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

<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font> being   <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>  st <font color="Olive" title="b3">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> holds <br/>( <font color="Olive" title="b3">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b2">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b2">M</font></span>)</span>,<font color="Olive" title="b3">f</font>) )</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font> being   <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>  st <font color="Olive" title="b2">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> holds <br/>( <font color="Olive" title="b2">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b1">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b1">M</font></span>)</span>,<font color="Olive" title="b2">f</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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>  st <font color="Olive" title="b1">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Olive" title="b1">f</font>) )</span><br/><span class="kw">let </span><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#K6" title="NUMBERS:func.6">ExtREAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies ( <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E4:44"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
(  <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#K2" title="MESFUNC2:func.2">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E5:44"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="mesfunc2.html#D2" onmouseover="rs('mesfunc2/D2')" onmouseout="rh()">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3" onmouseover="rs('mesfunc2/D3')" onmouseout="rh()">MESFUNC2:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">E1</font> = <font color="Maroon" title="c5">E</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T27" target="_self" onmouseover="rs('mesfun14/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E7:44"/>
(  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="mesfun11.html#T10" onmouseover="rs('mesfun11/T10')" onmouseout="rh()">MESFUN11:10</a></span>;<br/>
<a NAME="E8:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
(  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<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"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="mesfun14.html#T37" target="_self" onmouseover="rs('mesfun14/T37')" onmouseout="rh()">Th37</a>, <a class="ref" href="mesfun11.html#T5" onmouseover="rs('mesfun11/T5')" onmouseout="rh()">MESFUN11:5</a></span>;<br/>
<a NAME="E9:44"/>
<font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c6">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="mesfun14.html#T30" target="_self" onmouseover="rs('mesfun14/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:44"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>)</span><br/>

<a NAME="E11:44"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</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#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D16" onmouseover="rs('mesfunc5/D16')" onmouseout="rh()">MESFUNC5:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:44"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="mesfunc5.html#D16" onmouseover="rs('mesfunc5/D16')" onmouseout="rh()">MESFUNC5:def 16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
