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

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">f</font> = <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">h</font> = <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</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="E1:100"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>;<br/>
<i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> 
<b>by </b><i><a class="ref" href="sin_cos6.html#T2" target="_self">Th2</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T40">RFUNCT_2:40</a></i>
<br/>.= 
<a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</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>
<b>by </b><i><a class="ref" href="sin_cos6.html#T2" target="_self">Th2</a>, <a class="txt" href="sin_cos6.html#E33"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/>
<a NAME="E3:100"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E8:100"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sin_cos6.html#T1" target="_self">Th1</a>, <a class="ref" href="sin_cos6.html#T31" target="_self">Th31</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:100"/><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>;<br/>
<a NAME="E5:100"/><b>then </b><i><font color="Green">E32</font></i>: 
<a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:100_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><i><font color="Green">E35</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E12:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sin_cos6.html#E16:100"><i><font color="Green">E111</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="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E11:100"><i><font color="Green">E108</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E11:100"><i><font color="Green">E108</font></i></a>, <a class="txt" href="sin_cos6.html#E16:100"><i><font color="Green">E111</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</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">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>
;<br/><a NAME="E3:100_2"/>
 <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"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="sin_cos6.html#E16:100"><i><font color="Green">E111</font></i></a>, <a class="ref" href="comptrig.html#T23">COMPTRIG:23</a></i>;<br/><b>hence </b><a NAME="E4:100_2"/>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T49" target="_self">Th49</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:100"/><b>then </b><i><font color="Green">E36</font></i>: 
<a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</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> <a href="fdiff_1.html#R2">is_differentiable_on</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>
 
<b>by </b><i>, <a class="txt" href="sin_cos6.html#E8:100"><i><font color="Green">E106</font></i></a>, <a class="txt" href="sin_cos6.html#E10:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="sin_cos6.html#E12:100"><i><font color="Green">E109</font></i></a>, <a class="ref" href="fdiff_2.html#T48">FDIFF_2:48</a></i>;<br/>
<a NAME="E8:100"/><i><font color="Green">E106</font></i>: 
<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> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos6.html#K4" target="_self">arccos</a> 
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E9:100"/>
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> <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> holds <br/><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</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> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E16:100"><i><font color="Green">E111</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E10:100"/><i><font color="Green">E107</font></i>: 
 <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</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>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T49" target="_self">Th49</a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">x</font> = <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>;<br/>
<b>assume </b><a NAME="E11:100"/><i><font color="Green">E108</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E12:100"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E13:100"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:100"/>
<a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos6.html#K6" target="_self">arccos</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E15:100"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> &amp; <a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E86"><i><font color="Green">Lemma114</font></i></a>, </i>;<br/>
<a NAME="E16:100"/><b>then </b><i><font color="Green">E111</font></i>: 
<a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<i><font color="Green">E161</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E12:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sin_cos6.html#E87"><i><font color="Green">Lemma117</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="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E11:100"><i><font color="Green">E108</font></i></a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E11:100"><i><font color="Green">E108</font></i></a>, <a class="txt" href="sin_cos6.html#E87"><i><font color="Green">Lemma117</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</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> <span class="p2">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E86"><i><font color="Green">Lemma114</font></i></a>, <a class="ref" href="sin_cos6.html#T46" target="_self">Th46</a></i>
;<br/>
<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos6.html#K4" target="_self">arccos</a> ,<font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T50" target="_self">Th50</a>, , <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"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T50" target="_self">Th50</a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4" target="_self">arccos</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E16:100"><i><font color="Green">E111</font></i></a>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="sin_cos6.html#E8:100"><i><font color="Green">E106</font></i></a>, <a class="txt" href="sin_cos6.html#E10:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="sin_cos6.html#E12:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sin_cos6.html#E13:100"><i><font color="Green">E110</font></i></a>, , , <a class="txt" href="sin_cos6.html#E88"><i><font color="Green">Lemma118</font></i></a>, <a class="ref" href="fdiff_2.html#T48">FDIFF_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T193">XCMPLX_1:193</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>
;<br/>


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