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<div><span class="kw">theorem </span><a NAME="T40"><span class="comment"><font color="firebrick">:: MEASUR13:40</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> holds <br/> ex <font color="Olive" title="b7">Fk1</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">(<span class="default"><a href="measur13.html#K3" title="MEASUR13:func.3">CarProduct</a> <span class="p3">(<span class="default"><a href="measur13.html#K4" title="MEASUR13:func.4">SubFin</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measur13.html#K1" title="MEASUR13:func.1">ElmFin</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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ex <font color="Olive" title="b8">Gk1</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> ex <font color="Olive" title="b9">Fk</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( <font color="Olive" title="b8">Gk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">Fk1</font> &amp; <font color="Olive" title="b8">Gk1</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="b9">Fk</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> <span class="p2">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b9">Fk</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfun12.html#K7" title="MESFUN12:func.7">Integral2</a> (<span class="p1">(<span class="default"><a href="measur13.html#K8" title="MEASUR13:func.8">ElmFin</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>,<font color="Olive" title="b7">Fk1</font>) &amp; <font color="Olive" title="b8">Gk1</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> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p2">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</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>,<font color="Olive" title="b8">Gk1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> (<span class="p2">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p3">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measur13.html#K8" title="MEASUR13:func.8">ElmFin</a> (<font color="Olive" title="b4">M</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>,<font color="Olive" title="b7">Fk1</font>) &amp; (  for <font color="Olive" title="b10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="measur13.html#K3" title="MEASUR13:func.3">CarProduct</a> <span class="p1">(<span class="default"><a href="measur13.html#K4" title="MEASUR13:func.4">SubFin</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b6">k</font>)</span>)</span> holds   <a href="mesfun12.html#K2" title="MESFUN12:func.2">ProjPMap1</a> (<font color="Olive" title="b7">Fk1</font>,<font color="Olive" title="b10">x</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur13.html#K8" title="MEASUR13:func.8">ElmFin</a> (<font color="Olive" title="b4">M</font>,<span class="p1">(<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>) ) &amp; (  for <font color="Olive" title="b10">U</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measur13.html#K10" title="MEASUR13:func.10">Prod_Field</a> <span class="p1">(<span class="default"><a href="measur13.html#K5" title="MEASUR13:func.5">SubFin</a> (<font color="Olive" title="b3">S</font>,<font color="Olive" title="b6">k</font>)</span>)</span> holds  <font color="Olive" title="b9">Fk</font> is <font color="Olive" title="b10">U</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b9">Fk</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>,<font color="Olive" title="b6">k</font>)</span>)</span> &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> (<span class="p2">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p3">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measur13.html#K8" title="MEASUR13:func.8">ElmFin</a> (<font color="Olive" title="b4">M</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>,<font color="Olive" title="b7">Fk1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p2">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<font color="Olive" title="b9">Fk</font>) &amp; <font color="Olive" title="b9">Fk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <span class="p1">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p2">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p2">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</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>,<font color="Olive" title="b8">Gk1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measur13.html#K12" title="MEASUR13:func.12">Prod_Measure</a> <span class="p2">(<span class="default"><a href="measur13.html#K7" title="MEASUR13:func.7">SubFin</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b6">k</font>)</span>)</span></span>)</span>,<font color="Olive" title="b9">Fk</font>) )</div></div>
