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<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">E</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="b3">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b4">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Olive" title="b5">E</font>,<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="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> <font color="Olive" title="b5">E</font></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">E</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="b2">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b3">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Olive" title="b4">E</font>,<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="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> <font color="Olive" title="b4">E</font></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">E</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">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b2">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Olive" title="b3">E</font>,<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="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> <font color="Olive" title="b3">E</font></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">E</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="c5">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b1">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Olive" title="b2">E</font>,<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="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> <font color="Olive" title="b2">E</font></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">E</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="c5">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Olive" title="b1">E</font>,<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="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> <font color="Olive" title="b1">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">E</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="c5">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  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> (<font color="Maroon" title="c7">E</font>,<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="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> <font color="Maroon" title="c7">E</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:146"/><span class="lab"><font color="Green" title="E110">A1</font></span>: 
<font color="Maroon" title="c5">M1</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:146"/><span class="lab"><font color="Green" title="E111">A2</font></span>: 
<font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</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">  <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> (<font color="Maroon" title="c7">E</font>,<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="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> <font color="Maroon" title="c7">E</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">A</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c3">S1</font><span class="kw"> such that </span><br/><a NAME="E4:146"/><span class="lab"><font color="Green" title="E112">A3</font></span>: 
( <font color="Maroon" title="c8">A</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:146"><span class="lab"><font color="Green" title="E110">A1</font></span></a>, <a class="txt" href="measur11.html#E98"><span class="lab"><font color="Green" title="E75">LM0902a</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">B</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E6:146"/><span class="lab"><font color="Green" title="E113">A4</font></span>: 
( <font color="Maroon" title="c9">B</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="txt" href="measur11.html#E98"><span class="lab"><font color="Green" title="E75">LM0902a</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> $1</span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">C</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:146"/><span class="lab"><font color="Green" title="E114">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E9:146"/><span class="lab"><font color="Green" title="E115">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><br/>

<a NAME="E1:146_1"/>
<font color="Maroon" title="c11">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:146_1"/>
<font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:146"><span class="lab"><font color="Green" title="E114">A5</font></span></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="E10:146"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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></span><br/>

<a NAME="E1:146_2"/><span class="lab"><font color="Green" title="E116">A7</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a></span>;<br/>
<a NAME="E2:146_2"/>
( <font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> &amp; <font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E3:146_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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="txt" href="#E1:146_2"><span class="lab"><font color="Green" title="E116">A7</font></span></a></span>;<br/>
<a NAME="E4:146_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A8</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<a NAME="E5:146_2"/>
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:146_2"/>
<font color="Maroon" title="c10">C</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:146_2"><span class="lab"><font color="Green" title="E117">A8</font></span></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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c10">C</font> as    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E12:146"/><span class="lab"><font color="Green" title="E116">a9</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</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="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> implies <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c13">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_3"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c13">m</font></span><br/>

<a NAME="E2:146_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:146"><span class="lab"><font color="Green" title="E112">A3</font></span></a>, <a class="txt" href="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a></span>;<br/>
<a NAME="E3:146_3"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/>
<a NAME="E4:146_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:146_3"/>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></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="E13:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A9</font></span>: 
<font color="Maroon" title="c11">C</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a></span>;<br/>
<a NAME="E14:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">a10</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c11">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>;<br/>
<a NAME="E15:146"/><span class="lab"><font color="Green" title="E119">a11</font></span>: 
(  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c8">A</font> &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c9">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:146"><span class="lab"><font color="Green" title="E112">A3</font></span></a>, <a class="txt" href="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>;<br/>
<a NAME="E16:146"/><span class="lab"><font color="Green" title="E120">A15</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <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="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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"> <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="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><br/>

