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

<span class="kw">set </span><font color="Maroon" title="c1">X</font> = <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">g1</font> = <a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <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="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">g</font> = <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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E1:125"/><span class="lab"><font color="Green" title="E140">A1</font></span>: 
<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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a>, <a class="ref" href="xxreal_1.html#T22" title="XXREAL_1:th.22">XXREAL_1:22</a></span>;<br/>
<span class="lab"><font color="Green" title="E141">A2</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><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="p5"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p4"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" title="FUNCT_1:th.33">FUNCT_1:33</a></span>
<br/>.= 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</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#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125"><span class="lab"><font color="Green" title="E140">A1</font></span></a>, <a class="ref" href="relat_1.html#T72" title="RELAT_1:th.72">RELAT_1:72</a></span>
<br/>.= 
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>
;<br/>
<span class="lab"><font color="Green" title="E142">A3</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p4"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T72" title="RELAT_1:th.72">RELAT_1:72</a></span>
<br/>.= 
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T17" title="RFUNCT_2:th.17">RFUNCT_2:17</a></span>
<br/>.= 
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<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="p4"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>
<br/>.= 
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<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="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a>, <a class="ref" href="xxreal_1.html#T22" title="XXREAL_1:th.22">XXREAL_1:22</a></span>
<br/>.= 
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>
;<br/>
<a NAME="E4:125"/><span class="lab"><font color="Green" title="E143">A4</font></span>: 
<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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125"><span class="lab"><font color="Green" title="E140">A1</font></span></a>, <a class="ref" href="sincos10.html#T7" target="_self" title="SINCOS10:th.7">Th7</a>, <a class="ref" href="fdiff_2.html#T16" title="FDIFF_2:th.16">FDIFF_2:16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:125_3"/><span class="lab"><font color="Green" title="E144">A5</font></span>: 
 for <font color="Olive" title="b1">x0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<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> <font color="Olive" title="b1">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x0</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"> ( <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies  <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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:125_3_1"/><span class="lab"><font color="Green" title="E145">A6</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:125_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E3:125_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A7</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E4:125_3_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI: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><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T131" title="XXREAL_1:th.131">XXREAL_1:131</a></span>;<br/>
<a NAME="E5:125_3_1"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:125_3_1"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sincos10.html#E5"><span class="lab"><font color="Green" title="E5">Lm3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E7:125_3_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_3_1"><span class="lab"><font color="Green" title="E145">A6</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E8:125_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E9:125_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_3_1"><span class="lab"><font color="Green" title="E146">A7</font></span></a></span>;<br/>
<a NAME="E10:125_3_1"/><span class="kw">then </span>
<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"><font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T9" title="COMPTRIG:th.9">COMPTRIG:9</a></span>;<br/>
<a NAME="E11:125_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A8</font></span>: 
<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> <font color="Maroon" title="c5">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T78" title="SIN_COS:th.78">SIN_COS:78</a></span>;<br/>
<a NAME="E12:125_3_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T46" title="XXREAL_1:th.46">XXREAL_1:46</a></span>;<br/>
<a NAME="E13:125_3_1"/><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> <font color="Maroon" title="c5">x0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:125_3_1"><span class="lab"><font color="Green" title="E145">A6</font></span></a>, <a class="ref" href="comptrig.html#T11" title="COMPTRIG:th.11">COMPTRIG:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:125_3_1"/>
 <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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:125_3_1"><span class="lab"><font color="Green" title="E147">A8</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/>


</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c5">x0</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"> ( <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<font color="Maroon" title="c5">x0</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E2:125_3"/><span class="lab"><font color="Green" title="E145">A9</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<font color="Maroon" title="c5">x0</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<font color="Maroon" title="c5">x0</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p4"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E143">A4</font></span></a>, <a class="txt" href="#E2:125_3"><span class="lab"><font color="Green" title="E145">A9</font></span></a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a>, <a class="ref" href="xxreal_1.html#T22" title="XXREAL_1:th.22">XXREAL_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sincos10.html#T7" target="_self" title="SINCOS10:th.7">Th7</a>, <a class="ref" href="fdiff_2.html#T16" title="FDIFF_2:th.16">FDIFF_2:16</a></span>
<br/>.= 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>,<font color="Maroon" title="c5">x0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_3"><span class="lab"><font color="Green" title="E145">A9</font></span></a>, <a class="ref" href="sincos10.html#T7" target="_self" title="SINCOS10:th.7">Th7</a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>
<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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon" title="c5">x0</font></span>)</span> <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="#E2:125_3"><span class="lab"><font color="Green" title="E145">A9</font></span></a>, <a class="ref" href="sincos10.html#T7" target="_self" title="SINCOS10:th.7">Th7</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:125_3"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><span class="p2">(<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="p3"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<font color="Maroon" title="c5">x0</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_3"><span class="lab"><font color="Green" title="E145">A9</font></span></a>, <a class="txt" href="#E1:125_3"><span class="lab"><font color="Green" title="E144">A5</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/></div><span class="kw">end;</span></div>
<a NAME="E6:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A10</font></span>: 
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E141">A2</font></span></a>, <a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E143">A4</font></span></a>, <a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E142">A3</font></span></a>, <a class="txt" href="sincos10.html#E38"><span class="lab"><font color="Green" title="E31">Lm23</font></span></a>, <a class="ref" href="fdiff_2.html#T48" title="FDIFF_2:th.48">FDIFF_2:48</a></span>;<br/>
<a NAME="E7:125"/><span class="lab"><font color="Green" title="E145">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:125_4"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E2:125_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:125"><span class="lab"><font color="Green" title="E144">A10</font></span></a>, <a class="ref" href="fdiff_1.html#D6" title="FDIFF_1:def.6">FDIFF_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:125_4"/>
<a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T72" title="RELAT_1:th.72">RELAT_1:72</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/>


</div><span class="kw">end;</span></div>
<a NAME="E8:125"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></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="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E141">A2</font></span></a>, <a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E142">A3</font></span></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:125"/>
 <a href="sincos10.html#K3" title="SINCOS10:func.3">arccosec1</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E145">A11</font></span></a>, <a class="ref" href="fdiff_1.html#D6" title="FDIFF_1:def.6">FDIFF_1:def 6</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/>


</div>
