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<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>assume </b><a NAME="E1:123"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:123"><i><font color="Green">E115</font></i></a>, <a class="ref" href="mesfunc5.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E3:123"/>
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:123"><i><font color="Green">E115</font></i></a>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/>
<b>hence </b><a NAME="E4:123"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:123"><i><font color="Green">E115</font></i></a>, <a class="ref" href="mesfunc5.html#T94" target="_self">Th94</a></i>;<br/>

<b>hence </b><a NAME="E5:123"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T83" target="_self">Th83</a>, <a class="txt" href="mesfunc5.html#E1:123"><i><font color="Green">E115</font></i></a></i>;<br/>


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