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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">E</font>, <font color="Maroon" title="c6">A</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="E1:117"/><i><font color="Green" title="E111">A1</font></i>: 
( <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">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#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font>;<br/>
<a NAME="E2:117"/>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E3:117"/><b>then </b><i><font color="Green" title="E112">A2</font></i>: 
 <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">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">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> <font color="Maroon" title="c6">A</font></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="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:117"><i><font color="Green" title="E111">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T87" target="_self" title="MESFUNC5:th.87">Th87</a></i>;<br/>
<a NAME="E4:117"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<a NAME="E1:117_1"/>
<font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E2:117_1"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then </b><i><font color="Green" title="E113">A3</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:117"><i><font color="Green" title="E111">A1</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E4:117_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span></span>)</span> holds <br/><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">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:117_1"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:117_1"><i><font color="Green" title="E113">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:117"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="c5">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:117"><i><font color="Green" title="E111">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:117"><i><font color="Green" title="E112">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T88" target="_self" title="MESFUNC5:th.88">Th88</a></i>;<br/>
<b>hence </b><a NAME="E6:117"/>
 <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">E</font> <a href="measure1.html#K4" title="MEASURE1:func.4">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>


</div>
