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<b>let </b><font color="Maroon">X</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">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:101"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E2:101"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>and </b><br/><a NAME="E3:101"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E4:101"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">g</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>
<a NAME="E6:101"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E7:101"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:101"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:101"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:101"/><b>then </b>
 <a href="mesfunc1.html#K9">-</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T11">MESFUNC2:11</a></i>;<br/>
<a NAME="E11:101"/><b>then </b>
<font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E12:101"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T9">MESFUNC2:9</a></i>;<br/>
<a NAME="E13:101"/><i><font color="Green">E42</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><a NAME="E2:101_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:101_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E4:101_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:101"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>


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