<?xml version="1.0"?>
<div class="add">

<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">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="E1:113"/><i><font color="Green" title="E107">A1</font></i>: 
(  ex <font color="Olive" title="b1">E</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">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="Olive" title="b1">E</font> ) &amp; <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">E</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="E3:113"/><i><font color="Green" title="E108">A2</font></i>: 
( <font color="Maroon" title="c6">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="c6">E</font> )
 ;<br/>
<b>set </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>;<br/>
<a NAME="E4:113"/><i><font color="Green" title="E109">A3</font></i>: 
<font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:113"/><b>then </b><i><font color="Green" title="E110">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:113"><i><font color="Green" title="E108">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/>
<a NAME="E6:113"/><i><font color="Green" title="E111">A5</font></i>: 
<font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <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">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:113"><i><font color="Green" title="E108">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:113"/>
<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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:113"><i><font color="Green" title="E108">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:113"><i><font color="Green" title="E109">A3</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:113"/><b>then </b><i><font color="Green" title="E112">A6</font></i>: 
<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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:113"><i><font color="Green" title="E110">A4</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self" title="MESFUNC5:th.48">Th48</a></i>;<br/>
<i><font color="Green" title="E113">A7</font></i>:  <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> <font color="Maroon" title="c5">A</font></span>)</span> = 
<font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:113"><i><font color="Green" title="E108">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:113"><i><font color="Green" title="E108">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:113"><i><font color="Green" title="E109">A3</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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></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="E10:113"/>
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> <font color="Maroon" title="c5">A</font></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> <font color="Maroon" title="c5">A</font></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> <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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:113_2"/><i><font color="Green" title="E114">A8</font></i>: 
<font color="Maroon" title="c8">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> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>

<a NAME="E2:113_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">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="Maroon" title="c8">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="E3:113_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:113"><i><font color="Green" title="E113">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E1:113_2"><i><font color="Green" title="E114">A8</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:113"/><b>then </b><i><font color="Green" title="E114">A9</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:113"><i><font color="Green" title="E112">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E9:113"><i><font color="Green" title="E113">A7</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E12:113"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</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#E1:113"><i><font color="Green" title="E107">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E13:113"/>
 <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"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:113"><i><font color="Green" title="E111">A5</font></i></a>, <a class="txt" href="mesfunc5.html#E11:113"><i><font color="Green" title="E114">A9</font></i></a>, <a class="ref" href="mesfunc5.html#T85" target="_self" title="MESFUNC5:th.85">Th85</a></i>;<br/>


</div>
