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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>assume </b><a NAME="E1:122"/><i><font color="Green">E35</font></i>: 
(  ex <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">A</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="Olive">A</font> ) &amp; <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:122"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">A</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">A</font> )
 ;<br/>
<a NAME="E4:122"/><i><font color="Green">E38</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>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:122_1"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:122"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E7:122"/><i><font color="Green">E43</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#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:122_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E2:122_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E3:122_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc2.html#T21">MESFUNC2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:122"/><i><font color="Green">E47</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#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E10:122"/>
 <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<b>hence </b> <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, , , , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>
<br/>.= 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="measure6.html#T21">MEASURE6:21</a></i>
;<br/><br/>


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