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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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">f1</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>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="E1:51"/><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"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:51"/><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">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="real_1.html#K6">/</a> 2 &amp; <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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">x</font> ) ) &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E37</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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">x</font> ) &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E4:51"/>
<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> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, </i>;<br/>
<a NAME="E5:51"/><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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E6:51"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</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">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:51"/><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">f1</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"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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="E1:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then </b><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="real_1.html#K6">/</a> 2
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0

;<br/>
<b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:51"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f1</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">f1</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> 1 <a href="real_1.html#K6">/</a> 2 ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, </i>;<br/>
<a NAME="E9:51"/>
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"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></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"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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="E1:51_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></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="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</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">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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#K5">-</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">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<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"><span class="p2">(<span class="default"><font color="Maroon">f1</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#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</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></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"><span class="p2">(<span class="default"><font color="Maroon">f1</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#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</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#K24">cos</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <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#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/>
<b>hence </b><a NAME="E3:51_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></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"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:51"/>
( <font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></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"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></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"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  ) )
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a>, </i>;<br/>


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