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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:13"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a>, <a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a></i>;<br/>
<a NAME="E2:13"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>
 
;<br/>
<a NAME="E3:13"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
( <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a>  <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="integra5.html#T10" title="INTEGRA5:th.10">INTEGRA5:10</a>, <a class="ref" href="integra5.html#T11" title="INTEGRA5:th.11">INTEGRA5:11</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green" title="E18">A3</font></i>: 
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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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"> ( <font color="Maroon" title="c2">x</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">x</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="integr11.html#T1" target="_self" title="INTEGR11:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:13"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="integr11.html#T1" target="_self" title="INTEGR11:th.1">Th1</a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> 
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E7:13"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K41" title="VALUED_1:func.41">^2</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="integr11.html#T1" target="_self" title="INTEGR11:th.1">Th1</a>, <a class="ref" href="integra5.html#T13" title="INTEGRA5:th.13">INTEGRA5:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
