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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:116"/><i><font color="Green">E110</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:116"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T89">XBOOLE_1:89</a></i>;<br/>
<a NAME="E3:116"/><i><font color="Green">E112</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T87" target="_self">Th87</a>, <a class="ref" href="mesfunc5.html#T86" target="_self">Th86</a>, <a class="txt" href="mesfunc5.html#E2:116"><i><font color="Green">E111</font></i></a>, <a class="txt" href="mesfunc5.html#E1:116"><i><font color="Green">E110</font></i></a></i>;<br/>
<a NAME="E4:116"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="measure1.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:116"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<b>hence </b><a NAME="E5:116"/>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:116"><i><font color="Green">E112</font></i></a>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/>


</div>
