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<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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds   <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span></span><br/>



<a NAME="E1:95"/>
( <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra8.html#E45"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a>, <a class="ref" href="integra5.html#T10" onmouseover="rs('integra5/T10')" onmouseout="rh()">INTEGRA5:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:95"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="integra8.html#E45"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a>, <a class="ref" href="integra8.html#T26" target="_self" onmouseover="rs('integra8/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="integra8.html#T29" target="_self" onmouseover="rs('integra8/T29')" onmouseout="rh()">Th29</a>, <a class="ref" href="integra8.html#T68" target="_self" onmouseover="rs('integra8/T68')" onmouseout="rh()">Th68</a>, <a class="ref" href="integra5.html#T11" onmouseover="rs('integra5/T11')" onmouseout="rh()">INTEGRA5:11</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"> verum</span><br/>


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