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

<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E1:90"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K16" title="SIN_COS:func.16">sin</a> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E2:90"/><span class="kw">then </span>
(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>  &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra8.html#T37" target="_self" onmouseover="rs('integra8/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<span class="kw">then </span> <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c3">A</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra8.html#T78" target="_self" onmouseover="rs('integra8/T78')" onmouseout="rh()">Th78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" onmouseover="rs('valued_1/T8')" onmouseout="rh()">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" onmouseover="rs('valued_1/T8')" onmouseout="rh()">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T76" onmouseover="rs('sin_cos/T76')" onmouseout="rh()">SIN_COS:76</a>, <a class="ref" href="sin_cos.html#T78" onmouseover="rs('sin_cos/T78')" onmouseout="rh()">SIN_COS:78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T76" onmouseover="rs('sin_cos/T76')" onmouseout="rh()">SIN_COS:76</a>, <a class="ref" href="sin_cos.html#T78" onmouseover="rs('sin_cos/T78')" onmouseout="rh()">SIN_COS:78</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T76" onmouseover="rs('sin_cos/T76')" onmouseout="rh()">SIN_COS:76</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:90"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