<a NAME="E1:146_4"/><span class="lab"><font color="Green" title="E121">A12</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> &amp; <font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> &amp; <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E2:146_4"/>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a></span>;<br/>
<a NAME="E3:146_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A13</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="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:146_4"><span class="lab"><font color="Green" title="E121">A12</font></span></a>, <a class="ref" href="measur11.html#T16" target="_self" onmouseover="rs('measur11/T16')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E4:146_4"/>
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <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>  &amp; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <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>  &amp; <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:146"><span class="lab"><font color="Green" title="E112">A3</font></span></a>, <a class="txt" href="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:146_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="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="#E3:146_4"><span class="lab"><font color="Green" title="E122">A13</font></span></a>, <a class="ref" href="xxreal_3.html#T69" onmouseover="rs('xxreal_3/T69')" onmouseout="rh()">XXREAL_3:69</a>, <a class="ref" href="xxreal_0.html#T4" onmouseover="rs('xxreal_0/T4')" onmouseout="rh()">XXREAL_0:4</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>
<span class="kw">set </span><font color="Maroon" title="c12">C1</font> = <font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font>;<br/>
<a NAME="E17:146"/><span class="lab"><font color="Green" title="E121">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:146"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_5"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">n1</font> = <font color="Maroon" title="c13">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E3:146_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n1</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" onmouseover="rs('setlim_2/D5')" onmouseout="rh()">SETLIM_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:146_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">C1</font> = <font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:146"><span class="lab"><font color="Green" title="E121">A16</font></span></a>, <a class="ref" href="funct_2.html#T3" onmouseover="rs('funct_2/T3')" onmouseout="rh()">FUNCT_2:3</a></span>;<br/>
<a NAME="E20:146"/><span class="lab"><font color="Green" title="E122">A17</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span></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="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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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">  <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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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="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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:146_6"/><span class="lab"><font color="Green" title="E123">A18</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> &amp; <font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> &amp; <font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E2:146_6"/><span class="lab"><font color="Green" title="E124">A19</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a></span>;<br/>
<a NAME="E3:146_6"/><span class="lab"><font color="Green" title="E125">A20</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="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="#E16:146"><span class="lab"><font color="Green" title="E120">A15</font></span></a></span>;<br/>
<a NAME="E4:146_6"/>
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a></span>;<br/>
<a NAME="E5:146_6"/><span class="kw">then </span>
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <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 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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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="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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="txt" href="#E1:146_6"><span class="lab"><font color="Green" title="E123">A18</font></span></a>, <a class="txt" href="#E2:146_6"><span class="lab"><font color="Green" title="E124">A19</font></span></a>, <a class="txt" href="#E3:146_6"><span class="lab"><font color="Green" title="E125">A20</font></span></a>, <a class="ref" href="measur11.html#T114" target="_self" onmouseover="rs('measur11/T114')" onmouseout="rh()">Th114</a></span>;<br/>
<a NAME="E6:146_6"/><span class="kw">then </span>
<font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <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 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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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="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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span> </span> } </span> 
 
;<br/>
<a NAME="E7:146_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">E1</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 <br/>( <font color="Maroon" title="c7">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">E1</font> &amp;  <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#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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="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">E1</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E8:146_6"/>
 <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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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="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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> $1</span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>);<br/>
<a NAME="E21:146"/><span class="lab"><font color="Green" title="E123">A21</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">f</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  ex <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font>,<font color="Olive" title="b1">f</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">f1</font> =  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">f1</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">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c15">f1</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:146_7"/>
<font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c14">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E23:146"/><span class="lab"><font color="Green" title="E124">A22</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c14">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E21:146"><span class="lab"><font color="Green" title="E123">A21</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c15">F</font> = <font color="Maroon" title="c14">F</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>
<a NAME="E25:146"/><span class="lab"><font color="Green" title="E125">A23</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span><br/>

<a NAME="E1:146_8"/>
<font color="Maroon" title="c16">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:146_8"/>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:146"><span class="lab"><font color="Green" title="E124">A22</font></span></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>
<span class="kw">reconsider </span><font color="Maroon" title="c16">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">X12</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as    <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> <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="E28:146"/>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:146"><span class="lab"><font color="Green" title="E124">A22</font></span></a></span>;<br/>
<a NAME="E29:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A24</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XX1</font> &amp; <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E30:146"/><span class="lab"><font color="Green" title="E127">A25</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span><br/>



