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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> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<a NAME="E1:100"/>
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra8.html#E83"><i><font color="Green" title="E61">Lm14</font></i></a></i>;<br/>
<a NAME="E2:100"/><b>then </b><i><font color="Green" title="E66">A1</font></i>: 
(  <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><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="E3:100"/>
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T4" title="FCONT_3:th.4">FCONT_3:4</a>, <a class="ref" href="subset_1.html#D4" title="SUBSET_1:def.4">SUBSET_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:100"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</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"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra8.html#E2:100"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="integra8.html#T26" target="_self" title="INTEGRA8:th.26">Th26</a>, <a class="ref" href="integra8.html#T29" target="_self" title="INTEGRA8:th.29">Th29</a>, <a class="ref" href="integra8.html#T68" target="_self" title="INTEGRA8:th.68">Th68</a></i>;<br/>


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