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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E43">Th38</font></span>: <a NAME="T38"><span class="comment"><font color="firebrick">:: MEASUR13:38</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">X</font> being   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b3">S</font> being    <a href="measur13.html#M2" title="MEASUR13:mode.2">sigmaFieldFamily</a> of <font color="Olive" title="b2">X</font><br/>  for <font color="Olive" title="b4">M</font> being    <a href="measur13.html#M3" title="MEASUR13:mode.3">sigmaMeasureFamily</a> of <font color="Olive" title="b3">S</font><br/>  for <font color="Olive" title="b5">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="measur13.html#K3" title="MEASUR13:func.3">CarProduct</a> <font color="Olive" title="b2">X</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Olive" title="b4">M</font> is  <a href="measur13.html#V1" title="MEASUR13:attr.1">sigma_finite</a>  &amp; <font color="Olive" title="b5">f</font> <a href="measur13.html#R1" title="MEASUR13:pred.1">is_Sequentially_integrable_on</a> <font color="Olive" title="b4">M</font> &amp; <font color="Olive" title="b5">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <font color="Olive" title="b4">M</font> holds <br/> for <font color="Olive" title="b6">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> ex <font color="Olive" title="b7">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="measur13.html#K3" title="MEASUR13:func.3">CarProduct</a> <span class="p2">(<span class="default"><a href="measur13.html#K4" title="MEASUR13:func.4">SubFin</a> (<font color="Olive" title="b2">X</font>,<span class="p3">(<span class="default"><font color="Olive" title="b6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( <font color="Olive" title="b7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur13.html#K14" title="MEASUR13:func.14">FSqIntg</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b5">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Olive" title="b6">k</font></span>)</span> &amp; <font color="Olive" title="b7">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p1">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<span class="p2">(<span class="default"><font color="Olive" title="b6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )</div></div>
