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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E3:13"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:13_1"/>
 <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="txt" href="fdiff_7.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="sin_cos6.html#T108">SIN_COS6:108</a>, <a class="txt" href="fdiff_7.html#E1:13_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></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="E4:13"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E5:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:13_2"/><b>then </b><i><font color="Green">E15</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E3:13_2"/><i><font color="Green">E16</font></i>: 
 <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="txt" href="fdiff_7.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="sin_cos6.html#T108">SIN_COS6:108</a>, <a class="txt" href="fdiff_7.html#E1:13_2"><i><font color="Green">E14</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="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:13_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, <a class="txt" href="fdiff_7.html#E4:13"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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> <a href="sin_cos6.html#K4">arccos</a> ,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T3">TAYLOR_1:3</a>, <a class="txt" href="fdiff_7.html#E3:13_2"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T108">SIN_COS6:108</a>, <a class="txt" href="fdiff_7.html#E2:13_2"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E5:13_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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K1">#Z</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_7.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fdiff_7.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>


</div>
