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

<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</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">S1</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S2</font><br/>  for <font color="Olive" title="b5">E1</font>, <font color="Olive" title="b6">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span>  st <font color="Olive" title="b4">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b5">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b6">E2</font> holds <br/> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b5">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Olive" title="b6">E2</font></span>)</span>,<font color="Olive" title="b4">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b5">E1</font>,<font color="Olive" title="b4">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b6">E2</font>,<font color="Olive" title="b4">M2</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</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">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S2</font><br/>  for <font color="Olive" title="b4">E1</font>, <font color="Olive" title="b5">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span>  st <font color="Olive" title="b3">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b4">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b5">E2</font> holds <br/> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b4">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Olive" title="b5">E2</font></span>)</span>,<font color="Olive" title="b3">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b4">E1</font>,<font color="Olive" title="b3">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b5">E2</font>,<font color="Olive" title="b3">M2</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</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">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b3">E1</font>, <font color="Olive" title="b4">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="Olive" title="b2">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b3">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">E2</font> holds <br/> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b3">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Olive" title="b4">E2</font></span>)</span>,<font color="Olive" title="b2">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b3">E1</font>,<font color="Olive" title="b2">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b4">E2</font>,<font color="Olive" title="b2">M2</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</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">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b2">E1</font>, <font color="Olive" title="b3">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="Olive" title="b1">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b2">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">E2</font> holds <br/> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Olive" title="b3">E2</font></span>)</span>,<font color="Olive" title="b1">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b2">E1</font>,<font color="Olive" title="b1">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b3">E2</font>,<font color="Olive" title="b1">M2</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</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">E1</font>, <font color="Olive" title="b2">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b1">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E2</font> holds <br/> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Olive" title="b2">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b1">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Olive" title="b2">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">E1</font>, <font color="Maroon" title="c8">E2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</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"> ( <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c7">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">E2</font> implies  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:124"/><span class="lab"><font color="Green" title="E89">A1</font></span>: 
<font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:124"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
<font color="Maroon" title="c7">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">E2</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">  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">XX1</font> = <font color="Maroon" title="c1">X1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T7" onmouseover="rs('measure1/T7')" onmouseout="rh()">MEASURE1:7</a></span>;<br/>
<a NAME="E4:124"/><span class="lab"><font color="Green" title="E91">a3</font></span>: 
 <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:124"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="measur11.html#T93" target="_self" onmouseover="rs('measur11/T93')" onmouseout="rh()">Th92</a></span>;<br/>
<a NAME="E5:124"/><span class="lab"><font color="Green" title="E92">A3</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XX1</font> &amp;  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>) is <font color="Maroon" title="c9">XX1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:124"/><span class="lab"><font color="Green" title="E93">A4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XX1</font> &amp;  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>) is <font color="Maroon" title="c9">XX1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:124"/><span class="lab"><font color="Green" title="E94">A5</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XX1</font> &amp;  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) is <font color="Maroon" title="c9">XX1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Y1</font> =  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>) as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Y2</font> =  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>) as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>
<a NAME="E10:124"/>
 ex <font color="Olive" title="b1">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font> st <br/>( <font color="Olive" title="b1">Z</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"><span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span></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="c5">M1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">Y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">Z</font></span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">Y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">Z</font></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:124"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E6:124"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T78" onmouseover="rs('mesfunc5/T78')" onmouseout="rh()">MESFUNC5:78</a></span>;<br/>
<span class="kw">then </span> <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:124"><span class="lab"><font color="Green" title="E91">a3</font></span></a>, <a class="txt" href="#E7:124"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="ref" href="mesfunc5.html#T88" onmouseover="rs('mesfunc5/T88')" onmouseout="rh()">MESFUNC5:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:124"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="mesfunc5.html#T88" onmouseover="rs('mesfunc5/T88')" onmouseout="rh()">MESFUNC5:88</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:124"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">E2</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c8">E2</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:124"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T88" onmouseover="rs('mesfunc5/T88')" onmouseout="rh()">MESFUNC5:88</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>
