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<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"> ( <font color="Maroon" title="c1">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#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></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#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:85"/>
<font color="Maroon" title="c1">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#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <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#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:85"/><b>then </b>
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2 &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="integra8.html#T37" target="_self" title="INTEGRA8:th.37">Th37</a></i>;<br/>
<b>then </b> <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#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>,<font color="Maroon" title="c1">A</font> = 
<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="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="integra8.html#T69" target="_self" title="INTEGRA8:th.69">Th69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><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="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T83" title="SIN_COS:th.83">SIN_COS:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a>, <a class="ref" href="sin_cos.html#T83" title="SIN_COS:th.83">SIN_COS:83</a></i>
;<br/>
<b>hence </b><a NAME="E4:85"/>
 <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#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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