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<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>, <font color="Maroon" title="c5">g</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#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>







<b>assume </b><a NAME="E1:141"/><i><font color="Green" title="E126">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">g</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">A</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="E3:141"/><i><font color="Green" title="E127">A2</font></i>: 
( <font color="Maroon" title="c6">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> <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">A</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">B</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="E5:141"/><i><font color="Green" title="E128">A3</font></i>: 
( <font color="Maroon" title="c7">B</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="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a>, <a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a></i>;<br/>
<a NAME="E6:141"/>
( <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"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <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"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</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="E7:141"/><b>then </b>
( <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">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:141"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E5:141"><i><font color="Green" title="E128">A3</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/>
<a NAME="E8:141"/><b>then </b><i><font color="Green" title="E129">A4</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c7">B</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E5:141"><i><font color="Green" title="E128">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E3:141"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E46"><i><font color="Green" title="E42">Lm2</font></i></a></i>;<br/>
<a NAME="E9:141"/><i><font color="Green" title="E130">A5</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a></i>;<br/>
<div><i><font color="Green" title="E131">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>now </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="E1:141_1"/><i><font color="Green" title="E132">A7</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/><a NAME="E2:141_1"/><b>then </b><i><font color="Green" title="E133">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/><a NAME="E3:141_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E4:141_1"/><b>then </b><i><font color="Green" title="E134">A9</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T7" title="EXTREAL2:th.7">EXTREAL2:7</a>, <a class="txt" href="mesfunc5.html#E2:141_1"><i><font color="Green" title="E133">A8</font></i></a></i>;<br/><a NAME="E5:141_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141_1"><i><font color="Green" title="E132">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:141_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a>, <a class="txt" href="mesfunc5.html#E4:141_1"><i><font color="Green" title="E134">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:141"/><b>then </b><i><font color="Green" title="E132">A10</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>now </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="E1:141_2"/><i><font color="Green" title="E133">A11</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> <font color="Maroon" title="c5">g</font>
 ;<br/><a NAME="E2:141_2"/><i><font color="Green" title="E134">A12</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E3:141_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141_2"><i><font color="Green" title="E133">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:141_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a>, <a class="txt" href="mesfunc5.html#E2:141_2"><i><font color="Green" title="E134">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:141"/><b>then </b><i><font color="Green" title="E133">A13</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_3"><b>now </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="E1:141_3"/><i><font color="Green" title="E134">A14</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> <font color="Maroon" title="c4">f</font>
 ;<br/><a NAME="E2:141_3"/><i><font color="Green" title="E135">A15</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E3:141_3"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:141_3"><i><font color="Green" title="E134">A14</font></i></a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:141_3"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a>, <a class="txt" href="mesfunc5.html#E2:141_3"><i><font color="Green" title="E135">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:141"/><b>then </b><i><font color="Green" title="E134">A16</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="txt" href="mesfunc5.html#E9:141"><i><font color="Green" title="E130">A5</font></i></a></i>;<br/>
<div><i><font color="Green" title="E135">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:141_4"/>
<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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:141_4"/><b>then </b>
<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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></i>;<br/><a NAME="E3:141_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:141"><i><font color="Green" title="E131">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:141_4"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc2.html#T21" title="MESFUNC2:th.21">MESFUNC2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:141"/><i><font color="Green" title="E136">A18</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E18:141"/>
 <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:141"><i><font color="Green" title="E129">A4</font></i></a>, <a class="ref" href="mesfunc2.html#T28" title="MESFUNC2:th.28">MESFUNC2:28</a></i>;<br/>
<a NAME="E19:141"/><b>then </b><i><font color="Green" title="E137">A19</font></i>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T93" target="_self" title="MESFUNC5:th.93">Th93</a>, <a class="txt" href="mesfunc5.html#E16:141"><i><font color="Green" title="E135">A17</font></i></a>, <a class="txt" href="mesfunc5.html#E17:141"><i><font color="Green" title="E136">A18</font></i></a>, <a class="txt" href="mesfunc5.html#E8:141"><i><font color="Green" title="E129">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">C</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="E21:141"/><i><font color="Green" title="E138">A20</font></i>: 
( <font color="Maroon" title="c8">C</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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <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="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T84" target="_self" title="MESFUNC5:th.84">Th84</a>, <a class="txt" href="mesfunc5.html#E5:141"><i><font color="Green" title="E128">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E3:141"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a></i>;<br/>
<a NAME="E22:141"/>
( <font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:141"><i><font color="Green" title="E138">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E8:141"><i><font color="Green" title="E129">A4</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E23:141"/><b>then </b>
(  <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="c8">C</font></span>)</span> <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="c6">A</font></span>)</span> &amp;  <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="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">C</font></span>)</span> <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="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:141"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E5:141"><i><font color="Green" title="E128">A3</font></i></a>, <a class="ref" href="mesfunc5.html#T89" target="_self" title="MESFUNC5:th.89">Th89</a>, <a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a></i>;<br/>
<a NAME="E24:141"/><b>then </b><i><font color="Green" title="E139">A21</font></i>: 
(  <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="c8">C</font></span>)</span> <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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <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="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">C</font></span>)</span> <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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:141"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E5:141"><i><font color="Green" title="E128">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E13:141"><i><font color="Green" title="E133">A13</font></i></a>, <a class="txt" href="mesfunc5.html#E15:141"><i><font color="Green" title="E134">A16</font></i></a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></i>;<br/>
<a NAME="E25:141"/><i><font color="Green" title="E140">A22</font></i>: 
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:141"><i><font color="Green" title="E126">A1</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></i>;<br/>
<a NAME="E26:141"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T8" title="SUPINF_2:th.8">SUPINF_2:8</a></i>;<br/>
<a NAME="E27:141"/><b>then </b><i><font color="Green" title="E141">A23</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E28:141"/>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:141"><i><font color="Green" title="E138">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E11:141"><i><font color="Green" title="E132">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E24:141"><i><font color="Green" title="E139">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E25:141"><i><font color="Green" title="E140">A22</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/>
<a NAME="E29:141"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:141"><i><font color="Green" title="E141">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<b>hence </b><a NAME="E30:141"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:141"><i><font color="Green" title="E129">A4</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a>, <a class="txt" href="mesfunc5.html#E19:141"><i><font color="Green" title="E137">A19</font></i></a></i>;<br/>


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