<?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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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><br/>  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b3">M1</font>,<font color="Olive" title="b4">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b4">M2</font>)</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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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><br/>  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b2">M1</font>,<font color="Olive" title="b3">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b3">M2</font>)</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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b1">M1</font>,<font color="Olive" title="b2">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b2">M2</font>)</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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Olive" title="b1">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b1">M2</font>)</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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <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">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c7">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>) implies <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:129"/><span class="lab"><font color="Green" title="E94">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="E2:129"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
<font color="Maroon" title="c7">F</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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>)
 ; <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"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</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="E4:129"/>
(  not <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> implies <font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="129_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:129_1"/>
 not <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</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="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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><br/>

<a NAME="E2:129_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:129_1"/>
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D8" onmouseover="rs('setfam_1/D8')" onmouseout="rh()">SETFAM_1:def 8</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><span class="kw">end;</span></div>
<a NAME="E5:129"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E6:129"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c11">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E8:129"/><span class="lab"><font color="Green" title="E96">d4</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/>
<a NAME="E9:129"/><span class="lab"><font color="Green" title="E97">d5</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E96">d4</font></span></a>, <a class="ref" href="measur11.html#T16" target="_self" onmouseover="rs('measur11/T16')" onmouseout="rh()">Th10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:129_2"/>
( ( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_2_1"><span class="kw">suppose </span></a><a NAME="E1:129_2_1"/><span class="lab"><font color="Green" title="E98">d8</font></span>: 
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</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"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:129_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c1">X1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E96">d4</font></span></a>, <a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E97">d5</font></span></a>, <a class="ref" href="measur11.html#T97" target="_self" onmouseover="rs('measur11/T97')" onmouseout="rh()">Th96</a></span>;<br/><span class="kw">hence </span><a NAME="E3:129_2_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_2_1"><span class="lab"><font color="Green" title="E98">d8</font></span></a>, <a class="ref" href="measur10.html#T33" onmouseover="rs('measur10/T33')" onmouseout="rh()">MEASUR10:33</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_2_2"><span class="kw">suppose </span></a><a NAME="E1:129_2_2"/>
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</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"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:129_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E96">d4</font></span></a>, <a class="ref" href="measur11.html#T97" target="_self" onmouseover="rs('measur11/T97')" onmouseout="rh()">Th96</a></span>;<br/><span class="kw">hence </span><a NAME="E3:129_2_2"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E97">d5</font></span></a>, <a class="ref" href="measur11.html#T99" target="_self" onmouseover="rs('measur11/T99')" onmouseout="rh()">Th98</a>, <a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_2_3"><span class="kw">suppose </span></a><a NAME="E1:129_2_3"/><span class="lab"><font color="Green" title="E98">d6</font></span>: 
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</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"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:129_2_3"/>
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/><a NAME="E3:129_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">d7</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E97">d5</font></span></a>, <a class="txt" href="#E1:129_2_3"><span class="lab"><font color="Green" title="E98">d6</font></span></a>, <a class="ref" href="xxreal_3.html#D5" onmouseover="rs('xxreal_3/D5')" onmouseout="rh()">XXREAL_3:def 5</a></span>;<br/><a NAME="E4:129_2_3"/>
 <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c1">X1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E96">d4</font></span></a>, <a class="txt" href="#E1:129_2_3"><span class="lab"><font color="Green" title="E98">d6</font></span></a>, <a class="ref" href="measur11.html#T97" target="_self" onmouseover="rs('measur11/T97')" onmouseout="rh()">Th96</a></span>;<br/><span class="kw">hence </span><a NAME="E5:129_2_3"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:129_2_3"><span class="lab"><font color="Green" title="E99">d7</font></span></a>, <a class="txt" href="#E1:129_2_3"><span class="lab"><font color="Green" title="E98">d6</font></span></a>, <a class="ref" href="measur10.html#T33" onmouseover="rs('measur10/T33')" onmouseout="rh()">MEASUR10:33</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_2_4"><span class="kw">suppose </span></a><a NAME="E1:129_2_4"/>
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</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"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:129_2_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c11">Q</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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E96">d4</font></span></a>, <a class="ref" href="measur11.html#T97" target="_self" onmouseover="rs('measur11/T97')" onmouseout="rh()">Th96</a></span>;<br/><span class="kw">hence </span><a NAME="E3:129_2_4"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E97">d5</font></span></a>, <a class="ref" href="measur11.html#T99" target="_self" onmouseover="rs('measur11/T99')" onmouseout="rh()">Th98</a>, <a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
