br/>
<a NAME="E12:45"/><i><font color="Green">E49</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"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then </b><i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <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/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E3:45_1"/>
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:45_1"/><b>then </b>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:45_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </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="E1:45_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:45_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a>  <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="E3:45_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:45_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </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#T22">TAYLOR_1:22</a>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:45"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </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="E15:45"/>
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="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><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="E1:45_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:45_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a>  <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="E3:45_3"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</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#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</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#E1"><i><font color="Green">Lemma31</font></i></a></i>
;<br/>
<a NAME="E5:45_3"/><b>then </b><i><font color="Green">E60</font></i>: 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<i><font color="Green">E61</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </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>
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , , <a class="ref" href="fdiff_1.html#T26">FDIFF_1:26</a></i>
<br/>.= 
<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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, , <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <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#T68">SIN_COS:68</a></i>
;<br/>
<a NAME="E7:45_3"/><i><font color="Green">E62</font></i>: 
( <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></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> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</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"><span class="p3">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, , , <a class="txt" href="fdiff_6.html#E1"><i><font color="Green">Lemma31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T22">TAYLOR_1:22</a>, <a class="txt" href="fdiff_6.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="fdiff_6.html#E3"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:45"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fdiff_6.html#E2:45"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T90">PREPOWER:90</a>, <a class="txt" href="fdiff_6.html#E4"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i>, <a class="txt" href="fdiff_6.html#E3:45"><i><font color="Green">E37</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:45_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:45"/>
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></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="real_1.html#K1">-</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, </i>;<br/>


</div>
