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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">E</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><a NAME="E1:129"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>set </b><font color="Maroon">B</font> = <font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font>;<br/>
<a NAME="E2:129"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E3:129"/><i><font color="Green">E38</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:129"/><i><font color="Green">E39</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:129"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:129"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, , <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>
;<br/>
<b>hence </b><a NAME="E6:129"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, , <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/>


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