<a NAME="E1:146_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:146_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:146"><span class="lab"><font color="Green" title="E125">A23</font></span></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="E31:146"/><span class="lab"><font color="Green" title="E128">a26</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span><br/>

<a NAME="E1:146_10"/>
( <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>) &amp; <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:146"><span class="lab"><font color="Green" title="E125">A23</font></span></a></span>;<br/>
<a NAME="E2:146_10"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XX1</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XX1</font> )
 
<span class="kw">by</span> <span class="lab"><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">hence </span><a NAME="E3:146_10"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E32:146"/><span class="lab"><font color="Green" title="E129">A27</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c16">XX1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c16">XX1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E1:146_11"/>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:146"><span class="lab"><font color="Green" title="E125">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:146_11"/>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c16">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="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</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="E33:146"/><span class="lab"><font color="Green" title="E130">A28</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</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="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_12"/><span class="lab"><font color="Green" title="E131">A29</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:146_12"/>
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font></span><br/>

<a NAME="E3:146_12"/><span class="lab"><font color="Green" title="E132">A30</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:146"><span class="lab"><font color="Green" title="E116">a9</font></span></a>, <a class="txt" href="#E1:146_12"><span class="lab"><font color="Green" title="E131">A29</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/>
<span class="lab"><font color="Green" title="E133">A31</font></span>: <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> = 
<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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:146"><span class="lab"><font color="Green" title="E127">A25</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>
;<br/>
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span> = 
<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">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:146"><span class="lab"><font color="Green" title="E127">A25</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:146_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:146_12"><span class="lab"><font color="Green" title="E132">A30</font></span></a>, <a class="txt" href="#E4:146_12"><span class="lab"><font color="Green" title="E133">A31</font></span></a>, <a class="ref" href="measur11.html#T20" target="_self" onmouseover="rs('measur11/T20')" onmouseout="rh()">Th14</a>, <a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</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="E34:146"/><span class="lab"><font color="Green" title="E131">A32</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> holds <br/><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c18">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</font> implies <font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_13"/>
<font color="Maroon" title="c18">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">XX1</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="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_13_1"><span class="kw">now </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>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">n</font>, <font color="Maroon" title="c20">m</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="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:146_13_1"/>
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">n</font></span><br/><a NAME="E2:146_13_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:146"><span class="lab"><font color="Green" title="E130">A28</font></span></a></span>;<br/><a NAME="E3:146_13_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:146_13_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</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="E3:146_13"/><span class="kw">then </span>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T7" onmouseover="rs('rinfsup2/T7')" onmouseout="rh()">RINFSUP2:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:146_13"/>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T37" onmouseover="rs('rinfsup2/T37')" onmouseout="rh()">RINFSUP2:37</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>
<span class="kw">consider </span><font color="Maroon" title="c18">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><span class="kw"> such that </span><br/><a NAME="E36:146"/><span class="lab"><font color="Green" title="E132">A33</font></span>: 
( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c18">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>) ) &amp; <font color="Maroon" title="c18">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <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="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c18">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:146"><span class="lab"><font color="Green" title="E126">A24</font></span></a>, <a class="txt" href="#E31:146"><span class="lab"><font color="Green" title="E128">a26</font></span></a>, <a class="txt" href="#E32:146"><span class="lab"><font color="Green" title="E129">A27</font></span></a>, <a class="txt" href="#E33:146"><span class="lab"><font color="Green" title="E130">A28</font></span></a>, <a class="txt" href="#E34:146"><span class="lab"><font color="Green" title="E131">A32</font></span></a>, <a class="ref" href="mesfunc8.html#D2" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">MESFUNC8:def 2</a>, <a class="ref" href="mesfunc9.html#T52" onmouseover="rs('mesfunc9/T52')" onmouseout="rh()">MESFUNC9:52</a></span>;<br/>
<a NAME="E37:146"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <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"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D9" onmouseover="rs('mesfunc8/D9')" onmouseout="rh()">MESFUNC8:def 9</a></span>;<br/>
<a NAME="E38:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A34</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:146"><span class="lab"><font color="Green" title="E126">A24</font></span></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="E39:146"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_14"/>
<font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</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"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span><br/>

