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<div><span class="kw">theorem </span><a NAME="T68"><span class="comment"><font color="firebrick">:: MESFUN17:68</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">J</font>, <font color="Olive" title="b3">K</font> being   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><br/>  for <font color="Olive" title="b4">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><span class="p2"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a><br/>  for <font color="Olive" title="b5">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b6">Gzy</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>,<font color="Olive" title="b3">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>,<font color="Olive" title="b3">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> &amp; <font color="Olive" title="b6">Gzy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b5">g</font></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">I</font> holds <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b6">Gzy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b6">Gzy</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b6">Gzy</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">I</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b6">Gzy</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">I</font> &amp; <span class="p1">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b5">g</font></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">I</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p5">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b5">g</font></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">I</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<font color="Olive" title="b6">Gzy</font>,<font color="Olive" title="b1">I</font>) &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="mesfun12.html#K8" title="MESFUN12:func.8">Prod_Measure</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b5">g</font></span>)</span>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<font color="Olive" title="b6">Gzy</font>,<font color="Olive" title="b1">I</font>) )</div></div>
