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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 implies ( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span> &amp; <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">x</font> =  <a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font>;<br/>
<b>assume </b><a NAME="E1:117"/><i><font color="Green" title="E119">A1</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span> &amp; <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font> )</span><br/>

<a NAME="E2:117"/><b>then </b>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:117"/><b>then </b><i><font color="Green" title="E120">A2</font></i>: 
 <a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sincos10.html#E58"><i><font color="Green" title="E42">Lm14</font></i></a>, <a class="ref" href="sincos10.html#T86" target="_self" title="SINCOS10:th.86">Th86</a></i>;<br/>
<a NAME="E4:117"/><i><font color="Green" title="E121">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<i><font color="Green" title="E122">A5</font></i>: <font color="Maroon" title="c1">r</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E1:117"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="sincos10.html#T90" target="_self" title="SINCOS10:th.90">Th90</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E3:117"><i><font color="Green" title="E120">A2</font></i></a>, <a class="txt" href="sincos10.html#E4:117"><i><font color="Green" title="E121">A3</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>
;<br/>
<a NAME="E6:117"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sincos10.html#E1:117"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:117"/><b>then </b><i><font color="Green" title="E123">A7</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/>
<a NAME="E8:117"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></i>;<br/>
<b>then </b><i><font color="Green" title="E124">A8</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E5:117"><i><font color="Green" title="E122">A5</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T103" title="XCMPLX_1:th.103">XCMPLX_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E7:117"><i><font color="Green" title="E123">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

;<br/>
<a NAME="E10:117"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="sincos10.html#E1:117"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E11:117"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E12:117"/><b>then </b><i><font color="Green" title="E125">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E13:117"/><b>then </b><i><font color="Green" title="E126">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sincos10.html#E7:117"><i><font color="Green" title="E123">A7</font></i></a>, <a class="ref" href="xreal_1.html#T138" title="XREAL_1:th.138">XREAL_1:138</a></i>;<br/>
<a NAME="E14:117"/>
<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="sincos10.html#E46"><i><font color="Green" title="E30">Lm10</font></i></a>, <a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/>
<a NAME="E15:117"/><b>then </b><i><font color="Green" title="E127">A11</font></i>: 
<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:117"/>
<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="sincos10.html#E46"><i><font color="Green" title="E30">Lm10</font></i></a>, <a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/>
<a NAME="E17:117"/><b>then </b><i><font color="Green" title="E128">A12</font></i>: 
<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="sincos10.html#E15:117"><i><font color="Green" title="E127">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E18:117"/>
 <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E19:117"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="sincos10.html#E17:117"><i><font color="Green" title="E128">A12</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E20:117"/><b>then </b>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sincos10.html#E3:117"><i><font color="Green" title="E120">A2</font></i></a>, <a class="txt" href="sincos10.html#E58"><i><font color="Green" title="E42">Lm14</font></i></a>, <a class="ref" href="comptrig.html#T24" title="COMPTRIG:th.24">COMPTRIG:24</a></i>;<br/>
<b>then </b><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E9:117"><i><font color="Green" title="E124">A8</font></i></a>, <a class="txt" href="sincos10.html#E13:117"><i><font color="Green" title="E126">A10</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E7:117"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="sincos10.html#E12:117"><i><font color="Green" title="E125">A9</font></i></a>, <a class="ref" href="square_1.html#T99" title="SQUARE_1:th.99">SQUARE_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="txt" href="sincos10.html#E1:117"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="square_1.html#T90" title="SQUARE_1:th.90">SQUARE_1:90</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189" title="XCMPLX_1:th.189">XCMPLX_1:189</a></i>
;<br/>
<b>hence </b><a NAME="E22:117"/>
( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font></span>)</span> &amp; <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K10" title="SINCOS10:func.10">arcsec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">r</font> )
 <b>by </b><i><a class="txt" href="sincos10.html#E5:117"><i><font color="Green" title="E122">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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