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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>assume </b><a NAME="E1:135"/><i><font color="Green" title="E121">A1</font></i>: 
 ex <font color="Olive" title="b1">A</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">A</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="Olive" title="b1">A</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:135"/><i><font color="Green" title="E122">A2</font></i>: 
( <font color="Maroon" title="c5">A</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">A</font> )
 ;<br/>
<a NAME="E4:135"/><i><font color="Green" title="E123">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E5:135"/><i><font color="Green" title="E124">A4</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a>, <a class="ref" href="mesfunc2.html#T29" title="MESFUNC2:th.29">MESFUNC2:29</a></i>;<br/>
<a NAME="E6:135"/><b>then </b><i><font color="Green" title="E125">A5</font></i>: 
 <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T28" title="MESFUNC2:th.28">MESFUNC2:28</a></i>;<br/>
<a NAME="E7:135"/><i><font color="Green" title="E126">A6</font></i>: 
(  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <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">A</font> &amp;  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <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">A</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="ref" href="mesfunc2.html#T27" title="MESFUNC2:th.27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28" title="MESFUNC2:th.28">MESFUNC2:28</a></i>;<br/>
<a NAME="E8:135"/><i><font color="Green" title="E127">A7</font></i>: 
 <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green" title="E31">Lm1</font></i></a></i>;<br/>
<a NAME="E9:135"/><b>then </b><i><font color="Green" title="E128">A8</font></i>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:135"><i><font color="Green" title="E126">A6</font></i></a>, <a class="ref" href="mesfunc5.html#T85" target="_self" title="MESFUNC5:th.85">Th85</a></i>;<br/>
<a NAME="E10:135"/><i><font color="Green" title="E129">A9</font></i>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T26" title="MESFUNC2:th.26">MESFUNC2:26</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:135_1"/><i><font color="Green" title="E130">A10</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 ;<br/><div><i><font color="Green" title="E131">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:135_1_1"/><i><font color="Green" title="E132">A12</font></i>: 
<font color="Maroon" title="c6">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"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 ;<br/><a NAME="E2:135_1_1"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/><a NAME="E3:135_1_1"/><b>then </b><i><font color="Green" title="E133">A13</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> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/><a NAME="E4:135_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E1:135_1_1"><i><font color="Green" title="E132">A12</font></i></a>, <a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:135_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:135_1_1"><i><font color="Green" title="E133">A13</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div><a NAME="E3:135_1"/><b>then </b><i><font color="Green" title="E132">A14</font></i>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:135_1_2"/>
<font color="Maroon" title="c6">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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span>
 ;<br/><a NAME="E2:135_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E2:135_1"><i><font color="Green" title="E131">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:135_1_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="mesfunc2.html#T21" title="MESFUNC2:th.21">MESFUNC2:21</a></i>;<br/></div><b>end;</b></div><a NAME="E5:135_1"/><b>then </b><i><font color="Green" title="E133">A15</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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E5:135"><i><font color="Green" title="E124">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E6:135"><i><font color="Green" title="E125">A5</font></i></a>, <a class="ref" href="mesfunc5.html#T93" target="_self" title="MESFUNC5:th.93">Th93</a></i>;<br/><a NAME="E6:135_1"/><i><font color="Green" title="E134">A16</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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:135_1"><i><font color="Green" title="E130">A10</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></i>;<br/><a NAME="E7:135_1"/>
 <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green" title="E31">Lm1</font></i></a></i>;<br/><a NAME="E8:135_1"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:135"><i><font color="Green" title="E126">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E8:135"><i><font color="Green" title="E127">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E109"><i><font color="Green" title="E103">Lm11</font></i></a></i>;<br/><a NAME="E9:135_1"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:135_1"><i><font color="Green" title="E134">A16</font></i></a>, <a class="ref" href="supinf_2.html#T8" title="SUPINF_2:th.8">SUPINF_2:8</a></i>;<br/><a NAME="E10:135_1"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:135"><i><font color="Green" title="E129">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E3:135_1"><i><font color="Green" title="E132">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><b>hence </b><a NAME="E11:135_1"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:135"><i><font color="Green" title="E124">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E5:135_1"><i><font color="Green" title="E133">A15</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E12:135"/><i><font color="Green" title="E130">A17</font></i>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:135_2"/>
<font color="Maroon" title="c6">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"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 ;<br/><a NAME="E2:135_2"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:135_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:135"/><b>then </b><i><font color="Green" title="E131">A18</font></i>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/>
<a NAME="E15:135"/><i><font color="Green" title="E132">A19</font></i>: 
 <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green" title="E31">Lm1</font></i></a></i>;<br/>
<a NAME="E16:135"/><b>then </b><i><font color="Green" title="E133">A20</font></i>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:135"><i><font color="Green" title="E126">A6</font></i></a>, <a class="ref" href="mesfunc5.html#T85" target="_self" title="MESFUNC5:th.85">Th85</a></i>;<br/>
<a NAME="E17:135"/><i><font color="Green" title="E134">A21</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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:135"><i><font color="Green" title="E122">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:135"><i><font color="Green" title="E123">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:135"><i><font color="Green" title="E126">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E8:135"><i><font color="Green" title="E127">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E15:135"><i><font color="Green" title="E132">A19</font></i></a>, <a class="txt" href="mesfunc5.html#E109"><i><font color="Green" title="E103">Lm11</font></i></a></i>;<br/>
<a NAME="E18:135"/>
(  <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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span></span>)</span> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:135"><i><font color="Green" title="E130">A17</font></i></a>, <a class="ref" href="mesfunc5.html#T102" target="_self" title="MESFUNC5:th.102">Th102</a></i>;<br/>
<a NAME="E19:135"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:135"><i><font color="Green" title="E124">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E10:135"><i><font color="Green" title="E129">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E14:135"><i><font color="Green" title="E131">A18</font></i></a>, <a class="ref" href="mesfunc5.html#T94" target="_self" title="MESFUNC5:th.94">Th94</a></i>;<br/>
<a NAME="E20:135"/><b>then </b>
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:135"><i><font color="Green" title="E128">A8</font></i></a>, <a class="txt" href="mesfunc5.html#E16:135"><i><font color="Green" title="E133">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E17:135"><i><font color="Green" title="E134">A21</font></i></a>, <a class="ref" href="extreal2.html#T23" title="EXTREAL2:th.23">EXTREAL2:23</a></i>;<br/>
<a NAME="E21:135"/><b>then </b>
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<b>hence </b><a NAME="E22:135"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:135"><i><font color="Green" title="E121">A1</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></i>;<br/>


</div>