<a NAME="E2:146_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">L2</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D9" onmouseover="rs('mesfunc8/D9')" onmouseout="rh()">MESFUNC8:def 9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">G</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E4:146_14"/><span class="lab"><font color="Green" title="E135">L3</font></span>: 
( <font color="Maroon" title="c20">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c20">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c17">X12</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:146"><span class="lab"><font color="Green" title="E117">A9</font></span></a>, <a class="txt" href="#E15:146"><span class="lab"><font color="Green" title="E119">a11</font></span></a>, <a class="txt" href="#E4:146"><span class="lab"><font color="Green" title="E112">A3</font></span></a>, <a class="txt" href="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="txt" href="#E14:146"><span class="lab"><font color="Green" title="E118">a10</font></span></a>, <a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a>, <a class="ref" href="measur11.html#T116" target="_self" onmouseover="rs('measur11/T116')" onmouseout="rh()">Th116</a>, <a class="ref" href="measur11.html#T108" target="_self" onmouseover="rs('measur11/T108')" onmouseout="rh()">Th108</a></span>;<br/>
<a NAME="E5:146_14"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c20">G</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c20">G</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font></span><br/>

<a NAME="E1:146_14_1"/><span class="lab"><font color="Green" title="E136">L5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E137">L4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:146"><span class="lab"><font color="Green" title="E124">A22</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</a></span>
;<br/>
<a NAME="E3:146_14_1"/>
 <a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>,<font color="Maroon" title="c19">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c19">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T27" target="_self" onmouseover="rs('measur11/T27')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E4:146_14_1"/><span class="kw">then </span>
 <a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>,<font color="Maroon" title="c19">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c21">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:146_14"><span class="lab"><font color="Green" title="E135">L3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:146_14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c20">G</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_14_1"><span class="lab"><font color="Green" title="E137">L4</font></span></a>, <a class="txt" href="#E1:146_14_1"><span class="lab"><font color="Green" title="E136">L5</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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="E6:146_14"/><span class="kw">then </span>
<font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c19">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">M2</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c20">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<a NAME="E7:146_14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c17">X12</font>,<font color="Maroon" title="c19">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146_14"><span class="lab"><font color="Green" title="E134">L2</font></span></a>, <a class="txt" href="#E4:146_14"><span class="lab"><font color="Green" title="E135">L3</font></span></a>, <a class="ref" href="measure8.html#T26" onmouseover="rs('measure8/T26')" onmouseout="rh()">MEASURE8:26</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> = 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c17">X12</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T27" target="_self" onmouseover="rs('measur11/T27')" onmouseout="rh()">Th21</a></span>
<br/>.= 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c19">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:146_14"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:146"><span class="lab"><font color="Green" title="E111">A2</font></span></a>, <a class="ref" href="measur11.html#D13" target="_self" onmouseover="rs('measur11/D13')" onmouseout="rh()">DefYvol</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="E40:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">V3</font></span>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c15">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c6">M2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:146"><span class="lab"><font color="Green" title="E133">A34</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">J</font> = <font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font>;<br/>
<a NAME="E41:146"/><span class="lab"><font color="Green" title="E135">E1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E42:146"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:146_15"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">n1</font> = <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E3:146_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n1</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" onmouseover="rs('setlim_2/D5')" onmouseout="rh()">SETLIM_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:146_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">J</font> = <font color="Maroon" title="c7">E</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c11">C</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:146"><span class="lab"><font color="Green" title="E135">E1</font></span></a>, <a class="ref" href="funct_2.html#T3" onmouseover="rs('funct_2/T3')" onmouseout="rh()">FUNCT_2:3</a></span>;<br/>
<a NAME="E44:146"/><span class="lab"><font color="Green" title="E136">R11</font></span>: 
<font color="Maroon" title="c20">J</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:146"><span class="lab"><font color="Green" title="E117">A9</font></span></a>, <a class="ref" href="setlim_2.html#T22" onmouseover="rs('setlim_2/T22')" onmouseout="rh()">SETLIM_2:22</a></span>;<br/>
<a NAME="E45:146"/>
<font color="Maroon" title="c11">C</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:146"><span class="lab"><font color="Green" title="E117">A9</font></span></a>, <a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E137">R13</font></span>:  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c20">J</font> = 
<font color="Maroon" title="c7">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c11">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#T92" onmouseover="rs('setlim_2/T92')" onmouseout="rh()">SETLIM_2:92</a></span>
<br/>.= 
<font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:146"><span class="lab"><font color="Green" title="E119">a11</font></span></a>, <a class="txt" href="#E4:146"><span class="lab"><font color="Green" title="E112">A3</font></span></a>, <a class="txt" href="#E6:146"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="txt" href="#E14:146"><span class="lab"><font color="Green" title="E118">a10</font></span></a>, <a class="txt" href="#E9:146"><span class="lab"><font color="Green" title="E115">A6</font></span></a>, <a class="ref" href="measur11.html#T116" target="_self" onmouseover="rs('measur11/T116')" onmouseout="rh()">Th116</a></span>
<br/>.= 
<font color="Maroon" title="c7">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E47:146"/><span class="lab"><font color="Green" title="E138">R3</font></span>: 
 <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>) is   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p2">(<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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T8" target="_self" onmouseover="rs('measur11/T8')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E48:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">R4</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#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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E49:146"/>
 <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c20">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E50:146"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">R2</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">J</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:146"><span class="lab"><font color="Green" title="E139">R4</font></span></a>, <a class="ref" href="funct_2.html#D11" onmouseover="rs('funct_2/D11')" onmouseout="rh()">FUNCT_2:def 11</a></span>;<br/>
<a NAME="E51:146"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c18">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">J</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="146_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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="c18">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">J</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font></span><br/>

