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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <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="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2 implies ( <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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> &amp; <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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> <font color="Maroon" title="c1">r</font> ) )</span><br/>

<a NAME="E1:118"/>
<a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sincos10.html#E20"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<a NAME="E2:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A1</font></span>: 
<a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 <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="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
;<br/>
<a NAME="E3:118"/><span class="lab"><font color="Green" title="E138">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</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"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></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#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">x</font> =  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:118"/><span class="lab"><font color="Green" title="E139">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">and </span><br/><a NAME="E5:118"/><span class="lab"><font color="Green" title="E140">A4</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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> &amp; <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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> <font color="Maroon" title="c1">r</font> )</span><br/>

<a NAME="E6:118"/>
<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">1,<span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="txt" href="#E5:118"><span class="lab"><font color="Green" title="E140">A4</font></span></a></span>;<br/>
<a NAME="E7:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A5</font></span>: 
 <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</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"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sincos10.html#E80"><span class="lab"><font color="Green" title="E60">Lm32</font></span></a>, <a class="ref" href="sincos10.html#T88" target="_self" title="SINCOS10:th.88">Th88</a></span>;<br/>
<span class="lab"><font color="Green" title="E142">A6</font></span>: <font color="Maroon" title="c1">r</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="txt" href="#E5:118"><span class="lab"><font color="Green" title="E140">A4</font></span></a>, <a class="ref" href="sincos10.html#T92" target="_self" title="SINCOS10:th.92">Th92</a></span>
<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#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E141">A5</font></span></a>, <a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E138">A2</font></span></a>, <a class="ref" href="rfunct_1.html#D2" title="RFUNCT_1:def.2">RFUNCT_1:def 2</a></span>
;<br/>
<a NAME="E9:118"/>
<a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
;<br/>
<a NAME="E10:118"/><span class="kw">then </span>
<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="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></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"><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="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E137">A1</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<a NAME="E11:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A7</font></span>: 
<a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E141">A5</font></span></a>, <a class="txt" href="sincos10.html#E80"><span class="lab"><font color="Green" title="E60">Lm32</font></span></a>, <a class="ref" href="comptrig.html#T12" title="COMPTRIG:th.12">COMPTRIG:12</a></span>;<br/>
<a NAME="E12:118"/>
<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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="ref" href="square_1.html#T15" title="SQUARE_1:th.15">SQUARE_1:15</a></span>;<br/>
<a NAME="E13:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A8</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E14:118"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T28" title="SIN_COS:th.28">SIN_COS:28</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:118"><span class="lab"><font color="Green" title="E142">A6</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T102" title="XCMPLX_1:th.102">XCMPLX_1:102</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<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/>
<span class="kw">then </span><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:118"><span class="lab"><font color="Green" title="E143">A7</font></span></a>, <a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="txt" href="#E13:118"><span class="lab"><font color="Green" title="E144">A8</font></span></a>, <a class="ref" href="square_1.html#T30" title="SQUARE_1:th.30">SQUARE_1:30</a></span>
<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> <font color="Maroon" title="c1">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E139">A3</font></span></a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:118"/>
( <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</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> &amp; <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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> <font color="Maroon" title="c1">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:118"><span class="lab"><font color="Green" title="E142">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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