<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Olive" title="b2">M</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Olive" title="b1">M</font> )</span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )</span><br/>





<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funcop_1.html#T58" title="FUNCOP_1:th.58">FUNCOP_1:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">ND</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">ND</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ND</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:37"/>
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> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="#E4:37"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="lpspace1.html#E31"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:37"/>
( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E4:37"><i><font color="Green" title="E23">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