<a NAME="E1:146_17"/><span class="lab"><font color="Green" title="E141">R21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c18">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> = 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:146"><span class="lab"><font color="Green" title="E132">A33</font></span></a></span>
<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"><span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:146"><span class="lab"><font color="Green" title="E124">A22</font></span></a></span>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:146"><span class="lab"><font color="Green" title="E122">A17</font></span></a></span>
<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="c20">J</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" onmouseover="rs('setlim_2/D5')" onmouseout="rh()">SETLIM_2:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:146_17"/>
<font color="Maroon" title="c18">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">J</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:146"><span class="lab"><font color="Green" title="E140">R2</font></span></a>, <a class="txt" href="#E1:146_17"><span class="lab"><font color="Green" title="E141">R21</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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="E52:146"/><span class="kw">then </span>
<font color="Maroon" title="c18">I</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E53:146"/>
 <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> (<font color="Maroon" title="c7">E</font>,<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="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> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:146"><span class="lab"><font color="Green" title="E132">A33</font></span></a>, <a class="txt" href="#E40:146"><span class="lab"><font color="Green" title="E134">V3</font></span></a>, <a class="txt" href="#E46:146"><span class="lab"><font color="Green" title="E137">R13</font></span></a>, <a class="txt" href="#E44:146"><span class="lab"><font color="Green" title="E136">R11</font></span></a>, <a class="txt" href="#E50:146"><span class="lab"><font color="Green" title="E140">R2</font></span></a>, <a class="txt" href="#E47:146"><span class="lab"><font color="Green" title="E138">R3</font></span></a>, <a class="ref" href="measure8.html#T26" onmouseover="rs('measure8/T26')" onmouseout="rh()">MEASURE8:26</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>
