<?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">f</font>, <font color="Olive" title="b3">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b4">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b3">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b4">A</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="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Olive" title="b3">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">A</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">f</font>, <font color="Olive" title="b2">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b2">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b3">A</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="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">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#K7" title="NUMBERS:func.7">ExtREAL</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">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</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">g</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c5">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</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">g</font> implies <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c5">A</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">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E6:12"/>
( <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E4:12"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="#E5:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="mesfunc2.html#T12" target="_self" title="MESFUNC2:th.12">Th12</a>, <a class="ref" href="mesfunc1.html#T41" title="MESFUNC1:th.41">MESFUNC1:41</a></i>;<br/>
<a NAME="E7:12"/><b>then </b>
(  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp;  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T11" target="_self" title="MESFUNC2:th.11">Th11</a></i>;<br/>
<a NAME="E8:12"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E3:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="mesfunc2.html#T7" target="_self" title="MESFUNC2:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="mesfunc2.html#T9" target="_self" title="MESFUNC2:th.9">Th9</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>
