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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>









<b>assume </b><a NAME="E1:116"/><i><font color="Green" title="E110">A1</font></i>: 
(  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">A'</font> = <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font>;<br/>
<b>set </b><font color="Maroon" title="c8">B'</font> = <font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font>;<br/>
<a NAME="E2:116"/>
<font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T89" title="XBOOLE_1:th.89">XBOOLE_1:89</a></i>;<br/>
<a NAME="E3:116"/><b>then </b><i><font color="Green" title="E111">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:116"><i><font color="Green" title="E110">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T86" target="_self" title="MESFUNC5:th.86">Th86</a>, <a class="ref" href="mesfunc5.html#T87" target="_self" title="MESFUNC5:th.87">Th87</a></i>;<br/>
<a NAME="E4:116"/>
( <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:116"><i><font color="Green" title="E110">A1</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<b>hence </b><a NAME="E5:116"/>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:116"><i><font color="Green" title="E111">A2</font></i></a>, <a class="ref" href="supinf_2.html#T20" title="SUPINF_2:th.20">SUPINF_2:20</a></i>;<br/>


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