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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="E1:136"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 ;<br/>

<b>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="E3:136"/><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> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, </i>;<br/>
<a NAME="E4:136"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> &amp;  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:136"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a>, <a class="ref" href="mesfunc2.html#T29">MESFUNC2:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_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="E1:136_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>
 ;<br/><a NAME="E2:136_1"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:136_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51">EXTREAL2:51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:136"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<a NAME="E8:136"/><i><font color="Green">E41</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="E9:136"/><i><font color="Green">E42</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="E10:136"/><i><font color="Green">E43</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">A</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">A</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E11:136"/><b>then </b><i><font color="Green">E44</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:136"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</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#T26">MESFUNC2:26</a></i>;<br/>
<b>then </b><i><font color="Green">E47</font></i>:  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> = 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</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#K2">+</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> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , , </i>
;<br/>
<a NAME="E14:136"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<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="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T69">EXTREAL2:69</a>, </i>;<br/>
<a NAME="E15:136"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</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#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a>, </i>;<br/>
<b>hence </b><a NAME="E16:136"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a>, </i>;<br/>


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