<?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="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 ;<br/>

<a NAME="E4:60"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:60"><i><font color="Green">E58</font></i></a>, <a class="txt" href="mesfunc5.html#E3:60"><i><font color="Green">E59</font></i></a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:60"><i><font color="Green">E57</font></i></a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green">E61</font></i>: 
 <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:60"><i><font color="Green">E58</font></i></a>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a></i>;<br/>
<a NAME="E7:60"/>
(  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:60"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:60"><i><font color="Green">E60</font></i></a>, <a class="txt" href="mesfunc5.html#E6:60"><i><font color="Green">E61</font></i></a>, <a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma42</font></i></a></i>;<br/>


</div>
