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

<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> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>













<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c7">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E28">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>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">E</font> <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>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green" title="E30">B1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">E</font> <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>and </b><br/><a NAME="E5:42"/><i><font color="Green" title="E31">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/>( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c10">C</font> =  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E6:42"/><i><font color="Green" title="E32">P1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</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="mesfunc7.html#E2:42"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E7:42"/>
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</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="mesfunc7.html#E4:42"><i><font color="Green" title="E30">B1</font></i></a>, <a class="ref" href="mesfunc7.html#T24" target="_self" title="MESFUNC7:th.24">AG</a></i>;<br/>
<a NAME="E8:42"/><b>then </b><i><font color="Green" title="E33">P3</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</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="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E9:42"/><b>then </b><i><font color="Green" title="E34">P2</font></i>: 
( <font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <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="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <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="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>
<a NAME="E10:42"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:42_1"/><i><font color="Green" title="E35">M1</font></i>: 
<font color="Maroon" title="c11">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="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b><i><font color="Green" title="E36">M2</font></i>: 
<font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:42_1"/><b>then </b><i><font color="Green" title="E37">M3</font></i>: 
( <font color="Maroon" title="c11">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#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:42_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E5:42"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E6:42_1"/><b>then </b><i><font color="Green" title="E38">M4</font></i>: 
<font color="Maroon" title="c6">a</font> <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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:42_1"><i><font color="Green" title="E37">M3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:42_1"><i><font color="Green" title="E35">M1</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:42_1"><i><font color="Green" title="E36">M2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:42_1"><i><font color="Green" title="E37">M3</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E8:42_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> <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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:42_1"><i><font color="Green" title="E38">M4</font></i></a>, <a class="ref" href="extreal2.html#T4" title="EXTREAL2:th.4">EXTREAL2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:42"/><b>then </b>
<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="c5">E</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T1" target="_self" title="MESFUNC7:th.1">LMXZ</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">E1</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="E13:42"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c11">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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="c5">E</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c5">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">E1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:42"><i><font color="Green" title="E32">P1</font></i></a>, <a class="txt" href="mesfunc7.html#E9:42"><i><font color="Green" title="E34">P2</font></i></a>, <a class="ref" href="mesfunc7.html#T3" target="_self" title="MESFUNC7:th.3">LMXY</a></i>;<br/>
<a NAME="E14:42"/>
 <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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E15:42"/><b>then </b><i><font color="Green" title="E36">A6</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E3:42"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E16:42"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E17:42"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E18:42"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E19:42"/><b>then </b><i><font color="Green" title="E37">A8</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="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E20:42"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> holds <br/><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="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:42_2"/><i><font color="Green" title="E38">M0</font></i>: 
<font color="Maroon" title="c12">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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span>
 ;<br/>

<a NAME="E2:42_2"/><b>then </b><i><font color="Green" title="E39">M2</font></i>: 
<font color="Maroon" title="c12">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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E15:42"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="mesfunc7.html#E19:42"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:42_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:42_2"/><b>then </b><i><font color="Green" title="E40">M3</font></i>: 
( <font color="Maroon" title="c12">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#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:42_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E5:42"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E6:42_2"/><b>then </b><i><font color="Green" title="E41">M4</font></i>: 
<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="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:42_2"><i><font color="Green" title="E40">M3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:42_2"><i><font color="Green" title="E38">M0</font></i></a>, <a class="txt" href="mesfunc7.html#E15:42"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="mesfunc7.html#E19:42"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:42_2"><i><font color="Green" title="E39">M2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:42_2"><i><font color="Green" title="E40">M3</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E8:42_2"/>
<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="c5">E</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:42_2"><i><font color="Green" title="E41">M4</font></i></a>, <a class="ref" href="extreal2.html#T4" title="EXTREAL2:th.4">EXTREAL2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:42"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <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="c5">E</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T1" target="_self" title="MESFUNC7:th.1">LMXZ</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">E2</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="E23:42"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c12">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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="c5">E</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c5">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">E2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">E2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:42"><i><font color="Green" title="E32">P1</font></i></a>, <a class="txt" href="mesfunc7.html#E9:42"><i><font color="Green" title="E34">P2</font></i></a>, <a class="ref" href="mesfunc7.html#T3" target="_self" title="MESFUNC7:th.3">LMXY</a></i>;<br/>
<a NAME="E24:42"/><i><font color="Green" title="E39">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">a1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">b1</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> &amp; <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="c5">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> &amp; <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="c5">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E13:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E23:42"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="mesfunc7.html#E15:42"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="mesfunc7.html#E19:42"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<a NAME="E25:42"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">E</font>
 
;<br/>
<a NAME="E26:42"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:42"><i><font color="Green" title="E30">B1</font></i></a>, <a class="ref" href="mesfunc7.html#T25" target="_self" title="MESFUNC7:th.25">AH</a></i>;<br/>
<b>hence </b><a NAME="E27:42"/>
( <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">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="c5">E</font></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">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="c5">E</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">E</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E13:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E23:42"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="mesfunc7.html#E24:42"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="mesfunc7.html#E8:42"><i><font color="Green" title="E33">P3</font></i></a>, <a class="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>


</div>
